\label{thmintro:delocalised-trace-identification-simplified}
  Let $\cH$ be an Iwahori Hecke algebra of right-angled type $(W,S)$.  Let $\cO \subseteq W$ be a conjugacy class.  Then there is a unique trace $\vphi_\cO$ on $\cH$ that satisfies $\vphi_\cO(T_w) = \mathbb{1}_\cO(w)$ for every element $w \in W$ that is of minimal length in its conjugacy class.

\begin{remark}
  \label{rem:finite-order-element-cliques}
  Let $\Gamma$ be the commutation graph of a right-angled Coxeter system $(W, S)$.  Denote by $\mathrm{Cliq}(\Gamma)$ the set of cliques in $\Gamma$, that is the set of complete subgraphs of $\Gamma$ (including the empty graph).  Identifying vertices of $\Gamma$ with $S$, we consider the map $C \mapsto \prod_{s \in \rV(C)} s$.  By Remark~\ref{rem:finite-order-element-conjugacy} every finite order element is conjugate to an element in the image of this map.  Further, if $\prod_{s \in \rV(C)} s$ and $\prod_{s \in \rV(D)} s$ are conjugate, they must be equal by Tits solution to the word problem.  It follows that $C \mapsto \prod_{s \in \rV(C)} s$ defines a bijection between $\mathrm{Cliq}(\Gamma)$ and the set of conjugacy classes of finite order elements of $W$.
\end{remark}

\begin{theorem}
  \label{thm:right-angled-cocentre-basis}
  Let $\cH$ be a generic Hecke algebra of right-angled type $(W,S)$ over an algebraically closed field $K$ of characteristic different from $2$. Let $(a_s)_{s\in S}$, $(b_s)_{s\in S}$ be the deformation parameters of $\cH$ and assume that $b_s \neq -\frac{a_s^2}{4}$ for each $s \in S$.  Then $(T_{\cO})_{\cO}$ where $\cO$ runs through the conjugacy classes of $W$ is a basis of the cocentre of $\cH$.
\end{theorem}

\label{sec:introduction}

The study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \cite{lott1999}, who investigated for example delocalised $\Ltwo$-Betti numbers and delocalised $\Ltwo$-torsion of hyperbolic manifolds. The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint \cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts. More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.

While this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory. They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively. Delorme-Opdam's work in \cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context. Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed. These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \cite{opdamsolleveld2010-discrete-series} and generalised principal series \cite{delormeopdam2011} of affine Hecke algebras.

Hecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \cite{davisdymarajanusykiewiczokun07} around two decades ago. Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field. Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \cite{caspersklisselarsen2021}, simplicity results \cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \cite{raumskalski2022} for Hecke C*-algebras. Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \cite{klisseperovic25}. Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups. See e.g.\ \cite[Corollary B]{raumskalski2023}.

The cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \cite{baumnistor2002} in the purely algebraic setup and by Solleveld \cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.

In this work we consider Hecke algebras of right-angled, hyperbolic type. In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras. This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\ref{thm:delocalised-trace-identification}. For simplicity we state it in a simplified form for Iwahori-Hecke algebras.

\begin{theorem}
  \label{thm:delocalised-trace-identification}
  Let $\cH$ be a generic Hecke algebra of right-angled type $(W,S)$.  For every conjugacy class $\cO \subseteq W$, the map $\vphi_\cO$ is a trace on $\cH$.  If $(T_{\cU})_{\cU}$ denotes the basis of the cocentre of $\cH$ satisfying $T_{\cU} = T_w + [\cH, \cH]$ for every element of minimal length $w \in \cU$, then $\vphi_\cO(T_\cU) = \delta_{\cO, \cU}$.
\end{theorem}

The next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.
\begin{introtheorem}
 \label{introthm:Schwartz-algebra-existence}
 Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \in S} \in \RR_{> 0}^S$. There is a subalgebra $\cS(W, S, q) \subseteq \Cstarred(W, S, q)$, which
 \begin{itemize}
 \item is dense and closed under holomorphic functional calculus,
 \item and such that for every conjugacy class $\cO \subseteq W$, the trace $\vphi_\cO$ continuously extends to $\cS(W, S, q)$.
 \end{itemize}
\end{introtheorem}

The proof of \cite[Theorem 5.3]{henie2014} applies verbatim to show the following result: the collection $(T_\cO)_{\cO}$ linearly generates the cocentre of any generic Hecke algebra. We will use the following description of the coefficients appearing in linear combinations of these elements.
\begin{lemma}
 \label{lem:non-minimal-element-cocentre}
 Let $\cH$ be a generic Hecke algebra of type $(W,S)$. Let $\cO \subseteq W$ be a conjugacy class and let $w \in \cO$. Then
 \begin{gather*}
 T_w + [\cH, \cH]
 =
 \sum_\cK c_{\cK, w} T_\cK
 \end{gather*}
 for certain coefficients $c_{\cK, w}$, where $\cK$ runs over all conjugacy classes of elements that are Bruhat subordinate to $w$.
\end{lemma}
\begin{proof}
 Let $\ell$ be the minimal length of an element in $\cO$. We prove the claim by induction on $|w| - \ell = k \in \NN_0$. For $k = 0$ there is nothing to prove.

\begin{notation}
 \label{not:delocalised-trace}
 Let $(W, S)$ be a Coxeter system of right-angled type and let $\cH$ be a generic Hecke algebra of type $(W, S)$ defined over $R$. For a conjugacy class $\cO \subseteq W$, we denote by $\Sigma_\cO\colon \cH \lra R$ the sum over all coefficients of minimal length elements in $\cO$, that is
 \begin{gather*}
 \Sigma_\cO(T_w)
 =
 \begin{cases}
 1 & \text{ if $w \in \cO$ and $w$ is of minimal length in $\cO$} \\
 0 & \text{ otherwise}\eqstop
 \end{cases}
 \end{gather*}
 Combining this with Notation~\ref{not:cyclic-reduction-map}, we will write $\vphi_\cO = \Sigma_\cO \circ \Phi$.
\end{notation}

\begin{theorem}
 \label{thm:delocalised-trace-identification}
 Let $\cH$ be a generic Hecke algebra of right-angled type $(W,S)$. For every conjugacy class $\cO \subseteq W$, the map $\vphi_\cO$ is a trace on $\cH$. If $(T_{\cU})_{\cU}$ denotes the basis of the cocentre of $\cH$ satisfying $T_{\cU} = T_w + [\cH, \cH]$ for every element of minimal length $w \in \cU$, then $\vphi_\cO(T_\cU) = \delta_{\cO, \cU}$.
\end{theorem}
\begin{proof}
 In order to show that $\vphi_\cO$ is a trace, it suffices to show that for all $w \in W$ and all $s \in S$ the equality $\vphi_\cO(T_s T_w) = \vphi_\cO(T_w T_s)$ holds. We prove this statement by induction on $|w| \in \NN_0$ and observe that the case $|w| = 0$ is clear. We distinguish several cases.

\begin{theorem}
 \label{thm:delocalised-trace-existence}
 Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and let $\cH$ be the Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \in S} \in \RR_{> 0}^S$. Let $\cS'(W, S, q), \cS(W, S, q) \subseteq \Cstarred(W, S, q)$ be smooth subalgebras, such that $\cS'(W, S, q)$ is an unconditional completion of $\CC(W, S, q)$ and the natural quasi-derivation
 \begin{gather*}
 \Delta\colon T_w \mapsto \sum_{\substack{w = w_1w_2 \\ \text{ reduced decompositon}}} T_{w_1} \ot T_{w_2}
 \end{gather*}
 continuously extends to a map into the unconditional tensor product $\cS(W, S, q) \lra \cS'(W, S, q) \ot \cS'(W, S, q)$. Then for every conjugacy class $\cO \subseteq W$ the trace $\vphi_\cO$ continuously extends to $\cS(W, S, q)$.
\end{theorem}
\begin{proof}
 Let us first observe that the map $\Sigma_\cO$ summing over the finitely many coefficients of elements of minimal length in $\cO$ extends to a continuous map on $\cS'(W, S, q)$, as $\cS'(W, S, q)$ is an unconditional completion of $\CC(W, S, q)$. Consider maps $\gamma, \rho\colon W \lra W$ defined as in Lemma~\ref{lem:cyclic-reduction-hecke-algebra}. Since $\Delta$ extends continuously to a map $\cS(W, S, q) \lra \cS'(W, S, q) \ot \cS'(W, S, q)$ and since $\cS'(W, S, q)$ is an unconditional completion of $\CC(W, S, q)$, the map $\Delta'\colon \cS(W, S, q) \lra \cS'(W, S, q) \ot \cS'(W, S, q)$ satisfying $\Delta'(T_w) = T_{\gamma(w) \rho(w)} \ot T_{\gamma(w)^{-1}}$ is well-defined and continuous. Let $\Phi\colon \CC(W, S, q) \lra \CC(W, S, q)$ be the unique linear map satisfying $\Phi(T_w) = T_{\gamma(w)^{-1}} T_{\gamma(w)} T_{\rho(w)}$. Then we have $\vphi_\cO = \Sigma_\cO \circ \Phi$ as shown in Theorem~\ref{thm:delocalised-trace-identification}. So a continuous extension of $\vphi_\cO$ to $\cS(W, S, q)$ is obtained by the composition
 \begin{multline*}
 \cS(W, S, q) \stackrel{\Delta'}{\lra}
 \cS'(W, S, q) \ot \cS'(W, S, q) \stackrel{\text{tensor flip}}{\lra} \\
 \cS'(W, S, q) \ot \cS'(W, S, q) \stackrel{\text{multiplication}}{\lra}
 \cS'(W, S, q) \stackrel{\Sigma_\cO}{\lra}
 \CC
 \eqcomma
 \end{multline*}
This finishes the proof.
\end{proof}

Before proving Theorem~\ref{thmintro:pairing}, let us clarify its context and notation beyond what we have done in the introduction already. Recall from \cite[Theorem A.2.1]{bost1990-principe-oka} that the inclusion of a smooth subalgebra into a unital C*-algebra induces an isomorphism in K-theory. We also recall the K-theory calculations for right-angled Hecke C*-algebras from \cite[Section 4]{raumskalski2022}. To this end we use the description of conjugacy classes of finite order elements from Remark~\ref{rem:finite-order-element-cliques}. Denote by $\mathrm{Cliq}(\Gamma)$ the set of cliques in the commutation graph $\Gamma$ of a right-angled Coxeter system $(W, S)$, that is subsets $C \subseteq S$ such that $st = ts$ for all $s,t \in C$. For a clique $C \in \mathrm{Cliq}(\Gamma)$, we write $p_C = \prod_{s \in C} \chi_1^s$. Here $\chi_1^s = \pi_s(\chi_1)$ denotes the image of the projection $\chi_1 \in \CC[\ZZ/2]$ associated with the trivial representation under the linear map extending the group homomorphism $\pi_s\colon \ZZ/2 \lra \CC(W, S, q)^\times$ which satisfies $\pi_s(1) = \frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. Then for every deformation parameter $q \in \RR_{> 0}^S$ the map $\mathrm{Cliq}(\Gamma) \lra \rK_0(\Cstarred(W, S, q))\colon C \mapsto [p_C]$ induces an isomorphism $\ZZ^{\mathrm{Cliq}(\Gamma)} \lra \rK_0(\Cstarred(W, S, q))$.
\begin{proof}[Proof of Theorem~\ref{thmintro:pairing}]
 As described above, for $D \in \mathrm{Cliq}(\Gamma)$ we have $p_D = \prod_{s \in D} \chi_1^s$ where $\chi_1^s$ is the projection associated with the trivial representation of the unitary representation $\ZZ/2 \mapsto \CC(W, S, q)$ mapping $1$ to $\frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. We have $\chi_1^s = \mathbb{1}_{(0,\infty)}(T_s) = \frac{1}{q_s^{1/2} + q_s^{-1/2}} (T_s + q_s^{-1/2}) =\frac{1}{q_s^{1/2} + q_s^{-1/2}} T_s + \frac{1}{1 + q_s}$. Writing $T_D = \prod_{s \in D} T_s$, we find that
 \begin{gather*}
 p_D = \sum_{C \subseteq D} \bigl ( \prod_{s \in C} \frac{1}{q_s^{1/2} + q_s^{-1/2}} \bigr ) \bigl ( \prod_{s \in D \setminus C} \frac{1}{1 + q_s} \bigr ) T_C
 \eqstop
 \end{gather*}
 Since all elements in the support of $p_D$ have minimal length and are pairwise non-conjugate, we find that
 \begin{gather*}
 \vphi_C(p_D)
 =
 \Sigma_C(p_D)
 =
 \begin{cases}
 0 & C \nsubseteq D \eqcomma \\
 \prod_{s \in C} \frac{1}{q_s^{1/2} + q_s^{-1/2}} \prod_{s \in D \setminus C} \frac{1}{1 + q_s} & C \subseteq D \eqstop
 \end{cases}
 \end{gather*}
 Now the claimed formula for $\vphi_C(p_D)$ follows.

We mention that in order to prove this result, we establish in Theorem~\ref{thm:right-angled-cocentre-basis} that the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.  The problem of describing the cocentre of Hecke algebras is well-known in the context of character theory.  See e.g. \cite{geckpfeiffer2000-characters} for spherical Coxeter types and \cite{henie2014} for affine Coxeter types.  The recent work \cite{chen2025-centralizers} explicitly states folklore conjectures about the cocentre for indefinite Coxeter types.

The next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.
\begin{introtheorem}
  \label{introthm:Schwartz-algebra-existence}
  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \in S} \in \RR_{> 0}^S$.  There is a  subalgebra $\cS(W, S, q) \subseteq \Cstarred(W, S, q)$, which
  \begin{itemize}
  \item is dense and closed under holomorphic functional calculus,
  \item and such that for every conjugacy class $\cO \subseteq W$, the trace $\vphi_\cO$ continuously extends to $\cS(W, S, q)$.
  \end{itemize}
\end{introtheorem}

From the point of view of operator algebras, one of the main purposes of delocalised traces is to determine K-theory classes via their pairing.  The K-theory of right-angled Hecke C*-algebras was calculated in \cite{raumskalski2022}.  We briefly recall that for every clique $C$ in the commutation graph of a right-angled Coxeter system there is a projection $p_C$, such that the classes $([p_C])_C$ freely generate K-theory.  In Remark~\ref{rem:finite-order-element-cliques}, we observe that these cliques are in natural bijection with finite order elements of a right-angled Coxeter system.  We obtain the following result, which determines the pairing of K-theory and delocalised traces.
\begin{introtheorem}
  \label{thmintro:pairing}
  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $(q_s)_{s \in S} \in \RR_{> 0}^S$ a deformation parameter.  Denote by $\mathrm{Cliq}(\Gamma)$ the set of cliques of the commutation graph $\Gamma$ of $(W, S)$, identified with the set of conjugacy classes of finite order elements in $W$.  For a clique $C \in \mathrm{Cliq}(\Gamma)$ let $p_C$ be the associated projection and let $\vphi_C$ be the delocalised trace associated with $C$.  Then the pairing between K-theory of $\Cstarred(W, S, q)$ and delocalised traces of finite order elements is determined by the formula
  \begin{gather*}
    \vphi_C(p_D) =
    \begin{cases}
      0 & C \nsubseteq D\eqcomma \\
      \prod_{s \in C} \frac{1}{q_s^{1/2} + q_s^{-1/2}} \prod_{s \in D \setminus C} \frac{1}{1 + q_s} & C \subseteq D \eqstop
    \end{cases}
  \end{gather*}
  In particular, the pairing $\rK_0(\Cstarred(W,S,q)) \times \RR^{\mathrm{Cliq(\Gamma)}} \lra \RR$ is non-degenerate.
\end{introtheorem}

As mentioned above, in the last section of \cite{puschnigg2010} Puschnigg used his main results to obtain certain decompositions of cyclic cohomology for his smooth algebras and the group C*-algebras of hyperbolic groups, close in spirit to the computation of cyclic homology of group rings due to Burghelea \cite{burghelea1985, burghelea2023-erratum-for-1985}. This line of research was later continued, using the Jolissaint's rapid decay subalgebras, for example in \cite{jiogleramsey14} and in \cite{engel20}. Our results in principle open the way to obtaining similar decompositions of cyclic cohomology in the context of Hecke algebras, such as the Fr{\'e}chet algebra $\cS(W, S, q)$ appearing in Theorem \ref{introthm:Schwartz-algebra-existence}. However, the setting of right-angled hyperbolic Coxeter types remains too narrow to make such an endeavour worthwhile at present, since higher cyclic cohomology is expected to vanish.  Further work in this direction, would chime with the recent progress in understanding noncommutative geometric properties of the Iwahori-Hecke algebras such as in the work \cite{klisseperovic25} already mentioned.

This paper is organised as follows: after this introduction, in the preliminaries we describe background material and fix notation, focusing on Coxeter groups and Hecke algebras.  Then in the third section, we investigate the combinatorics of the multiplication in right-angled Hecke algebras, which plays a major role in this work.  In the fourth section, we adapt work of Blackadar-Cuntz and Puschnigg on differential seminorms and quasi-derivations to the needed generality.  In the fifth section, we use this framework in order to exhibit specific smooth subalgebras of the Hecke algebras under consideration, to which the natural quasi-derivation extends.  In the sixth section, we describe traces on right-angled Hecke algebras, and in that last section combine this with the previous results on smooth subalgebras, to obtain delocalised traces on Hecke-Schwartz algebras and calculate their pairing with K-theory.

\subsection*{Acknowledgements}

SP was partially supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 677120-INDEX) and the Knut and Alice Wallenberg Foundation (grant number KAW 2020.0252).

SR was partially supported by the Swedish Research Council (grant number 2018-04243) and the German Research Foundation (DFG project no. 550184791).

PN was supported by the National Science Center Grant Maestro-13 UMO-2021/42/A/ST1/00306.

The authors would like to thank IMPAN and its B{\c e}dlewo Conference Center as well as the University to Potsdam for their hospitality during respective visits.

To prove Theorem~\ref{thmintro:delocalised-trace-identification-simplified}, the introduction says that the authors first establish in Theorem~\ref{thm:right-angled-cocentre-basis} that “the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.”

Question: Let \(\mathcal H\) be the Iwahori Hecke algebra associated with a right-angled Coxeter system \((W,S)\), and let \(\{T_w\}_{w\in W}\) denote its standard basis indexed by \(W\). Let \(\mathcal O\subseteq W\) be a conjugacy class. A trace on \(\mathcal H\) means a linear functional \(\varphi:\mathcal H\to\mathbb C\) such that \(\varphi(xy)=\varphi(yx)\) for all \(x,y\in\mathcal H\). For \(w\in W\), say that \(w\) is of minimal length in its conjugacy class if its Coxeter length is minimal among all elements conjugate to \(w\). Which existence statement holds under these assumptions?

There exists a unique trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) that is of minimal length in its conjugacy class, \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.

Choices:
B. There exists a unique trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\), \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.
C. There exists a trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) that is of minimal length in its conjugacy class, \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.
D. For each conjugacy class \(\mathcal O\subseteq W\), there exists a unique linear functional \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) that is of minimal length in its conjugacy class, \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\).
E. There exists a trace \(\varphi:\mathcal H\to\mathbb C\) such that for every conjugacy class \(\mathcal O\subseteq W\) and every \(w\in W\) that is of minimal length in its conjugacy class, \(\varphi(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.

trace property replaced by mere linearity

dependence on the conjugacy class parameter \(\mathcal O\)

The stem gives only background definitions and asks which existence statement is valid; it does not explicitly state the theorem or directly reveal the correct option.

The item is very close to a theorem-recall question: the correct choice is essentially the precise theorem statement, though the alternatives introduce meaningful variations in quantifiers, uniqueness, and the trace condition.

Some reasoning is needed to distinguish the exact valid statement from stronger or weaker variants, but the task is mainly precise theorem recognition rather than substantial mathematical generation or derivation.

The distractors are plausible and mathematically targeted: they test common confusions about minimal-length restriction, uniqueness, dependence on the conjugacy class parameter, and trace versus arbitrary linear functional.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it remains largely a near-restatement/recall item rather than a high-generative-reasoning question.


\begin{thm}\label{mainthm}
    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.
\end{thm}

\par In slightly more detail,
given a boundaryless surface with negative Euler characteristic
and a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\colon S \to S$, one can construct the veering triangulation of the mapping torus of $f$, whose
flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.

\subsection{Connections to literature}\label{discuss}

\par Work of Penner-Papadopoulos \cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.
\par In the setting of \cite[Theorem 4.1]{PP},
the support map $\sigma$ produced for the $f$-invariant train track $\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\tau$ could e.g. be mapped by $\sigma \circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\sigma \circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\sigma \circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\tau) \prec \tau$ has irreducible transition matrix while $f(\tau) \prec \tau$ does not.

\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \textit{\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \Cref{mainthm} necessarily produces train tracks which are not generic.
This motivates the following question:

\begin{qn}
    For all $f\colon S \to S$ pseudo-Anosov, does there exist a \textit{generic} invariant train track for $f$ with irreducible transition matrix?
\end{qn}

\par \textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.

\section{Background}\label{background}

\par A \textit{\textbf{strongly connected component}} of a directed graph $G$ is a maximal subset $C$ of the vertices of $G$ such that for any two vertices in $C$, there is a path in $G$ from the first vertex to the second.
If $G$ has only one strongly connected component, it is called \textit{\textbf{strongly connected}}.
A nonnegative square matrix $A$ is called \textit{\textbf{irreducible}} if for every pair of indices $i$, $j$ there is an $n>0$ such that the $(i,j)$-th entry of $A^n$ is positive. $A$ is irreducible if and only if it is the adjacency matrix of a strongly connected graph.
\par
    Given an orientable finite-type surface $S$ without boundary and negative Euler characteristic, a homeomorphism $f\colon S \to S$ is called \textit{\textbf{pseudo-Anosov}} if there is a pair of transverse measured singular foliations $(\Lambda^u,\mu_u)$ and $(\Lambda^s,\mu_s)$, called the \textit{\textbf{unstable}} and \textit{\textbf{stable}} foliations, and a constant $\lambda > 1$, called the \textit{\textbf{stretch factor}}, such that $f(\Lambda^u)=\Lambda^u$, $f(\Lambda^s)=\Lambda^s$ and $f(\mu^u) = \lambda \mu^u$, $f(\mu^s) = \lambda^{-1} \mu^s$. No nontrivial power of a pseudo-Anosov map fixes the homotopy class of any essential closed curve in $S$. A pseudo-Anosov map is a diffeomorphism except at the singular points of $\Lambda^u$.

    \par The \textit{\textbf{mapping torus of $f$}} is the 3-manifold
    $$ M_f = \faktor{(S \times [0,1])}{ (f(p),0) \sim (p,1)} $$ which fibers over the circle with fiber $S$ and monodromy $f$.

    \par If all of the singularities of the foliations are at the punctures of $S$, we say $f$ is \textit{\textbf{fully punctured}}. We will study general pseudo-Anosovs by deleting the singularities of the foliations to obtain a new surface $S^\circ$ on which the restriction of $f$ is fully punctured.
For full background on pseudo-Anosov homeomorphisms, see Fathi--Laudenbach--Poénaru \cite{FLP}.

\subsection{Train tracks}\label{tts}
\par A \textit{\textbf{train track}} $\tau$ is a closed 1-complex embedded in a surface $S$ with a ``smoothing" at each vertex so that $\tau$ has a well-defined tangent space at each vertex, and $S \setminus \tau$ contains no nullgons, unpunctured monogons or unpunctured bigons. See Penner-Harer \cite{PH}. The vertices of $\tau$ are called \textit{\textbf{switches}} and the edges are called \textit{\textbf{branches}}. If the switches of $\tau$ all have degree three, $\tau$ is called \textit{\textbf{generic}}.

\par Given two train tracks $\tau_1$ and $\tau_2$ on $S$, $\tau_2$ is said to \textit{\textbf{carry}} $\tau_1$ (or $\tau_1$ \textit{\textbf{is carried by}} $\tau_2$), denoted $\tau_1 \prec \tau_2$, if there is a $C^1$ map $\sigma\colon S \to S$ called the \textit{\textbf{support map}}
such that
    \begin{enumerate}
        \item $\sigma$ is homotopic to the identity,
        \item $\sigma(\tau_1) \subseteq \tau_2$,
        \item for all points $x$ in $\tau_1$,
    $d_x \sigma_{|\tau_1}\colon T_x \tau_1 \to T_{h(x)}\tau_2$ is an isomorphism of tangent spaces,
    \end{enumerate}
Note that carrying is transitive: if $\tau_1 \prec \tau_2$ and $\tau_2 \prec \tau_3$, then
$\tau_1 \prec \tau_3$.
We also write $\tau_1 \prec_\sigma \tau_2$ to indicate that $\sigma$ is the support map for $\tau_1 \prec \tau_2$. A carrying map $\sigma$ for $\tau_1 \prec \tau_2$  is called \textit{\textbf{combinatorial}} if for every switch $v$ in $\tau_1$, $\sigma(v)$ is a switch in $\tau_2$.
    \par
    Given $\tau_1$ with branches $\{b^1_j\}_{j=1\ldots n}$ and $\tau_2$ with branches $\{b^2_i\}_{i=1\ldots m}$ such that $\tau_1$ is carried by $\tau_2$ with combinatorial support map $\sigma$,
    the \textit{\textbf{transition matrix}} is the $m \times n$ matrix $T = [t_{i,j}]$ where
    $t_{i,j}$ is the number of times the image of $b^1_j$ under $\sigma$ passes over $b^2_i$.
\par
    A \textit{\textbf{train path}} is a $C^1$ immersion $p\colon [0,1] \to \tau$ so that the endpoints $\{0,1\} = \partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\tau$. For $t \in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\colon [0,t] \to \tau$, $p_1 = p_{|[t,1]}\colon [t,1] \to \tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\partial_t)$ and $d_0 p_1(\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\colon [0,2] \to \tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\ldots p_{n-1})$, where $p_i\colon\left[\frac{i}{n},\frac{i+1}{n}\right] \to \tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \textit{\textbf{embedded}}.
\par
    A \textit{\textbf{fold}} is a particular combinatorial support map $\phi$ for $\tau_1 \prec \tau_2$ which induces a bijection between the branches of $\tau_1$ and $\tau_2$
    except on a particular connected subgraph where it is defined according to either of the two pictures in \Cref{fig:fold}. The three branches of $\tau_1$ which are mapped over the small branch $e'$ in the domain of \Cref{fig:fold} are said to \textit{\textbf{fold to}} $e'$.
    If there is a $\tau_1 \prec_\phi \tau_2$
    we say $\tau_1$ \textit{\textbf{folds to}} $\tau_2$, denoted $\tau_1 \leftharpoonup \tau_2$. If $\tau_1$ folds to $\tau_2$, $\tau_1$ is carried by $\tau_2$. If $\tau_1 \prec \tau_2$ with support map given by a sequence of folds we also say $\tau_1$ folds to $\tau_2$.
    \begin{figure}[h]
    \centering
    \includegraphics[width=0.66\textwidth]{foldgood.pdf}
    \caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}
    \label{fig:fold}
\end{figure}
    \par
    A train track $\tau$ is an \textit{\textbf{invariant}} track for a map $f: S \to S$ with support map $\sigma$ if $f(\tau)$ is carried by $\tau$ with support map $\sigma$.
    If the transition matrix for $f(\tau) \prec_\sigma \tau$ is irreducible, we also say $\tau$ is \textit{\textbf{irreducible}}.
By smoothly isotoping $\tau$ if necessary, we may assume it is disjoint from the singularities of $\Lambda^u$. This will be convenient later when we remove the singularities of $\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:

\begin{thm}[\protect{\cite[Theorem 3.5]{A}}]\label{ttfs} Given $f\colon S \to S$ pseudo-Anosov, there exists a generic invariant train track $\tau$ for $f$ such that $f(\tau)$ folds to $\tau$, i.e. there exist a finite sequence of train tracks
$$f(\tau) = \tau_0 \leftharpoonup \tau_1 \leftharpoonup \ldots \leftharpoonup \tau_n = \tau,$$
where $\tau_i$ is carried by $\tau_{i+1}$ with support map given by a single fold.
\end{thm}

\label{intro}
Given a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.
The existence of invariant train tracks with irreducible transition matrix
allows the application of Perron-Frobenius theory to the study of the properties of these maps.
For example, when an invariant train track is irreducible it can be used to build a Markov partition for the map.
The existence of irreducible invariant train tracks is often taken for granted
(See e.g. \cite{FR}, \cite{F}, \cite{J}, \cite{EF}, \cite{M}, \cite{Pe})
but to our knowledge no complete proof is currently in the literature. See \Cref{discuss} for more details.

\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.
One important datum attached to a veering triangulation is its \textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \cite{LMT1}.
When the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with
irreducible transition matrices.
\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component" from which there are paths to every vertex, and some number of ``infinitesimal cycles" which do not have any paths back to the rest of the graph \cite{AT}.
By studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:

\par In slightly more detail,
given a boundaryless surface with negative Euler characteristic
and a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\colon S \to S$, one can construct the veering triangulation of the mapping torus of $f$, whose
flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.

\begin{qn}
 For all $f\colon S \to S$ pseudo-Anosov, does there exist a \textit{generic} invariant train track for $f$ with irreducible transition matrix?
\end{qn}

\par Given two train tracks $\tau_1$ and $\tau_2$ on $S$, $\tau_2$ is said to \textit{\textbf{carry}} $\tau_1$ (or $\tau_1$ \textit{\textbf{is carried by}} $\tau_2$), denoted $\tau_1 \prec \tau_2$, if there is a $C^1$ map $\sigma\colon S \to S$ called the \textit{\textbf{support map}}
such that
 \begin{enumerate}
 \item $\sigma$ is homotopic to the identity,
 \item $\sigma(\tau_1) \subseteq \tau_2$,
 \item for all points $x$ in $\tau_1$,
 $d_x \sigma_{|\tau_1}\colon T_x \tau_1 \to T_{h(x)}\tau_2$ is an isomorphism of tangent spaces,
 \end{enumerate}
Note that carrying is transitive: if $\tau_1 \prec \tau_2$ and $\tau_2 \prec \tau_3$, then
$\tau_1 \prec \tau_3$.
We also write $\tau_1 \prec_\sigma \tau_2$ to indicate that $\sigma$ is the support map for $\tau_1 \prec \tau_2$. A carrying map $\sigma$ for $\tau_1 \prec \tau_2$ is called \textit{\textbf{combinatorial}} if for every switch $v$ in $\tau_1$, $\sigma(v)$ is a switch in $\tau_2$.
 \par
 Given $\tau_1$ with branches $\{b^1_j\}_{j=1\ldots n}$ and $\tau_2$ with branches $\{b^2_i\}_{i=1\ldots m}$ such that $\tau_1$ is carried by $\tau_2$ with combinatorial support map $\sigma$,
 the \textit{\textbf{transition matrix}} is the $m \times n$ matrix $T = [t_{i,j}]$ where
 $t_{i,j}$ is the number of times the image of $b^1_j$ under $\sigma$ passes over $b^2_i$.
\par
 A \textit{\textbf{train path}} is a $C^1$ immersion $p\colon [0,1] \to \tau$ so that the endpoints $\{0,1\} = \partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\tau$. For $t \in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\colon [0,t] \to \tau$, $p_1 = p_{|[t,1]}\colon [t,1] \to \tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\partial_t)$ and $d_0 p_1(\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\colon [0,2] \to \tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\ldots p_{n-1})$, where $p_i\colon\left[\frac{i}{n},\frac{i+1}{n}\right] \to \tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \textit{\textbf{embedded}}.
\par
 A \textit{\textbf{fold}} is a particular combinatorial support map $\phi$ for $\tau_1 \prec \tau_2$ which induces a bijection between the branches of $\tau_1$ and $\tau_2$
 except on a particular connected subgraph where it is defined according to either of the two pictures in \Cref{fig:fold}. The three branches of $\tau_1$ which are mapped over the small branch $e'$ in the domain of \Cref{fig:fold} are said to \textit{\textbf{fold to}} $e'$.
 If there is a $\tau_1 \prec_\phi \tau_2$
 we say $\tau_1$ \textit{\textbf{folds to}} $\tau_2$, denoted $\tau_1 \leftharpoonup \tau_2$. If $\tau_1$ folds to $\tau_2$, $\tau_1$ is carried by $\tau_2$. If $\tau_1 \prec \tau_2$ with support map given by a sequence of folds we also say $\tau_1$ folds to $\tau_2$.
 \begin{figure}[h]
 \centering
 \includegraphics[width=0.66\textwidth]{foldgood.pdf}
 \caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}
 \label{fig:fold}
\end{figure}
 \par
 A train track $\tau$ is an \textit{\textbf{invariant}} track for a map $f: S \to S$ with support map $\sigma$ if $f(\tau)$ is carried by $\tau$ with support map $\sigma$.
 If the transition matrix for $f(\tau) \prec_\sigma \tau$ is irreducible, we also say $\tau$ is \textit{\textbf{irreducible}}.
By smoothly isotoping $\tau$ if necessary, we may assume it is disjoint from the singularities of $\Lambda^u$. This will be convenient later when we remove the singularities of $\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:

\par Note that the referenced theorem is stated in terms of \textit{splits} rather than folds, which are the combinatorial inverse of folds.
 Additionally, the referenced theorem is stronger than what we state here because the splitting sequence is shown to be canonical up to ``commuting maximal splits": at each step in the sequence, the train track is split at a branch carrying maximal measure of the invariant lamination, and the splits of maximal branches commute if there is not a unique branch carrying maximal measure.
 This theorem motivates the following definition:
\begin{defn}
 An invariant train track $\tau$ for a pseudo-Anosov map $f$ is called \textit{\textbf{veering}} if it
 is generic and $\tau$ can be obtained from $f(\tau)$ by a sequence of folds, in which case there is a sequence of train tracks
 $$f(\tau) = \tau_0 \leftharpoonup \tau_1 \leftharpoonup \ldots \leftharpoonup \tau_n = \tau$$
 where $\tau_i$ is carried by $\tau_{i+1}$ with support map consisting of a single fold.
\end{defn}
\par In \Cref{clvts} we will see that a veering train track as defined here is sufficient to produce the veering triangulation of the fully punctured mapping torus of $f$. While veering train tracks are not necessarily irreducible, they will be the beginning of our construction of irreducible tracks.
\par
 Given two train tracks $\tau_1$, $\tau_2$ on $S$, a (surface) \textit{\textbf{train track map}} is a map $t: S \to S$ so that $t(\tau_1)$ is contained in $\tau_2$, and the restriction of $t$ to $\tau_1$ is
 such that for any train path $p\colon [0,1] \to \tau_1$, $t \circ p\colon [0,1] \to \tau_2$ is also a train path. In other words, a train track map is like a combinatorial support map except that it does not have to be homotopic to the identity map.
 \par The following lemma will be useful for producing support maps:
\begin{lemma}\label{supplemma}
 Let $t\colon S \to S$ be a train track map taking $\tau_1$ to $\tau_2$, and $g\colon S \to S$ be a diffeomorphism.
 If $t$ is homotopic to $g$, then $g(\tau_1) \prec \tau_2$ with support map given by $t \circ g^{-1}$.
\end{lemma}
\begin{proof}
 Since $t$ and $g$ are homotopic, $t \circ g^{-1}$ is homotopic to the identity. Since $t(\tau_1) \subseteq \tau_2$, $(t \circ g^{-1})(g(\tau_1)) = t(\tau_1) \subseteq \tau_2$. Let $p \in g(\tau_1)$. Since $g$ is a diffeomorphism, the restriction of the differential
 $$d_p g^{-1}_{|g(\tau_1)}\colon T_p g(\tau_1) \to T_{g^{-1}(p)} \tau_1$$
 is an isomorphism. Since $t$ is a train track map, $d_{g^{-1}(p)} t_{| \tau_1}$ is an isomorphism, so the composition
 $$d_p (t \circ g^{-1})_{|g(\tau_1)} = (d_{g^{-1}(p)} t_{| \tau_1}) \circ (d_p g^{-1}_{|g(\tau_1)})\colon T_p g(\tau_1) \to T_{(t \circ g^{-1})(p)} \tau_2$$
 is also an isomorphism.
\end{proof}

Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\colon S\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.

\label{mainthm}
    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.,

Question: Let \(f\colon S\to S\) be a pseudo-Anosov homeomorphism. A train track \(\tau\) on \(S\) is called invariant for \(f\) if \(f(\tau)\) is carried by \(\tau\) via a combinatorial support map. If the branches of \(f(\tau)\) are \(\{b^1_j\}\) and the branches of \(\tau\) are \(\{b^2_i\}\), the associated transition matrix is the nonnegative matrix \(T=[t_{i,j}]\), where \(t_{i,j}\) is the number of times the image of \(b^1_j\) under the support map passes over \(b^2_i\). A nonnegative matrix is irreducible if for every \(i,j\) there exists \(n\ge 1\) such that \((T^n)_{i,j}>0\). Under these assumptions, which existence statement holds?

For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that the transition matrix associated to \(f(\tau)\prec \tau\) is irreducible.

Choices:
B. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), every invariant train track \(\tau\) for \(f\) has the property that the transition matrix associated to \(f(\tau)\prec \tau\) is irreducible.
C. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that the transition matrix associated to \(f(\tau)\prec \tau\) is nonzero.
D. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that for every pair of branches \(b^2_i,b^2_j\) of \(\tau\), one has \(t_{i,j}>0\) in the transition matrix associated to \(f(\tau)\prec \tau\).
E. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that each strongly connected component of the transition matrix associated to \(f(\tau)\prec \tau\) is irreducible.

existential choice of modified track after contracting obstructing branches

irreducibility requirement dropped to mere existence of a nontrivial transition matrix

irreducible replaced by strictly positive one-step transition matrix

infinitesimal-cycle obstruction bypassed only after contracting branches, not for the original component decomposition

The stem gives definitions of invariant train tracks and irreducibility, but it does not explicitly state the target existence theorem or uniquely point to choice A. The correct answer is not leaked, though the setup clearly narrows the topic.

The item is close to a theorem-recall question: after introducing the relevant definitions, it asks which existence statement holds. It is not a verbatim restatement, because the alternatives vary in quantifier strength and matrix properties, but it still largely tests recognition of a standard result.

There is some reasoning pressure in separating 'there exists' from 'for every,' and irreducible from merely nonzero or strictly positive. However, for a student who knows the theorem, the answer is mainly direct identification rather than substantial derivation.

The distractors are mathematically plausible and target natural failure modes: overstrengthening to every invariant track (B), weakening to nonzero (C), confusing irreducibility with positivity in one step (D), and using a subtle component-wise condition (E). They are distinct and well aligned with common misunderstandings.

A solid MCQ with strong distractors and little answer leakage, but it leans toward theorem recognition rather than deep generative reasoning.


\begin{Definition}\label{def:phi homogenization}
Fix $\lambda = \lambda(n) > 0$ sufficiently small such that $\sigma = C_1(n)\lambda$ satisfies $4^{n+1}\sigma < c(n)$, where $C_1(n)$ is the one in Lemma \ref{lem:covering Lemma} and $c(n)$ is the one in Proposition \ref{prop:basic}. For $\Lambda\ge \max\{\Lambda_0,\Lambda_1,\Lambda_2\}$, and any nonnegative functions $v \in C(\Omega)$, we define its $\phi$-homogenization $M_{\phi,\Lambda}v(x):=M_{\phi,\lambda,\Lambda}v(x)$ in $\frac{7}{8}\Omega$ through the following maximal averaging type operator:
\begin{equation}\label{eq:phi homogenization}
\begin{split}
M_{\phi,\Lambda}v(x)=  \sup\left\{ t\in\R:\; \exists \ \check{h}(x)<h<  \hat{h}(x),\ s.t., \
\frac{\mu ( \{v> t\} \cap S_h (x)  )}{\mu(S_h (x))} > \lambda \right\}.
\end{split}
\end{equation}
\end{Definition}

\begin{Theorem}\label{thm:uniqueness within periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ be a quadratic function with $A\in \mathcal{S}_+^{n\times n}$ satisfying \eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \eqref{eq:periodic equation ma} with quadratic component $P$.
\end{Theorem}

In this paper, we study convex solutions to the Monge-Amp\`ere equation
\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \mathbb{R}^n,
\end{equation}
where $\mu\not\equiv 0$ is a nonnegative locally finite Borel measure on $\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\mu $ is periodic with respect to the integer lattice $\mathbb{Z}^n := \{(k_1,\dots,k_n) \mid k_i \in \mathbb{Z}\} \subset \mathbb{R}^n$, i.e., for any Borel set $E \subset \mathbb{R}^n$,
\begin{equation}\label{eq:periodic f}
\mu (E+z)=\mu (E),\quad \forall \ z \in \mathbb{Z}^n.
\end{equation}

Let $\mathcal{S}_+^{n\times n}$ denote the set of positive definite symmetric $n\times n$ matrices.
\begin{Definition}[Periodic solution]\label{def:periodic solution}
A \emph{periodic solution} to \eqref{eq:periodic equation ma} is a convex function $u \in C(\mathbb{R}^n)$ satisfying \eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as
\begin{equation}\label{eq:periodic solution}
u(x) = v(x) +P(x),
\end{equation}
where $v$ is periodic with respect to $\mathbb{Z}^n$ and $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ is a quadratic function with $A\in \mathcal{S}_+^{n\times n}$, $b\in\mathbb{R}^n$ and $c\in\mathbb{R}$.
\end{Definition}

Note that if $u$ is a periodic solution to \eqref{eq:periodic equation ma} and is expressed as \eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition
\begin{equation}\label{eq:A compatible}
 \det A =\mu (\mathbb{T}^n),
\end{equation}
where $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$.

Our first result is the uniqueness of periodic solutions.

\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \R^n,
\end{equation}

Let $\mu \not\equiv 0$ be a nonnegative locally finite periodic Borel measure on $\R^n$. We show that any convex solution to the Monge-Amp\`ere equation
\[
\det D^2 u = \mu \quad \text{in } \R^n
\]
admits a unique decomposition (up to addition of constants) as the sum of a quadratic polynomial and a periodic function. This result extends,
in full generality, the earlier works for the case $\mu=f(x)\,\ud x$: when $\log f \in C^\alpha$, it was established by Caffarelli and Li; and
when $\log f$ is merely bounded, it was proved by Li and Lu. Our result thus answers a question raised by Li and Lu. A key ingredient in the proof is a new dichotomous Harnack-type inequality for linearized Monge-Amp\`ere equations with nonnegative periodic measures.

Let $\mathcal{S}_+^{n\times n}$ denote the set of positive definite symmetric $n\times n$ matrices.
\begin{Definition}[Periodic solution]\label{def:periodic solution}
A \emph{periodic solution} to \eqref{eq:periodic equation ma} is a convex function $u \in C(\mathbb{R}^n)$ satisfying \eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as
\begin{equation}\label{eq:periodic solution}
u(x) = v(x) +P(x),
\end{equation}
where $v$ is periodic with respect to $\mathbb{Z}^n$ and $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ is a quadratic function with $A\in \mathcal{S}_+^{n\times n}$, $b\in\R^n$ and $c\in\R$.
\end{Definition}

Our first result is the uniqueness of periodic solutions.

The existence of periodic solutions follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof. For the uniqueness, when $\mu=f(x)\,\ud x$ is a positive measure, the case $\log f \in C^\alpha$ was proved by Li \cite{li1990existence}, and the case $\log f\in L^\infty$ was shown by Li-Lu \cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.

\begin{Theorem}\label{thm:solutions are periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $u$ be a convex solution of \eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \ref{def:periodic solution}.
\end{Theorem}

\begin{Proposition}\label{prop:semiconcave}
Let $u$ be a convex solution to \eqref{eq:periodic equation ma} with $\mu$ being a $\mathbb{Z}^n/8n$-periodic measure (where $\mathbb{Z}^n := \Gamma_{\{e_1,\cdots,e_n\}}$) satisfying $\mu(\mathbb{Z}^n) = 1$.
Under the nondegeneracy condition
\begin{equation}\label{eq:nondegenerate}
\Delta^2_{i} u(0):=\Delta^2_{e_i} u(0) > 0 \quad \text{for all } 1 \leq i \leq n,
\end{equation}
there exists a unique compatible matrix $A$ satisfying:
\[
\sup_{ \R^n}\Delta_z^2 u=z^{\top}Az ,\quad \forall z \in \mathbb{Z}^n.
\]
\end{Proposition}
\begin{proof}
Without loss of generality, we may assume $u(0) = 0$ and $u \geq 0$.
Let us define $S_h := \{x \in \mathbb{R}^n : u(x) \leq h\}$, and select $\Lambda_u>0$ such that the section $S_{\Lambda_u}$ satisfies $\kappa_{\mathbb{Z}^n}(S_{\Lambda_u}) \geq \Lambda_0^2$.
Let us now normalize $S_h$ such that $B_1(0) \subset T_h^{-1} S_h \subset B_n(0)$ and consider the normalized function
\[
u_{h}(x) =\frac{u(T_h x)}{(\det T_h)^{\frac{2}{n}}} \quad \text{with } \det D^2 u_h= \mu_h=\mu \circ T_h .
\]
Due to the periodicity of $\mu$ and large $h$, we know that
\[
c(n)h^{\frac{n}{2}} \leq \det T_h \leq C(n)h^{\frac{n}{2}},
\]
and thus, $c(n)h^{\frac{n}{2}} \leq |S_h| \leq C(n)h^{\frac{n}{2}}$. Applying Proposition \ref{prop:basic} to $\phi=u_h-h(\det T_h)^{-\frac{2}{n}}$, we obtain that for sufficiently large $h$,
\[
c(n) (h/\Lambda_u)^{\frac{1}{3}} S_{\Lambda_u} \subset S_h \subset C(n) (h/\Lambda_u)^{\frac{2}{3}} S_{\Lambda_u},
\]
and consequently, $B_{c(n) h^{\frac{1}{4}}}\subset S_h\subset B_{C(n) h^{\frac{3}{4}}}$ for all large $h$. That is, for sufficiently large $h$, we now have
\[
\quad c(n)h^{\frac{1}{4}} I\leq T_h \leq C(n)h^{\frac{3}{4}} I.
\]
The transformation law \eqref{eq:transformation law} then yields the following estimates at corresponding points
\begin{equation}\label{eq:transformh}
c(n)h^{-\frac{1}{2}}\Delta^2_{e_i} u \leq \Delta^2_{T_h^{-1} e_i} u_h \leq C(n)h^{\frac{1}{2}}\Delta^2_{e_i} u.
\end{equation}

\textbf{Step 4.} Suppose for some sequence $h \to \infty$, the rescaled functions $\tilde{u}_h$ locally converge to a quadratic form $P_A(x) = \frac{1}{2}x^{\top}A x$ for $A \in \mathcal{S}_+^{n\times n}$ with $\det A = 1$.
For each fixed direction $i$, we denote
\[
\alpha=\sup_{\R^n}\Delta^2_{e_i} u,\quad \beta= e_i^{\top} A e_i.
\]
Note that the estimate \eqref{eq:w21 estimate} continues to hold for $\tilde{u}_{h}$, from which we derive $\alpha \geq \beta$.
We claim
\begin{equation}\label{eq:w2infty sup}
\alpha=\beta.
\end{equation}
Assume by contradiction that $\alpha = \beta + 4s$ for some $s > 0$.
Through rescaling from infinity, we may assume without loss of generality that there exist points $x_h \in B_{1/8}$ satisfying
\[
\alpha- \Delta^2_{h^{\frac{1}{2}}e_i} \tilde{u}_{h}(x_h)= \inf_{B_{1/8}} (\alpha- \Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_{h} ) \leq a_h,
\]
where $a_h \to 0 $ as $h \to \infty$. Define the normalized function $v = (\alpha - \Delta^2_{h^{\frac{1}{2}}e_i} \tilde{u}_{h})/a_h$ and set $\delta = s/[C(\beta + 2s)]$. Applying Lemma \ref{lem:Theorem 7.3.1 modify} yields with $\phi=\tilde{u}_h-1$ and $\Omega=S_{1,h}=\{\tilde{u}_h \leq 1\} $, we obtain that
\[
\tilde{\mu}_h ( \{v\leq M_1(n,\delta) \} \cap S_{1,h} ) \geq (1-\delta) \tilde{\mu}_h(S_{1,h} ),
\]
where $\tilde{\mu}_h=\tilde{f}_{h}\,\ud x$. For sufficiently small $a_h$, we consequently obtain
\[
\tilde{\mu}_h\left(\left\{\Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_h>\beta +2 s\right\} \cap S_{1,h}\right)= \tilde{\mu}_h\left(\left\{v <\frac{2s}{a_h}\right\} \cap S_{1,h}\right)\geq (1-\delta) \tilde{\mu}_h (S_{1,h}).
\]
This leads to the integral lower bound
\[
\int_{S_{1,h}} \Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_h \ud \tilde{\mu}_h \geq (1-\delta) (\beta+2s) \tilde{\mu}_h (S_{1,h})\geq
(\beta+s) \tilde{\mu}_h (S_{1,h}).
\]
This contradicts the uniform estimate \eqref{eq:w21 estimate} after performing a suitable rescaling.

\begin{proof}[Proof of Theorems \ref{thm:uniqueness within periodic} and \ref{thm:solutions are periodic}]
Let $u$ be a global solution of \eqref{eq:periodic equation ma}.
Clearly, $u$ cannot be linear on any ray segment. Indeed, if it were linear on a ray, after subtracting its supporting hyperplane on this ray and by using the convexity of $u$, $u$ would be bounded within any cylinder having that ray as its axis. This would imply that the associated measure $\mu$ is finite on any strictly smaller cylinder, contradicting the assumptions that $\mu$ is periodic and $\mu \not\equiv 0$. Therefore, there exists a sufficiently large integer $R$ such that $S_1(0) \subset B_R(0)$.
We now rescale $u$ by defining $u(8nRx)/64n^2R^2$. For simplicity, we continue to denote this rescaled function as $u$. Under this scaling, one can verify that condition \eqref{eq:nondegenerate} in Proposition \ref{prop:semiconcave} is satisfied. Hence, by Proposition \ref{prop:semiconcave}, there exists a compatible matrix $A \in \mathcal{S}_+^{n \times n}$ such that
\[
\sup_{ \R^n}\Delta_z^2 u=z^{\top}Az,\quad \forall z \in \mathbb{Z}^n.
\]
Let $w$ be a periodic solution to \eqref{eq:periodic equation ma} in the sense of Definition \ref{def:periodic solution} with quadratic part $\frac{1}{2}z^{\top} A z $. Then $\Delta_z^2 w=z^{\top}Az$. Let
\[
v(x)=u(x)-w(x).
\]
By subtracting a suitable linear function, we now assume that $v(0)=0$ and
\[
v(e_i)=v(-e_i),\quad 1\leq i\leq n.
\]
Noting that $\sup_{\R^n} \Delta_z^2 v = \sup_{\R^n} \Delta_z^2 (u - w) = 0$, we then find that
\[
v(\pm ke_i)\leq 0 \quad \text{for all } k\in \mathbb{Z},\ 1\leq i\leq n.
\]

The existence of periodic solutions follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof.  For the uniqueness, when $\mu=f(x)\,\mathrm{d} x$ is a positive measure, the case $\log f \in C^\alpha$ was proved by Li \cite{li1990existence}, and the case  $\log f\in L^\infty$ was shown by Li-Lu \cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.

Our second result is the classification below, which extends the main results  of Caffarelli-Li \cite{caffarelli2004liouville} and Li-Lu \cite{li2019monge} to the full generality. In particular, it provides a complete answer to Question 1.1 in \cite{li2019monge}.

\begin{Theorem}\label{thm:solutions are periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $u$ be a convex solution of \eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \ref{def:periodic solution}.
\end{Theorem}

Caffarelli-Li \cite{caffarelli2004liouville} first proved Theorem \ref{thm:solutions are periodic} for positive measures $\mu=f(x)\,\mathrm{d} x$ with $\log f \in C^\alpha$, and conjectured the extension to merely bounded $\log f$. This conjecture was resolved by Li-Lu \cite{li2019monge}, who further asked whether the result holds for the nonnegative case $f \geq 0$ with $f \not\equiv 0$. Theorem \ref{thm:solutions are periodic} affirmatively answers this question -- and indeed establishes it in full generality.

\begin{Remark}
When $\mu \equiv 0$, equation \eqref{eq:periodic equation ma} reduces to the homogeneous Monge-Amp\`ere equation $\det D^2u = 0$. The solutions are precisely those functions that are linear along certain directions (see, e.g., Caffarelli-Nirenberg-Spruck \cite{caffarelli1986dirichlet}).
In particular, writing $x = (x',x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}$, any convex function of the form $u(x) = w(x')$ is a solution, and such solutions are not necessarily periodic.
\end{Remark}

Since $\mu$ is assumed to be nonnegative,  equation \eqref{eq:periodic equation ma} represents a possibly degenerate Monge-Amp\`ere equation, for which many arguments in  \cite{caffarelli2004liouville,li2019monge} do not apply.  The principal challenge stems from the behavior of $u$ on small sections: the associated measure $\mu$ may not satisfy the doubling condition. This failure implies that sections lose the engulfing property, consequently invalidating the Vitali or  Besicovitch covering lemma which is essential for the Calder\'on-Zygmund decomposition on sections.
Therefore, we cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez \cite{caffarelli1997properties} for subsolutions or supersolutions of linearized Monge-Amp\`ere equations.

Nonetheless, by exploiting the equation's periodic structure, we derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator (see Definition \ref{def:phi homogenization}). This constitutes one of the key innovations of the present work. For supersolutions, we address the challenge by excluding contributions from small sections. Theorem \ref{thm:harnack super} establishes decay estimates for homogenized level-sets of supersolutions within our framework. For subsolutions, for which standard $L^{\infty}$ bounds fail, Theorem \ref{thm:harnack sub} establishes an alternative dichotomy regarding their growth behavior: either the $L^{\infty}$ bound holds or the $L^{\infty}$ norm grows exponentially. Finally, we obtain  in Theorem \ref{thm:harnack} a Harnack inequality with a quantitatively controlled error term.

After overcoming this critical difficulty arising from the degeneracy of equation \eqref{eq:periodic equation ma}, the rest proof focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart. As observed in \cite{caffarelli2004liouville}, these quantities naturally arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations. This is another point where the periodicity of the measure plays a crucial role. We then establish the quadratic behavior of solutions near infinity, extending the arguments in \cite{caffarelli2004liouville,li2019monge} for positive measures $\mu=f(x)\mathrm{d} x$ to  the degenerate case considered here.  In this step, we must also overcome additional difficulties due to the degeneracy of
 \eqref{eq:periodic equation ma}. Consequently, when Theorem \ref{thm:harnack} is applied to the difference of two solutions, exponential growth is precluded. Hence, the classical Harnack inequality holds for this difference, and our theorems would follow.

This paper is organized as follows. In Section \ref{sec:harnack}, we study convex functions whose Monge-Amp\`ere measure has small period, establishing a dichotomous Harnack type  inequality for linearized Monge-Amp\`ere equations. Section \ref{sec:semiconvavity} investigates  the uniqueness of compatible quadratic component of each solution in the sense of Proposition \ref{prop:semiconcave}. Finally, in Section \ref{sec:main theorem}, we prove  Theorems \ref{thm:uniqueness within periodic} and \ref{thm:solutions are periodic}.

The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof,” while Theorem~\ref{thm:uniqueness within periodic} “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including Theorem~\ref{thm:uniqueness within periodic}) in the nonnegative/degenerate setting: “the associated measure $\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez.”

To overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” For supersolutions they do so “by excluding contributions from small sections,” obtaining “decay estimates for homogenized level-sets” (Theorem~\ref{thm:harnack super}); for subsolutions (where “standard $L^{\infty}$ bounds fail”), they prove a “dichotomy” (Theorem~\ref{thm:harnack sub}): “either the $L^{\infty}$ bound holds or the $L^{\infty}$ norm grows exponentially”; combining these yields “a Harnack inequality with a quantitatively controlled error term” (Theorem~\ref{thm:harnack}).

“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying Theorem~\ref{thm:harnack} “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and “our theorems would follow.”

The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof,” while in establishing the main theorem it “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including the main theorem) in the nonnegative/degenerate setting: “the associated measure $\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez.”

To overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” We first use the following theorem for supersolutions:
\begin{Theorem}\label{thm:harnack super}
Assume that $v \in C^2(\Omega)$ is a nonnegative supersolution of \eqref{eq:linearized ma phi} in $\Omega$.
Then there exist positive constants  $\Lambda_6(n)$, $\epsilon(n) $ and $C(n) $ such that if
\[
\inf_{B_{1/4}} v \leq 1,
\]
then for $\Lambda\ge \Lambda_6(n)$, we have
\begin{equation}\label{eq:harnack super}
\mu(\{M_{\phi,\Lambda}v >  t\} \cap B_{1/2})< Ct^{-\epsilon} .
\end{equation}
\end{Theorem}
For subsolutions (where “standard $L^{\infty}$ bounds fail”), they prove the following “dichotomy”:
\begin{Theorem}\label{thm:harnack sub}
Assume that $v \in C^2(\Omega)$ with $v \geq 0$ is a subsolution of \eqref{eq:linearized ma phi} in $\Omega$.
Then, there exist positive constants $\Lambda_7(n)$ and $\beta(n)$ such that for any $\epsilon > 0$, there is a constant $C_2(n, \epsilon)$ with the following property: If $\Lambda\ge \Lambda_7(n)$ and
\[
\min\{\mu(\{v >  t\}  \cap B_{1/2}), \mu(\{M_{\phi,\Lambda}v >  t\} \cap B_{1/2})  \} \leq t^{-\epsilon}   \quad  \forall t>0,
\]
then either
\[
\sup_{B_{1/4} } v  \leq C_2(n,\epsilon),
\]
or for $\kappa =\kappa(B_1)\geq \Lambda^2$ that
\[
\sup_{B_{1/2} } v  \geq e^{\beta \kappa^{\frac{1}{3}} } \sup_{B_{1/4} }v.
\]
\end{Theorem}
Combining these yields “a Harnack inequality with a quantitatively controlled error term,” namely the following theorem:
\begin{Theorem}\label{thm:harnack}
Let $\Omega$ and $\widetilde{\Omega}$ be open convex subsets of $\mathbb{R}^n$ satisfying $B_1\subset \Omega \subset B_n, B_1\subset \widetilde{\Omega} \subset B_n$, $n\geq 2$, and let $\phi\in C^2(\overline{\Omega})$ and $\tilde{\phi}\in C^2(\overline{\widetilde{\Omega}})$ be convex functions satisfying,
\[
\mu_\phi =\mu \quad \text{in } \Omega, \quad \phi=0 \quad \text{on } \partial \Omega,
\]
\[
\mu_{\tilde{\phi}} =\mu \quad \text{in }  \widetilde{\Omega}, \quad \tilde{\phi}=0 \quad \text{on } \partial \widetilde{\Omega},
\]
where $\mu$ is a periodic Borel measure defined on $\R^n$ satisfying \eqref{eq:period gamma},  \eqref{eq:normalized assumption}, and \eqref{eq:small period assumption}. Assume that $v\in C^2(\Omega) \cap C^2(\widetilde{\Omega})$ with $v\geq 0$ satisfies
\[
L_{\phi}v \leq 0   \quad  \text{in } \Omega, \quad L_{\tilde{\phi}}v \geq 0   \quad  \text{in }\widetilde{\Omega}.
\]

Then there exist positive constants $\beta(n)$, $\Lambda_9(n)$,  and $C(n)$ such that if $\Lambda>\Lambda_9(n)$, then either
\[
\sup_{B_{1/4}}v\leq C\inf_{B_{1/4}}v,
\]
or for $\kappa =\kappa(B_1)\geq \Lambda^2$ that
\[
\sup_{B_{1/2}} v  \geq e^{\beta \kappa^{\frac{1}{6}}}\sup_{B_{1/4}}v.
\]
Consequently, we have
\[
\sup_{B_{1/4}}v \leq C\inf_{B_{1/4}}v+e^{-\beta \kappa^{\frac{1}{6}}}\sup_{B_{1/2}} v.
\]
\end{Theorem}

“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying the preceding theorem (the Harnack inequality with controlled error term) “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and this completes the proof strategy for the main theorem.

\label{thm:uniqueness within periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying
\begin{equation}\label{eq:periodic f}
\mu (E+z)=\mu (E),\quad \forall \ z \in \mathbb{Z}^n.
\end{equation}
and let $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ be a quadratic function with $A\in \mathcal{S}_+^{n\times n}$ satisfying
\begin{equation}\label{eq:A compatible}
 \det A =\mu (\mathbb{T}^n),
\end{equation}
Then, to prove the main theorem, we show that there exists a unique periodic solution (up to addition of constants) to
\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \R^n,
\end{equation}
with quadratic component $P$.

Question: Let $\mu\not\equiv 0$ be a nonnegative locally finite Borel measure on $\mathbb{R}^n$ that is $\mathbb{Z}^n$-periodic in the sense that for every Borel set $E\subset \mathbb{R}^n$ and every $z\in \mathbb{Z}^n$,\n\[\n\mu(E+z)=\mu(E).\n\]\nLet\n\[\nP(x)=\tfrac12 x^\top A x+b\cdot x+c,\n\]\nwhere $A\in \mathcal S_+^{n\times n}$ is positive definite symmetric, $b\in \mathbb{R}^n$, $c\in \mathbb{R}$, and\n\[\n\det A=\mu(\mathbb{T}^n),\qquad \mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n.\n\]\nA periodic solution of\n\[\n\det D^2u=\mu\quad\text{in }\mathbb{R}^n\n\]\nmeans a convex function $u\in C(\mathbb{R}^n)$ satisfying the equation in the Alexandrov sense and admitting a decomposition $u(x)=v(x)+P(x)$ with $v$ $\mathbb{Z}^n$-periodic. Under these assumptions, which statement holds?

There exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, and it is unique up to addition of constants; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1-u_2$ is constant on $\mathbb{R}^n$.

Choices:
B. There exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, and it is unique without any ambiguity; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1=u_2$ on $\mathbb{R}^n$.
C. There exists at most one periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$ up to addition of constants; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1-u_2$ is constant on $\mathbb{R}^n$.
D. For every quadratic function $P(x)=\tfrac12 x^\top A x+b\cdot x+c$ with $A\in \mathcal S_+^{n\times n}$ satisfying $\det A\ge \mu(\mathbb{T}^n)$, there exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, unique up to addition of constants.
E. There exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, and for any two periodic solutions $u_1,u_2$ with quadratic components having the same matrix part $A$, the difference $u_1-u_2$ is an affine function on $\mathbb{R}^n$.

constant-shift ambiguity in comparing two solutions

dropped existence while retaining uniqueness up to constants

exact compatibility condition $\det A=\mu(\mathbb{T}^n)$

normalization removing linear parts before applying Harnack to the difference

The stem gives the hypotheses and asks for the valid conclusion, but it does not explicitly reveal the correct option. There is no direct wording that singles out existence plus uniqueness up to constants.

The item is quite close to a theorem-recall question: the assumptions are laid out in theorem style and the correct answer is essentially the theorem’s conclusion. However, the alternatives introduce meaningful nearby variants, so it is not a pure verbatim restatement.

Some reasoning is needed to distinguish exact uniqueness up to constants from strict uniqueness, mere uniqueness without existence, and an incorrect compatibility condition. Still, the task is driven more by precise theorem recognition than by substantial generative mathematical reasoning.

The distractors are strong and mathematically targeted: they test common mistakes about constant-shift ambiguity, omission of existence, weakening/altering the determinant compatibility condition, and overgeneralizing the difference of two solutions.

A solid MCQ with little answer leakage and strong distractors, but it leans heavily on theorem recall rather than deeper generative reasoning.


\begin{prop}[{cf.~\cite[Corollary~3.4]{S78}}]\label{prop:ci}
Let $R$ be a standard graded $\kk$-algebra.
If $R$ is a complete intersection, then the $h$-polynomial of $R$ is of type~CI, that is, $h_R(t)=\prod_{i=1}^r\Psi_{m_i}(t)$ for some non-negative integer $r$ and positive integers $m_1, \dots, m_r$.
\end{prop}

Throughout this paper, let $\Bbbk$ denote a field. We say that $R$ is a \emph{standard graded} $\Bbbk$-algebra
if $R = \bigoplus_{n \ge 0} R_n$ is a commutative graded $\Bbbk$-algebra with $R_0 = \Bbbk$ and finitely generated in degree one.
(A standard graded $\Bbbk$-algebra is also called a \emph{homogeneous} $\Bbbk$-algebra in other literature.)

\subsection{Background}
The Hilbert series of a (not necessarily commutative) graded $\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\Bbbk$-algebras.
It provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\Bbbk$-algebra as a $\Bbbk$-vector space.
It is a classical and well-known fact that if a standard graded $\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \textit{$h$-polynomial}, has all positive coefficients.
This fact is one of the cornerstones in graded commutative algebra.
Furthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series.
A prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\cite{S78}),
which has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness,
such as almost Gorensteinness (\cite{H}), nearly Gorensteinness (\cite{M}), and Artin-Schelter Gorensteinness (\cite{JZ}).

It is also known that when a standard graded $\Bbbk$-algebra is a complete intersection, its Hilbert series takes a particularly simple form.
In fact, for a positive integer $m$, let \[\Psi_m(t)=1+t+\cdots+t^{m-1}.\]
Then the $h$-polynomial of a standard graded complete intersection is of the form of a product of a finite collection of $\Psi_m(t)$'s (see Proposition~\ref{prop:ci}).
In addition, the Hilbert series also serves as an essential tool in investigating noncommutative graded complete intersections (see \cite{KKZ, KKZ2}).

\subsection{Cyclotomic standard graded $\Bbbk$-algebras and polynomials of type CI}

In recent years, increasing attention has been directed to the finer analytic behavior of the Hilbert series of standard graded $\Bbbk$-algebras, particularly the roots of their $h$-polynomials.
These roots encode subtle structural information and sometimes admit striking combinatorial interpretations
(see, e.g., the survey \cite{Br} for combinatorial consequences of the real-rootedness of the numerator polynomials of generating series).

Motivated by this perspective, we focus on the following notion for standard graded $\Bbbk$-algebras:

\begin{dfn}[cf. \cite{BD, KKZ}]
A standard graded $\Bbbk$-algebra $R$ is called \textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane.
\end{dfn}

Monic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \textit{Kronecker polynomials}.
It is known that every such polynomial can be expressed as a product of cyclotomic polynomials.
Note that the $h$-polynomials of cyclotomic $\Bbbk$-algebras are \textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials.
We say that a polynomial is \textit{of type CI} if it can be written as a product of $\Psi_m(t)$'s.
In particular, the $h$-polynomials of standard graded complete intersections are of type CI.
However, the converse does not hold in general; see, for example, \cite[Example~3.7]{S78}.
Borzi and D'Al\`i showed that the converse does hold when $R$ is Koszul~\cite[Theorem~3.9]{BD}.
Moreover, it was proved that a cyclotomic polynomial $\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\cite[Theorem~4.3]{BD}.

Despite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial.
This naturally leads to the following question:
\begin{q}
Which polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\Bbbk$-algebras, other than those of type CI?
\end{q}
For example, \cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\Bbbk$-algebra.

\subsection{Main Results}

The aim of this paper is to make a contribution to this question.
Our main result is stated as follows.

\subsection{Background}
The Hilbert series of a (not necessarily commutative) graded $\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\Bbbk$-algebras.
It provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\Bbbk$-algebra as a $\Bbbk$-vector space.
It is a classical and well-known fact that if a standard graded $\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \textit{$h$-polynomial}, has all positive coefficients.
This fact is one of the cornerstones in graded commutative algebra.
Furthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series.
A prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\cite{S78}),
which has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness,
such as almost Gorensteinness (\cite{H}), nearly Gorensteinness (\cite{M}), and Artin-Schelter Gorensteinness (\cite{JZ}).

\begin{dfn}[cf. \cite{BD, KKZ}]
A standard graded $\Bbbk$-algebra $R$ is called \textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane.
\end{dfn}

Monic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \textit{Kronecker polynomials}.
It is known that every such polynomial can be expressed as a product of cyclotomic polynomials.
Note that the $h$-polynomials of cyclotomic $\Bbbk$-algebras are \textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials.
We say that a polynomial is \textit{of type CI} if it can be written as a product of $\Psi_m(t)$'s.
In particular, the $h$-polynomials of standard graded complete intersections are of type CI.
However, the converse does not hold in general; see, for example, \cite[Example~3.7]{S78}.
Borzi and D'Al\`i showed that the converse does hold when $R$ is Koszul~\cite[Theorem~3.9]{BD}.
Moreover, it was proved that a cyclotomic polynomial $\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\cite[Theorem~4.3]{BD}.

Despite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial.
This naturally leads to the following question:
\begin{q}
Which polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\Bbbk$-algebras, other than those of type CI?
\end{q}
For example, \cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\Bbbk$-algebra.

\subsection{Main Results}

The aim of this paper is to make a contribution to this question.
Our main result is stated as follows.

\begin{abstract}
We call a standard graded commutative $\kk$-algebra \textit{cyclotomic} if its $h$-polynomial has all its roots on the unit circle in the complex plane.
Complete intersections provide typical examples of cyclotomic algebras,
since the $h$-polynomial of any standard graded complete intersection is a product of polynomials of the form $1 + t + \cdots + t^{m-1}$.
We refer to such polynomials as being \textit{of type CI}.
A natural question is whether there exists a cyclotomic standard graded $\kk$-algebra whose $h$-polynomial is not of type CI.
In this paper, we give a partial answer to this question.
We show that the $h$-polynomial $h_R(t)$ of a cyclotomic standard graded $\kk$-algebra $R$ is of type CI whenever $h_R(1) \in \{1, 4, 6\}$ or $h_R(1)$ is prime.
On the other hand, if $n \ge 8$ and $n$ is not prime, then there exists a cyclotomic standard graded $\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1) = n$.
\end{abstract}

Monic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \textit{Kronecker polynomials}.
It is known that every such polynomial can be expressed as a product of cyclotomic polynomials.
Note that the $h$-polynomials of cyclotomic $\kk$-algebras are \textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials.
We say that a polynomial is \textit{of type CI} if it can be written as a product of $\Psi_m(t)$'s.
In particular, the $h$-polynomials of standard graded complete intersections are of type CI.
However, the converse does not hold in general; see, for example, \cite[Example~3.7]{S78}.
Borzi and D'Al\`i showed that the converse does hold when $R$ is Koszul~\cite[Theorem~3.9]{BD}.
Moreover, it was proved that a cyclotomic polynomial $\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\kk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\cite[Theorem~4.3]{BD}.

The aim of this paper is to make a contribution to this question.
Our main result is stated as follows.

Note that $h_R(1)$ is nothing but the multiplicity of $R$ (see \cite[Proposition 4.1.9]{BH}), which is one of the important invariants of graded modules.

\section{Proof of Theorem~\ref{thm:main} (2)}\label{sec:(2)}
The goal of this section is to give a proof of Theorem~\ref{thm:main} (2).
Specifically, for any non-prime $n$ with $n \geq 8$, we construct a cyclotomic standard graded $\kk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.
Such $\kk$-algebras are realized as Stanley-Reisner rings of certain simplicial complexes.

We prove the following theorem, which implies Theorem~\ref{thm:main} (2).
\begin{thm}\label{thm:p=3}
{\em (1)} Let $q \ge 5$ be an odd integer.
Then there exists a cyclotomic standard graded $\kk$-algebra $R$ such that $h_R(t)$ is not of type CI with $h_R(1)=2q$.

\noindent
{\em (3)} Let $5 \le p \le q$ be odd integers.
Then there exists a cyclotomic standard graded $\kk$-algebra $R$ such that $h_R(t)$ is not of type CI with $h_R(1)=pq$.
\end{thm}

We postpone the proof of this theorem. So far, assume Theorem~\ref{thm:p=3} holds. Then we can prove the desired result. In fact,
let $R$ be a cyclotomic standard graded $\kk$-algebra whose $h$-polynomial is not of type CI.
Then, for any positive integer $a$, the standard graded $\kk$-algebra $R'=R[x]/(x^a)$ has the $h$-polynomial $h_{R'}(t)$ which is not of type CI and $h_{R'}(1)=a\cdot h_R(1)$.
(In fact, we have $h_{R'}(t)=h_R(t)\Psi_a(t)$.)

\begin{thm}[Main Result]\label{thm:main}
{\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\kk$-algebra $R$ is of type CI if $h_R(1) \in \{1,4,6\}$ or $h_R(1)$ is prime.

\noindent
{\em (2)} There exists a cyclotomic standard graded $\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.
\end{thm}

Note that $h_R(1)$ is nothing but the multiplicity of $R$ (see \cite[Proposition 4.1.9]{BH}), which is one of the important invariants of graded modules.

\subsection{Structure of the paper}
We briefly describe the structure of this paper.
In Section~\ref{sec:pre}, we fix our notation (Hilbert series, $h$-polynomials, simplicial complexes, etc.),
review fundamental notions (cyclotomic polynomials, Stanley-Reisner theory, and related topics),
and recall classical results, namely Macaulay's theorem and the $g$-theorem, which will be used in the proof of Theorem~\ref{thm:main}.
Sections~\ref{sec:(1)} and~\ref{sec:(2)} are devoted to the proofs of parts~(1) and~(2) of Theorem~\ref{thm:main}, respectively.

The post-theorem introduction gives only a high-level roadmap: it says that Section~\ref{sec:pre} recalls classical results—"namely Macaulay's theorem and the $g$-theorem"—"which will be used in the proof of Theorem~\ref{thm:main}," and that Sections~\ref{sec:(1)} and~\ref{sec:(2)} contain the proofs of parts~(1) and~(2) of Theorem~\ref{thm:main}, respectively.

The post-theorem introduction gives only a high-level roadmap: it says that next it recalls classical results—"namely Macaulay's theorem and the $g$-theorem"—"which will be used in establishing the main theorem," and that later it contains the proofs of parts~(1) and~(2) of the main theorem, respectively.

[Main Result]\label{thm:main}
{\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\Bbbk$-algebra $R$ is of type CI if $h_R(1) \in \{1,4,6\}$ or $h_R(1)$ is prime.

\noindent
{\em (2)} There exists a cyclotomic standard graded $\Bbbk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.,

Question: Let $R$ be a standard graded commutative $\Bbbk$-algebra with Hilbert series of the form $H_R(t)=\dfrac{h_R(t)}{(1-t)^d}$, where $h_R(t)$ is its $h$-polynomial. Call $R$ cyclotomic if all roots of $h_R(t)$ lie on the unit circle in $\mathbb{C}$. A polynomial is said to be of type CI if it can be written as a product of polynomials $\Psi_m(t)=1+t+\cdots+t^{m-1}$. Which statement holds for such algebras and $h$-polynomials?

For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, for every non-prime integer $n\ge 8$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.

Choices:
B. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6,8\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, for every non-prime integer $n\ge 9$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.
C. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI.
D. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, there exists a non-prime integer $N\ge 8$ such that for every non-prime integer $n\ge N$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ with $h_R(t)$ not of type CI and $h_R(1)=n$.
E. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is composite, then $h_R(t)$ is of type CI. Moreover, for every prime integer $n\ge 8$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.

sharp exceptional values and threshold $n\ge 8$

dropped the existence clause in part (2)

uniform bound replaced exact range of all non-prime $n\ge 8$

prime/non-prime dichotomy in both directions

The stem only defines the terms and asks which statement is true; it does not reveal the specific exceptional values, quantifiers, or existence claim appearing in the correct option.

The item is largely theorem-recognition: the correct answer is essentially the exact statement of a result. However, it is not a pure tautology because the options introduce meaningful competing variants with altered thresholds and quantifiers.

Selecting the best answer requires moderate reasoning or careful recall, especially to track the sharp set {1,4,6}, the prime condition, and the exact existence clause for every non-prime n >= 8. Still, it does not require constructing a proof or applying the theorem in a new setting.

The distractors are strong: they are close to the target statement, differ in mathematically meaningful ways, and reflect common failure modes such as weakening the conclusion, shifting thresholds, or confusing prime/composite cases.

A solid MCQ with little answer leakage and strong distractors, but it mainly tests precise theorem recall/comparison rather than deeper generative mathematical reasoning.