\documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{mathrsfs} \usepackage{xcolor} \usepackage[margin=1.08in]{geometry} \usepackage{enumitem} \usepackage{fvextra} %\usepackage{graphicx} % figures %\usepackage[cmtip,all]{xy} % commutative diagrams \usepackage[ colorlinks=true, linkcolor=magenta, citecolor=cyan, urlcolor=magenta, pdftitle={The fifteenth cohomology of moduli spaces of stable curves}, pdfauthor={} ]{hyperref} \setlist[enumerate]{leftmargin=2.2em,itemsep=2pt,topsep=4pt} \setlist[itemize]{leftmargin=2em,itemsep=2pt,topsep=4pt} \allowdisplaybreaks \emergencystretch=2em \numberwithin{equation}{section} \setcounter{tocdepth}{1} % One shared counter per section: Theorem 2.1, Definition 2.2, Remark 2.3, … % The trailing comment gives the label kind of each environment % (\label{thm:main}, \label{def:admissible-pair}, …). \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] % label kind: thm \newtheorem{proposition}[theorem]{Proposition} % label kind: prop \newtheorem{lemma}[theorem]{Lemma} % label kind: lem \newtheorem{corollary}[theorem]{Corollary} % label kind: cor \newtheorem{conjecture}[theorem]{Conjecture} % label kind: conj \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} % label kind: def \newtheorem{example}[theorem]{Example} % label kind: ex \newtheorem{notation}[theorem]{Notation} % label kind: not \newtheorem{construction}[theorem]{Construction} % label kind: constr \newtheorem{setup}[theorem]{Set-up} % label kind: setup \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} % label kind: rem %----------------------------------------------------------------------- % Macros. Everything in this block is a suggestion and stays commented out. % The project's own notation — the macro block at the top of the outline — % is added HERE, in the project copy, and nowhere else: a section file is a % bare body and carries no preamble and no macro of its own. %----------------------------------------------------------------------- % house macros, for example: %\newcommand{\card}[1]{\lvert #1\rvert} %\newcommand{\set}[1]{\left\{#1\right\}} %\newcommand{\suchthat}{\,\middle|\,} % project macros, for example: %\newcommand{\F}{\mathbb F} %\newcommand{\rk}{\operatorname{rk}} % Mbar15 project notation copied from the outline. \providecommand{\Mbar}{\overline{\mathcal M}} \providecommand{\Mopen}{\mathcal M} \providecommand{\Lmot}{\mathsf L} \providecommand{\Soneone}{{\mathsf S_{12}}} \providecommand{\Sonesix}{{\mathsf S_{16}}} \providecommand{\Pmot}{\mathsf P} \providecommand{\En}{\mathcal E} \providecommand{\Fn}{F} \providecommand{\Bg}{B_g} \providecommand{\Khook}{K^{15}} \providecommand{\Sn}{\mathfrak S} \providecommand{\ch}{\operatorname{ch}} \providecommand{\Aut}{\operatorname{Aut}} \providecommand{\rank}{\operatorname{rank}} \providecommand{\tr}{\operatorname{tr}} \providecommand{\Gal}{\operatorname{Gal}} \providecommand{\OpenSerre}{\mathfrak a} % open equivariant Serre series \providecommand{\CompactSerre}{\mathfrak b} % compact equivariant Serre series %----------------------------------------------------------------------- % Top matter. The date is part of the author block; \date{} stays empty. %----------------------------------------------------------------------- \title{The fifteenth cohomology of moduli spaces of stable curves} \author{Zeming Sun} \address{Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China} \email{zeming@kurims.kyoto-u.ac.jp} \begin{document} \maketitle \begin{abstract} We determine the fifteenth cohomology of the moduli stack of stable curves $\Mbar_{g,n}$ in every genus $g$ and number of markings $n$, with its symmetric-group action, by providing explicit multiplicity formulas for its simple constituents. We also prove the generalized Hodge conjecture in this case. % The proof identifies independent families of decorated boundary classes, % establishes restriction and lifting for induction in the genus, and uses % a finite formula in terms of a rational Gram matrix in genus three. % The boundary-generation results also determine the geometric coniveau % filtration and prove the generalized Hodge conjecture in degree fifteen. The main result of this paper was obtained using the Pharos system. The author has reviewed the arguments and developed an understanding of the proof. On the basis of this review, the author believes the paper to be correct. \end{abstract} \tableofcontents %----------------------------------------------------------------------- % The sections, in order: the introduction first — the style file gives its % shape, and it closes with the acknowledgements — then the technical % sections. The build replaces the placeholder line below by one \input per % section, in sorted name order. %----------------------------------------------------------------------- \input{sections/01-introduction.tex} \input{sections/02-notation.tex} \input{sections/03-motivic-types.tex} \input{sections/04-low-genus.tex} \input{sections/05-genus-three-generation.tex} \input{sections/06-genus-three-formula.tex} \input{sections/07-ordinary-image.tex} \input{sections/08-geometric-bases.tex} \input{sections/09-marking-supports.tex} \input{sections/10-genus-two-detectors.tex} \input{sections/11-top-coefficients.tex} \input{sections/12-nine-row-coefficients.tex} \input{sections/13-eight-row-coefficients.tex} \input{sections/15-ordinary-lifting.tex} \input{sections/16-main-proof.tex} \input{sections/17-geometric-coniveau.tex} \appendix \clearpage \section{Use of generative AI}\label{app:ai} \begin{center} by Guoxiong Gao\footnote{School of Mathematical Sciences, Peking University, Beijing 100871, China. Email: \texttt{samggx@stu.pku.edu.cn}}, Shurui Liu\footnote{Department of Mathematics, Stanford University, Palo Alto, California, 94305, United States. Email: \texttt{srliu@stanford.edu}}, and Zeming Sun\footnote{Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China. Email: \texttt{zeming@kurims.kyoto-u.ac.jp}} \end{center} The results of this paper were obtained using Pharos. Pharos is an automated mathematical reasoning system that inherits selected design principles from Rethlas~\cite{Ju26} and Danus~\cite{Liu26}, especially the fact graph and multi-agent architecture, while substantially redesigning and extending the remaining system components. Qizheng Yin supplied the problem: for which stable pairs $(g,n)$ is $H^{15}(\overline{\mathcal{M}}_{g,n})$ nonzero, and what is its explicit $\mathfrak{S}_n$-equivariant decomposition as a Hodge structure and an $\ell$-adic Galois representation? Working fully autonomously and without human intervention, Pharos answered the question as posed and produced the main body of the paper. Shurui Liu and Zeming Sun participated in checking the correctness of the paper. No step of the proof relies on a numerical computation or on the system's internal acceptance decision. In the run reported here, agents of every role ran on GPT-6 Astra, with the main agent at effort level ``ultra'', three workers at ``xhigh'' and six at ``max'', and the verifier at ``xhigh''. It took 76 hours and 24 minutes from the creation of the project to the acceptance of the target, and a further 12 hours and 7 minutes to write and check its own manuscript. Its token usage is recorded in Table~\ref{tab:tokens}; in all, the run consumed approximately 5.4 times the weekly usage limit of a \$200 ChatGPT Pro subscription. The final fact graph contains 667 facts verified by the system's verifier. The global memory contains 710 conclusions, 276 identified obstacles, 137 directions, 500 proof attempts, 281 plans, 66 counterexamples, 44 recorded dead ends, and 1342 verification records, of which 379 are rejections. Table~\ref{tab:closure} records how the 667 facts divide: 214 lie in the supporting closure of the final target facts, computed from the dependencies recorded with each fact, and the remaining 453 lie outside it. \begin{table}[ht] \centering\small \begin{tabular}{lrrrr} \hline & Sessions & Input (M) & of which cached (M) & Output (M)\\ \hline Main agent & 3 & 579.3 & 563.4 & 1.8\\ Workers & 1610 & 4842.1 & 4602.0 & 21.4\\ Verifier & 1593 & 1323.9 & 1200.8 & 11.7\\ Helper agents & 333 & 658.0 & 601.1 & 4.8\\ Manuscript and reports & 36 & 115.5 & 106.2 & 0.8\\ Total & 3575 & 7518.8 & 7073.5 & 40.5\\ \hline \end{tabular} \vspace{6pt} \caption{Token usage of the run, in millions of tokens.} \label{tab:tokens} \end{table} \begin{table}[ht] \centering\small \begin{tabular}{lrr} \hline & Count & Share of all facts\\ \hline Main-theorem closure & 214 & 32.1\%\\ Outside the closure & 453 & 67.9\%\\ Total & 667 & 100\%\\ \hline \end{tabular} \vspace{6pt} \caption{Distribution of facts in the fact graph.} \label{tab:closure} \end{table} % 开始写作 Table~\ref{tab:main-facts} lists facts underlying the main results. After computing the genus-one character and the ordinary Gysin image, Pharos established the genus-five base and compact injectivity for induction. It then completed the genus-three and Schur-positive genus-two formulas, the uniform high-genus formula, and the decomposition for all stable pairs. \begin{table}[ht] \centering\small \begin{tabular}{@{}llp{0.47\textwidth}r@{}} \hline Fact & Corresponding theorem & Mathematical content & Time\\ \hline % A: 50cb4eb228b1f184; B: d2c2d910152ef53b. % C: 5b16539f40aa9061; D: fb59ff48454b58fb. % E: f813b1449a25aa12; F: 192f70dc4c0c79c9. % G: c10ed9b12da4e3d9; H: 160fd7a07ac3724c. A & Theorem~\ref{thm:genus-one} & Equivariant character in genus one. & T+3:03\\ B & Theorem~\ref{thm:ordinary-image} & Character of the ordinary Gysin image, $g\geq4$. & T+23:45\\ C & Corollary~\ref{thm:genus-five-base} & Ordinary Gysin generation in genus five. & T+74:56\\ D & Theorem~\ref{thm:compact-injectivity} & Nonseparating pullback injectivity, $g\geq6$. & T+75:53\\ E & Theorem~\ref{thm:genus-three} & Finite Gram formula in genus three. & T+75:57\\ F & Theorem~\ref{thm:genus-two} & Schur-positive character in genus two. & T+76:00\\ G & Theorem~\ref{thm:uniform-high-genus} & Uniform character for every $g\geq4$. & T+76:12\\ H & Theorem~\ref{thm:main} & Decomposition and nonvanishing for all stable $(g,n)$. & T+76:24\\ \hline \end{tabular} \vspace{6pt} \caption{Selected facts and their acceptance times, in hours and minutes from project creation.} \label{tab:main-facts} \end{table} Table~\ref{tab:research-writing} records the research and writing timeline. Parallel exploration led from image calculations to the genus-five reduction and the final assembly. During writing, proof gaps identified in reviews led the main agent to expand V1 into V2; a further audit led to V3, separating shared detectors, coefficient proofs, and the inputs to descent. \begin{table}[ht] \centering\small \begin{tabular}{@{}lp{0.66\textwidth}r@{}} \hline Stage & Description & Time\\ \hline \multicolumn{3}{@{}l}{\emph{Research}}\\ Image & The ordinary Gysin image and its character are determined (B). & T+23:45\\ Reduction & Genus-five generation is reduced to seven coefficient vanishings (inputs to C, G). & T+42:56\\ Coefficients & All seven vanishings needed for the genus-five base are established (inputs to C, G). & T+71:05\\ Assembly & The genus formulas give decomposition and nonvanishing for every stable pair (H). & T+76:24\\ \hline \multicolumn{3}{@{}l}{\emph{Writing}}\\ V1 & Seven sections organize the low-genus formulas, genus-three Gram formula, and high-genus induction. & T+83:22\\ V2 & Ten sections separate genus-three generation from its formula, and the ordinary image and base cases from high-genus descent. & T+84:38\\ V3 & Sixteen sections separate shared detectors and genus-five coefficient proofs, then compact restriction, ordinary lifting, and final assembly. & T+85:09\\ Review & The assembled manuscript passes the system's final review. & T+88:32\\ \hline \end{tabular} \vspace{6pt} \caption{Research and writing milestones, timed from project creation. Letters refer to facts in Table~\ref{tab:main-facts}.} \label{tab:research-writing} \end{table} % 结束写作 % \subsection{How the system was run}\label{app:ai-run} %The retained main-session records document the base model %\texttt{gpt-6-astra} with \texttt{ultra} reasoning effort in three %sessions. Nine proof workers used the same base model, three at %\texttt{xhigh} effort and six at \texttt{max} effort. The final verifier %run used \texttt{gpt-6-astra} at \texttt{xhigh} effort. These settings %describe the retained run metadata. %The project started on 4 September 2026 at 15:35:03 UTC. The system's verifier %accepted the final proof on 7 September 2026 at 19:59:54 UTC, %approximately three days, four hours, and twenty-five minutes after %the start. No comparison run is recorded. Before editorial polishing, %the fourteen proof sections and the assembled body underwent separate %paper-verification checks. \section{The problem as posed}\label{app:ai-problem} The following is the full supplied problem, including its original background and bibliographic conventions. \begin{quote} \begin{Verbatim}[breaklines,breakanywhere,fontsize=\footnotesize] # The Odd Cohomology of $\overline{\mathcal{M}}_{g,n}$: Known Results and the Problem for Degree 15 --- ## 1. Setup ### 1.1 The moduli space $\overline{\mathcal{M}}_{g,n}$ For integers $g \geq 0$, $n \geq 0$ with $2g - 2 + n > 0$, the **Deligne--Mumford compactification** $\overline{\mathcal{M}}_{g,n}$ is the moduli stack of **stable curves**: connected, at-worst-nodal algebraic curves of arithmetic genus $g$ equipped with $n$ distinct ordered marked points, satisfying the stability condition that the automorphism group is finite. It is a smooth proper Deligne--Mumford stack over $\operatorname{Spec} \mathbb{Z}$ of dimension $$\dim \overline{\mathcal{M}}_{g,n} = 3g - 3 + n.$$ The **boundary** $\partial \mathcal{M}_{g,n} := \overline{\mathcal{M}}_{g,n} \setminus \mathcal{M}_{g,n}$ is a normal crossing divisor. Its irreducible components are images of **gluing maps**: - **Non-separating:** $\xi \colon \overline{\mathcal{M}}_{g-1, n+2} \to \overline{\mathcal{M}}_{g,n}$, identifying two new marked points to form a node. - **Separating:** $\vartheta \colon \overline{\mathcal{M}}_{g_1, S \cup p} \times \overline{\mathcal{M}}_{g_2, S^c \cup p'} \to \overline{\mathcal{M}}_{g,n}$, for a partition $\{1,\ldots,n\} = S \sqcup S^c$ and $g_1 + g_2 = g$, gluing at the points $p$ and $p'$. ### 1.2 Motivic structures on cohomology The rational singular cohomology $H^k(\overline{\mathcal{M}}_{g,n}) := H^k((\overline{\mathcal{M}}_{g,n})_{\mathbb{C}}, \mathbb{Q})$ is a finite-dimensional $\mathbb{Q}$-vector space. Because $\overline{\mathcal{M}}_{g,n}$ is smooth and proper over $\mathbb{Q}$, each $H^k(\overline{\mathcal{M}}_{g,n})$ carries two compatible structures: 1. A **pure Hodge structure** of weight $k$: a decomposition $H^k \otimes \mathbb{C} = \bigoplus_{p+q=k} H^{p,q}$ with $\overline{H^{p,q}} = H^{q,p}$. 2. A continuous **$\ell$-adic Galois representation**: an action of $\operatorname{Gal}(\overline{\mathbb{Q}}|\mathbb{Q})$ on $H^k_{\text{\'et}}((\overline{\mathcal{M}}_{g,n})_{\overline{\mathbb{Q}}}, \mathbb{Q}_\ell)$, pure of weight $k$. Both are encoded by saying $H^k(\overline{\mathcal{M}}_{g,n})$ is a **pure motive** of weight $k$. The symmetric group $\mathfrak{S}_n$ acts on $\overline{\mathcal{M}}_{g,n}$ by permuting marked points, making $H^k(\overline{\mathcal{M}}_{g,n})$ an $\mathfrak{S}_n$-equivariant motive. ### 1.3 The Tate motive The **Tate motive** $\mathsf{L} := H^2(\mathbb{P}^1)$ is a 1-dimensional pure motive of weight 2 and Hodge type $(1,1)$. Each Frobenius $\operatorname{Frob}_p$ acts on $\mathsf{L}_{\mathbb{Q}_\ell}$ by multiplication by $p$. We say $H^k(\overline{\mathcal{M}}_{g,n})$ is **pure Hodge--Tate** if it is isomorphic to a direct sum of copies of $\mathsf{L}^{k/2}$ (possible only when $k$ is even). ### 1.4 Cusp form motives For each even integer $w \geq 2$, let $S_w(\operatorname{SL}_2(\mathbb{Z}))$ denote the space of cuspidal modular forms of weight $w$ for $\operatorname{SL}_2(\mathbb{Z})$. The ring of modular forms for $\operatorname{SL}_2(\mathbb{Z})$ is $\mathbb{C}[E_4, E_6]$, and the space of cusp forms is the principal ideal $S_w = \Delta \cdot M_{w-12}$, where $\Delta$ is the Ramanujan discriminant of weight 12 and $M_k$ is the space of modular forms of weight $k$. In particular: $$\dim S_w(\operatorname{SL}_2(\mathbb{Z})) = \begin{cases} \lfloor w/12 \rfloor & \text{if } w \equiv 2 \pmod{12}, \\ \lfloor w/12 \rfloor - 1 & \text{if } w \not\equiv 2 \pmod{12} \text{ and } w \geq 12, \\ 0 & \text{if } w < 12. \end{cases}$$ (More precisely, $\dim S_w = 0$ for $w < 12$ and $w = 14$; $\dim S_w = 1$ for $w = 12, 16, 18, 20, 22, 26$.) To each normalized Hecke eigenform $f \in S_w(\operatorname{SL}_2(\mathbb{Z}))$, Deligne [Del71] attaches a 2-dimensional pure motive $\mathsf{S}_w(f)$ of weight $w - 1$ and Hodge type $\{(w-1, 0), (0, w-1)\}$. When $\dim S_w = 1$, we write $\mathsf{S}_w$ for the unique such motive. | Weight $w$ | $\dim S_w$ | Motivic weight $w - 1$ | Hodge type | |---|---|---|---| | 12 | 1 | 11 | $(11,0) + (0,11)$ | | 14 | 0 | --- | --- | | 16 | 1 | 15 | $(15,0) + (0,15)$ | | 18 | 1 | 17 | $(17,0) + (0,17)$ | | 20 | 1 | 19 | $(19,0) + (0,19)$ | The motive $\mathsf{S}_{12}$ has Betti and $\ell$-adic realizations equal to $H^{11}(\overline{\mathcal{M}}_{1,11})$ [CLP23]. ### 1.5 Gysin pushforward If $\iota \colon D \hookrightarrow X$ is a codimension-$c$ closed embedding of smooth Deligne--Mumford stacks, the **Gysin pushforward** is: $$\iota_* \colon H^k(D) \to H^{k+2c}(X),$$ a morphism of pure Hodge structures that shifts weight by $2c$ (i.e., tensors with $\mathsf{L}^c$). ### 1.6 Specht modules and hook representations For a partition $\lambda$ of $n$, the **Specht module** $V_\lambda$ is the corresponding irreducible $\mathbb{Q}$-representation of $\mathfrak{S}_n$. The **hook representation** $$K_n^m := V_{n-m+1, 1^{m-1}}$$ has dimension $\binom{n-1}{m-1}$ and is generated by elements $k_P$ indexed by ordered subsets $P \subset \{1,\ldots,n\}$ of size $m$, with relations: $$k_{\sigma(P)} = \operatorname{sign}(\sigma) k_P, \qquad \sum_{j=0}^{m} (-1)^j k_{\{a_0, \ldots, \hat{a}_j, \ldots, a_m\}} = 0.$$ For subgroups $\mathfrak{S}_m \times \mathfrak{S}_{n-m} \subset \mathfrak{S}_n$ and representations $U$ of $\mathfrak{S}_m$, $W$ of $\mathfrak{S}_{n-m}$, the **induced representation** is: $$\operatorname{Ind}^{\mathfrak{S}_n}_{\mathfrak{S}_m \times \mathfrak{S}_{n-m}}(U \boxtimes W) := \mathbb{Q}[\mathfrak{S}_n] \otimes_{\mathbb{Q}[\mathfrak{S}_m \times \mathfrak{S}_{n-m}]} (U \otimes W).$$ --- ## 2. Known Results: Odd Cohomology of $\overline{\mathcal{M}}_{g,n}$ ### 2.1 Vanishing for $k = 1, 3, 5$ **Theorem (Arbarello--Cornalba [AC98]).** For all $g, n$ with $2g - 2 + n > 0$: $$H^k(\overline{\mathcal{M}}_{g,n}) = 0 \quad \text{for } k = 1, 3, 5.$$ Arbarello and Cornalba introduced an inductive method based on excision of the boundary and pullback to its normalization. They applied it to establish the vanishing for $k \in \{1, 3, 5\}$. ### 2.2 Vanishing for $k = 7, 9$ **Theorem (Bergstr\"om--Faber--Payne [BFP22]).** For all $g, n$ with $2g - 2 + n > 0$: $$H^k(\overline{\mathcal{M}}_{g,n}) = 0 \quad \text{for } k = 7, 9.$$ The same inductive method of Arbarello--Cornalba was used, after establishing the additional base cases needed for the induction via point counting over finite fields. ### 2.3 Even degree cohomology for $k \leq 12$ **Theorem 1.2 (Canning--Larson--Payne [CLP23]).** For any even $k \leq 12$, $H^k(\overline{\mathcal{M}}_{g,n})$ is pure Hodge--Tate for all $g, n$. ### 2.4 Degree $k = 11$ **Theorem 1.1 (Canning--Larson--Payne [CLP23]).** The cohomology group $H^{11}(\overline{\mathcal{M}}_{g,n})$ is nonzero if and only if $g = 1$ and $n \geq 11$. For $g = 1$ and $n \geq 11$, there is an $\mathfrak{S}_n$-equivariant isomorphism of Hodge structures (or $\ell$-adic Galois representations): $$H^{11}(\overline{\mathcal{M}}_{1,n}) \cong K_n^{11} \otimes_{\mathbb{Q}} \mathsf{S}_{12},$$ where $K_n^{11} = V_{n-10, 1^{10}}$ is the hook Specht module of dimension $\binom{n-1}{10}$, and $\mathsf{S}_{12}$ is the motive associated to the Ramanujan $\Delta$-function. The proof uses: (i) Getzler's generating functions [Get98] to compute $H^{11}(\overline{\mathcal{M}}_{1,n})$ and identify the $\mathfrak{S}_n$-action; (ii) the Arbarello--Cornalba induction to reduce to the genus 1 case; (iii) the weight spectral sequence to prove vanishing for $g = 2$. ### 2.5 Degree $k = 13$ **Theorem 1.7 (Canning--Larson--Payne--Willwacher [CLPW]).** The cohomology group $H^{13}(\overline{\mathcal{M}}_{g,n})$ is spanned by Gysin pushforward maps $\iota_{A*}$ from boundary divisors of the form $\overline{\mathcal{M}}_{1, A \cup p} \times \overline{\mathcal{M}}_{g-1, A^c \cup p'}$, for $A \subset \{1, \ldots, n\}$. If $g \geq 2$, then $\bigoplus_A \iota_{A*}$ is also injective, and there is an $\mathfrak{S}_n$-equivariant isomorphism of Hodge structures: $$H^{13}(\overline{\mathcal{M}}_{g,n}) \cong \left(\bigoplus_{m=10}^{n} \operatorname{Ind}^{\mathfrak{S}_n}_{\mathfrak{S}_m \times \mathfrak{S}_{n-m}} \left((\operatorname{Res}^{\mathfrak{S}_{m+1}}_{\mathfrak{S}_m} K^{11}_{m+1}) \boxtimes \mathbf{1}\right)\right) \otimes_{\mathbb{Q}} \mathsf{L}\mathsf{S}_{12}.$$ In particular, $H^{13}(\overline{\mathcal{M}}_{g,n})$ is a direct sum of copies of $\mathsf{L}\mathsf{S}_{12} := \mathsf{L} \otimes \mathsf{S}_{12}$. For $g = 0$: $H^{13}(\overline{\mathcal{M}}_{0,n}) = 0$. For $g = 1$: the kernel of $\bigoplus_A \iota_{A*}$ and the resulting presentation of $H^{13}(\overline{\mathcal{M}}_{1,n})$ are described in [CLPW, Section 4.4 and Corollary 4.9]. For $n < 10$ and any $g$: $H^{13}(\overline{\mathcal{M}}_{g,n}) = 0$. For $g \geq 2$: $\dim H^{13}(\overline{\mathcal{M}}_{g,n}) = 2\sum_{m=10}^{n} \binom{n}{m}\binom{m}{10}$. ### 2.6 Nonvanishing of higher odd cohomology **Theorem 1.5 (Canning--Larson--Payne [CLP23]).** Let $g_1, \ldots, g_k$ be distinct positive integers, and $g = 1 + g_1 + \cdots + g_k$. Then: $$H^{11+2k}(\overline{\mathcal{M}}_{g,n}) \neq 0 \quad \text{for } n \geq 11 - k.$$ In particular: - $H^{13}(\overline{\mathcal{M}}_{g,n}) \neq 0$ for $g \geq 2$ and $n \geq 10$. - $H^{15}(\overline{\mathcal{M}}_{g,n}) \neq 0$ for $g = 1 + g_1 + g_2$ (with $g_1 \neq g_2$, $g_1, g_2 \geq 1$) and $n \geq 9$. - $H^{17}(\overline{\mathcal{M}}_{g,n}) \neq 0$ for $g = 1 + g_1 + g_2 + g_3$ (with $g_i$ distinct, $g_i \geq 1$) and $n \geq 8$. The proof uses the push-pull formula (self-intersection formula) along codimension-$k$ gluing maps $\overline{\mathcal{M}}_{1,11} \times \overline{\mathcal{M}}_{g_1,1} \times \cdots \times \overline{\mathcal{M}}_{g_k,1} \to \overline{\mathcal{M}}_{g, 11-k}$ to show the Gysin pushforward of $\alpha \otimes 1 \otimes \cdots \otimes 1$ is nonzero, where $\alpha \in H^{11,0}(\overline{\mathcal{M}}_{1,11})$. ### 2.7 Tate type for open moduli spaces **Theorem 1.5 (Canning--Larson--Payne--Willwacher [CLPW]).** The rational cohomology of $\mathcal{M}_{g,n}$ is of Tate type if and only if $g = 0$ or $3g + 2n < 25$. ### 2.8 Weight 13 Euler characteristics **Corollary 1.8 ([CLPW]).** The weight 13 Euler characteristics of $\mathcal{M}_{12}$ and $\mathcal{M}_{8,1}$ are: $$\chi_{13}(\mathcal{M}_{12}) = -6, \qquad \chi_{13}(\mathcal{M}_{8,1}) = -2.$$ These nonzero values are used to show non-polynomiality of the point counts $N_{g,n}(q)$ in cases where the weight 11 Euler characteristic vanishes. --- ## 3. Summary | Degree $k$ (odd) | $H^k(\overline{\mathcal{M}}_{g,n})$ | Status | Reference | |---|---|---|---| | 1 | $= 0$ for all $g, n$ | Proved | [AC98] | | 3 | $= 0$ for all $g, n$ | Proved | [AC98] | | 5 | $= 0$ for all $g, n$ | Proved | [AC98] | | 7 | $= 0$ for all $g, n$ | Proved | [BFP22] | | 9 | $= 0$ for all $g, n$ | Proved | [BFP22] | | 11 | $\neq 0$ iff $g = 1, n \geq 11$; all copies of $\mathsf{S}_{12}$ | Fully computed | [CLP23] | | 13 | All copies of $\mathsf{L}\mathsf{S}_{12}$; explicit $\mathfrak{S}_n$-structure | Fully computed | [CLPW] | | **15** | **Nonzero for some $(g,n)$ with $g \geq 3$** | **Partial (nonvanishing only)** | [CLP23, Thm 1.5] | | 17 | Nonzero for some $(g,n)$ with $g \geq 4$ | Open | [CLP23, Thm 1.5] | --- ## 4. The Problem **Problem.** Determine the cohomology group $H^{15}(\overline{\mathcal{M}}_{g,n})$ for all $g \geq 0$ and $n \geq 0$ (with $2g - 2 + n > 0$), as an $\mathfrak{S}_n$-equivariant pure motive of weight 15. More precisely: **(A)** For which $(g, n)$ is $H^{15}(\overline{\mathcal{M}}_{g,n})$ nonzero? **(B)** Give an explicit decomposition of $H^{15}(\overline{\mathcal{M}}_{g,n})$ into simple motivic pieces, together with their $\mathfrak{S}_n$-equivariant multiplicity spaces. Note that $H^{15}$ is the first odd-degree cohomology group of $\overline{\mathcal{M}}_{g,n}$ for which neither a full computation nor a complete vanishing result is known, and for which multiple independent types of motives (of different Hodge types) are expected to contribute simultaneously. The computation of $H^{15}$ depends on the computation of $H^{13}(\overline{\mathcal{M}}_{g,n})$, because Gysin pushforwards from codimension-1 boundary divisors map $H^{13}$ into $H^{15}$. --- ## References - **[AC98]** E. Arbarello and M. Cornalba, *Calculating cohomology groups of moduli spaces of curves via algebraic geometry*, Inst. Hautes \'Etudes Sci. Publ. Math. (1998) **88** (1999), 97--127. MR 1733327. - **[BFP22]** J. Bergstr\"om, C. Faber, and S. Payne, *Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves*, Preprint, 2022, arXiv:2206.07759. - **[CLP23]** S. Canning, H. Larson, and S. Payne, *The eleventh cohomology group of $\overline{\mathcal{M}}_{g,n}$*, Forum of Mathematics, Sigma **11** (2023), e62, 1--18. - **[CLPW]** S. Canning, H. Larson, S. Payne, and T. Willwacher, *Moduli spaces of curves with polynomial point counts*, arXiv:2410.19913v2, 2026. - **[Del71]** P. Deligne, *Formes modulaires et repr\'esentations $\ell$-adiques*, S\'eminaire Bourbaki 1968/69, exp. 355, Lecture Notes in Math. **179**, Springer, 1971, 139--172. - **[Get98]** E. Getzler, *The semi-classical approximation for modular operads*, Comm. Math. Phys. **194**(2) (1998), 481--492. MR 1627677. \end{Verbatim} \end{quote} % \subsection{Guidance given during the run}\label{app:ai-guidance} % The recorded instructions were as follows. The two Chinese instructions % are given in English translation. % \begin{enumerate} % \item \textbf{4 September 2026, launch.} % \begin{quote} % Refer to the Mbar13 Acta paper. First conjecture a general structural % expression, and then prove it. The structure of its Theorem 1.7 is the % expected form of the theorem. Calculations at individual pairs $(g,n)$ % can be useful, but the final objective is to give and prove this general % structural expression; pointwise calculations alone are insufficient. % \end{quote} % This instruction directed the work toward a uniform equivariant % decomposition. The paper was identified during the run as % Canning--Larson--Payne--Willwacher, % \emph{Moduli spaces of curves with polynomial point counts}, % arXiv:2410.19913v2~\cite{canning2026}. % \item \textbf{8 September 2026, writing.} % \begin{quote} % Refer to the Mbar13 paper for the style of writing. Organize the paper % around giving and proving explicit structural expressions, and answer % the questions asked by the expert in the original problem. Please begin % writing carefully. % \end{quote} % This instruction led to the theorem-centered organization, the displayed % multiplicity formulas, and the explicit nonvanishing range in the introduction. % The original questions are answered by % Theorem~\ref{thm:introduction-degree-fifteen} and the formulas following it. % \item \textbf{8 September 2026, continuation.} % \begin{quote} % continue % \end{quote} % The authorized writing work continued through the section proofs, % whole-paper verification, editorial revision, and final compilation. % \item \textbf{8 September 2026, separate backup.} % The operator requested one additional version after the normal writing % pipeline was complete, with the author line \texttt{Anonymous} and all % acknowledgements, attribution, and production provenance removed. The % mathematical content and mathematical references were to remain identical, % and the completed normal version was to be preserved. This instruction % governed a separate backup artifact. % \end{enumerate} % \subsection{Computations}\label{app:ai-computations} % % Optional: only when the paper has a computation-backed step. One item per % % computation: what was computed, with which method (the algorithm, the % % software, the parameters) and the outcome — enough for a reader to redo % % it. The scripts live in computation/ and are never cited by path. % \begin{enumerate}[leftmargin=*] % \item\textbf{[What was computed.]} [The method.] [The outcome.] % \end{enumerate} % % \subsection{Discovery trajectory}\label{app:ai-trajectory} % % Reconstructed from the checkpoints and the shared memory: the literature % % phase and the initial reductions; the routes explored and what obstructed % % each, dead routes included; the key idea — where and when it appeared, and % % whether it came from a worker, a helper, the literature, or the authors' % % guidance; how the final argument was assembled. % \begin{enumerate}[leftmargin=*] % \item\textbf{Literature and initial reductions.} [...] % \item\textbf{Approaches and their obstructions.} [...] % \item\textbf{The key idea.} [...] % \item\textbf{Assembling the proof.} [...] % \end{enumerate} % % \subsection{Verification}\label{app:ai-verification} % % Every result entered the fact graph only after a cold-start verifier % % accepted its proof; the size of the verified graph and the depth of the % % final dependency chain; the rejections that mattered and how they were % % repaired; the human verification that followed. % [...] % % \subsection{What the authors did}\label{app:ai-authors} % % Posed the problem, gave the guidance above, read, checked, and rewrote the % % paper; and which results turned out to be already known — stated honestly, % % with the reference. % [...] %----------------------------------------------------------------------- % Bibliography. Alphabetic labels ([EPW16], [BHM$^+$22]). The entries % collated by the outline's last section are merged in here at the first % `danus paper assign` and kept in step with the outline afterwards, sorted % by label; polish checks the list and adds what the introduction cites. % The two system references are fixed: they keep their keys and their text. %----------------------------------------------------------------------- \clearpage \begin{thebibliography}{99} % Checked version: arXiv:1102.3425v3, 10 October 2011, % Theorem 3.2 and proof (coedge squares); Theorem 3.4 (second Voronoi). % Primary text: https://arxiv.org/html/1102.3425v3 % Local source: workers/max5/downloads/1102.3425v3-source/ab.tex \bibitem[AB12]{alexeev2011} V.~Alexeev and A.~Brunyate, \emph{Extending the Torelli map to toroidal compactifications of Siegel space}, Invent. Math. \textbf{188} (2012), no.~1, 175--196. \url{https://doi.org/10.1007/s00222-011-0347-2}. Preprint version cited: \url{https://arxiv.org/abs/1102.3425v3}. % exists: yes -- arXiv:math/9803001 and the published paper. % used by: the divisor presentation and compact boundary induction. % matches: Theorems 2.2 and 2.10 are audited in the local source. \bibitem[AC98]{arbarello1998} E.~Arbarello and M.~Cornalba, \emph{Calculating cohomology groups of moduli spaces of curves via algebraic geometry}, Publ. Math. IHES \textbf{88} (1998), 97--127. % Original 1982 edition, not the expanded 2018 BBDG reprint. % Primary full scan: IAS publications archive; printed pp. 101--103 checked. % In that scan printed p. 102 is PDF page 103 (one-based). \bibitem[BBD82]{bbd1982} A.~A.~Be\u{\i}linson, J.~Bernstein, and P.~Deligne, \emph{Faisceaux pervers}, in \emph{Analyse et topologie sur les espaces singuliers, I}, Ast\'erisque \textbf{100} (1982), 5--171. \url{https://publications.ias.edu/sites/default/files/Faisceaux%20pervers.pdf}. \bibitem[BF23]{bergstromfaber2023} J.~Bergstr\"om and C.~Faber, \emph{Cohomology of moduli spaces via a result of Chenevier and Lannes}, \'Epijournal G\'eom. Alg\'ebrique \textbf{7} (2023), Art.~20, 14 pp. \url{https://doi.org/10.46298/epiga.2023.10307}. % exists: yes -- arXiv:2206.07759v2 (2023), published 2024. \bibitem[BFP22]{bergstrom2024} J.~Bergstr\"om, C.~Faber, and S.~Payne, \emph{Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves}, Ann. of Math. (2) \textbf{199} (2024), 1323--1365; arXiv:2206.07759v2. \bibitem[BIKL26]{BIKL26} A.~Bradford, A.~Iribar L\'opez, A.~Kretschmer, and H.~Larson, \emph{The fifteenth cohomology group of moduli spaces of stable curves}, arXiv:2610.14377v1 (2026). \url{https://arxiv.org/abs/2610.14377v1}. % exists: yes -- arXiv:2012.09887v3; DOI 10.1017/fms.2022.21. % used by: stable forgetting and the correspondence in marking support. % matches: Proposition 3.10 and Lemma C.8 concern Chow, not odd cohomology. \bibitem[BS22]{bae2022} Y.~Bae and J.~Schmitt, \emph{Chow rings of stacks of prestable curves I}, arXiv:2012.09887v3, \url{https://doi.org/10.1017/fms.2022.21}. % exists: yes -- arXiv:math/0504418v2 and the published paper. % used by: the corrected genus-one Serre identity. % matches: Section 4, p. 306 corrects the semiclassical numerator. \bibitem[CF06]{consani2006} C.~Consani and C.~Faber, \emph{On the cusp form motives in genus 1 and level 1}, Adv. Stud. Pure Math. \textbf{45} (2006), 297--314. % exists: yes -- arXiv:2208.02357v2; primary source downloaded and checked. % used by: finite-gerbe descent, hyperelliptic and Hurwitz base inputs. % matches: Lemma 3.6, Theorem 6.2, Section 8.2, % Lemmas 9.1--9.2, 10.6, 10.9, 10.11. \bibitem[CL22]{canninglarson2022} S.~Canning and H.~Larson, \emph{On the Chow and cohomology rings of moduli spaces of stable curves}, arXiv:2208.02357v2. % exists: yes — Forum Math. Sigma 11 (2023), e62. \bibitem[CLP23]{canning2023} S.~Canning, H.~Larson, and S.~Payne, \emph{The eleventh cohomology group of $\Mbar_{g,n}$}, Forum Math. Sigma 11 (2023), e62. % exists: yes — arXiv:2307.08830v3. \bibitem[CLP24]{canning2024a} S.~Canning, H.~Larson, and S.~Payne, \emph{Extensions of tautological rings and motivic structures in the cohomology of $\Mbar_{g,n}$}, Forum Math. Pi \textbf{12} (2024), e23; arXiv:2307.08830v3. % exists: yes — arXiv:2410.19913v2; matches the degree-thirteen Gysin model. \bibitem[CLPW]{canning2026} S.~Canning, H.~Larson, S.~Payne, and T.~Willwacher, \emph{Moduli spaces of curves with polynomial point counts}, arXiv:2410.19913v2 (2026), to appear in Acta Mathematica. % exists: yes -- arXiv:2411.12652v1, 19 November 2024. % used by: the genus-one primitive-source input. % matches: Theorem 1.3 gives the actual primitive hook and a complement % whose semisimplification consists of copies of L^2 S12. \bibitem[CLPW24]{canning2024b} S.~Canning, H.~Larson, S.~Payne, and T.~Willwacher, \emph{The motivic structures $\Lmot\Soneone$ and $\Sonesix$ in the cohomology of moduli spaces of curves}, arXiv:2411.12652v1 (2024). \bibitem[CLPW25]{canning2025} S.~Canning, H.~Larson, S.~Payne, and T.~Willwacher, \emph{FA-modules of holomorphic forms on $\Mbar_{g,n}$}, arXiv:2509.08774v1 (2025). \url{https://arxiv.org/abs/2509.08774v1}. % exists: yes -- DOI 10.1515/crll.1993.443.11; primary source audited. % used by: thm:genus-three-ample-cited. % matches: Theorem 2.2 and Sections 2--4 give the required ampleness. \bibitem[Cor93]{cornalba1993} M.~Cornalba, \emph{On the projectivity of the moduli spaces of curves}, J. Reine Angew. Math. \textbf{443} (1993), 11--20. % exists: yes -- author's published scan IstLomb_95.pdf. % used by: thm:genus-three-ample-cited. % matches: Theorem 1 at r=d=0 identifies the logarithmic Deligne pairing. \bibitem[Cor95]{cornalba1995} M.~Cornalba, \emph{A simple proof of the projectivity of Kontsevich's space of maps}, (1995), \url{https://mate.unipv.it/~cornalba/papers/IstLomb_95.pdf}. % Published author offprint, Theorem 2.0.1, p. 311. % Necessary category-specific alternate for rational complex perverse sheaves. \bibitem[dC12]{decataldo2012} M.~A.~A.~de~Cataldo, \emph{The perverse filtration and the Lefschetz hyperplane theorem, II}, J. Algebraic Geom. \textbf{21} (2012), no.~2, 305--345. \url{https://doi.org/10.1090/S1056-3911-2011-00566-3}. Author's published offprint: \url{https://www.math.stonybrook.edu/~mde/MyPublishedPapers/PFII-JAG.pdf}. % exists: yes -- literature/papers/Deligne_HodgeII_1971.pdf. % used by: the normal-crossings weight spectral sequence. % matches: Sections 3.1 and 3.2 give the spectral sequence and degeneration. \bibitem[Del71]{deligne1971} P.~Deligne, \emph{Th\'eorie de Hodge. II}, Publ. Math. IHES \textbf{40} (1971), 5--57. % exists: yes -- arXiv:math/0311026v2. % used by: thm:genus-three-lefschetz-cited. % matches: Theorem 2.2(1) is ordinary orbifold Hard Lefschetz. \bibitem[Fer06]{fernandez2006} J.~Fern\'andez, \emph{Hodge structures for orbifold cohomology}, arXiv:math/0311026v2 (2006). % exists: yes -- arXiv:math/9908052v3, Theorem 1 and Section 0.5. % used by: thm:cited-lambda-g-integrals. % matches: nonnegative descendant exponents and the one-point generating series. \bibitem[FP03]{faberpandharipande2003} C.~Faber and R.~Pandharipande, \emph{Hodge integrals, partition matrices, and the $\lambda_g$ conjecture}, Ann. of Math. (2) \textbf{157} (2003), no.~1, 97--124; arXiv:math/9908052v3. % exists: yes -- arXiv:2606.13517v1; source Theorems A and B checked. % used by: thm:external-gaifullin and the owned low-degree coefficient lemma. % matches: q<=g-2, algebraicity without passing to finite index, first part<=q. \bibitem[Gai26]{gaifullin2026} A.~Gaifullin, \emph{Finite generation, algebraicity, and representation stability for homology of Torelli groups}, arXiv:2606.13517v1 (2026). % exists: yes — Comm. Math. Phys. 194 (1998), 481--492. \bibitem[Getzler98]{getzler1998} E.~Getzler, \emph{The semi-classical approximation for modular operads}, Comm. Math. Phys. 194 (1998), 481--492. \bibitem[GP03]{graberpandharipande2003} T.~Graber and R.~Pandharipande, \emph{Constructions of nontautological classes on moduli spaces of curves}, Michigan Math. J. \textbf{51} (2003), no.~1, 93--109; arXiv:math/0104057. \url{https://doi.org/10.1307/mmj/1049832895}. % Published Corollary 7, p. 479, checked; arXiv:math/0203057v2, % 11 March 2002. Decomposable geometry: published Lemma 8, % Proposition 10, Lemma 11, pp. 479--482. Do not substitute preprint % Proposition 8 for those published locators: the numbering changed. \bibitem[Hai02]{hain2002} R.~Hain, \emph{The rational cohomology ring of the moduli space of abelian 3-folds}, Math. Res. Lett. \textbf{9} (2002), no.~4, 473--491. \url{https://archive.intlpress.com/site/pub/files/_fulltext/journals/mrl/2002/0009/0004/MRL-2002-0009-0004-a007.pdf}. Preprint version: \url{https://arxiv.org/abs/math/0203057v2}. \bibitem[Ju$^+$26]{Ju26} H.~Ju, G.~Gao, J.~Jiang, B.~Wu, Z.~Sun, S.~Liu, L.~Chen, Yutong Wang, Yuefeng Wang, Z.~Wang, W.~He, P.~Wu, L.~Xiao, R.~Liu, B.~Dai, and B.~Dong, \emph{Automated conjecture resolution with formal verification}, arXiv:2604.03789v2 (2026), \url{https://arxiv.org/abs/2604.03789v2}. % exists: yes -- literature/papers/Keel1992.pdf. % used by: genus-zero vanishing and genus-zero divisor relations. % matches: Section 2, Theorem 1 gives the cycle-class isomorphism. \bibitem[Kee92]{keel1992} S.~Keel, \emph{Intersection theory of moduli space of stable $n$-pointed curves of genus zero}, Trans. Amer. Math. Soc. \textbf{330} (1992), 545--574. % Checked version: arXiv:1106.1588v2, 3 April 2012, Theorems 5.1--5.2. % Local text: workers/max4/downloads/Knudsen_1106.1588v2.txt % Correct publisher DOI ends in .03.021, not the arXiv metadata's .03.21. \bibitem[Knu12]{knudsen2012} F.~F.~Knudsen, \emph{A closer look at the stacks of stable pointed curves}, J. Pure Appl. Algebra \textbf{216} (2012), no.~11, 2377--2385. \url{https://doi.org/10.1016/j.jpaa.2012.03.021}. Preprint version cited: \url{https://arxiv.org/abs/1106.1588v2}. % Checked version: arXiv:math/0611412v2, 1 January 2009, % Theorem 1.3, preprint p. 3; use dominant transforms and the order condition. % Local text: workers/max4/downloads/Li_0611412v2.txt \bibitem[Li09]{li2009} L.~Li, \emph{Wonderful compactification of an arrangement of subvarieties}, Michigan Math. J. \textbf{58} (2009), no.~2, 535--563. \url{https://doi.org/10.1307/mmj/1250169076}. Preprint version cited: \url{https://arxiv.org/abs/math/0611412v2}. \bibitem[Liu$^+$26]{Liu26} J.~Liu, G.~Gao, Z.~Sun, B.~Wu, S.~Liu, J.~Jiang, H.~Ju, L.~Chen, R.~Cheng, X.~Zhang, and B.~Dong, \emph{Danus: Orchestrating mathematical reasoning agents with fact-graph memory}, arXiv:2607.06447v2 (2026), \url{https://arxiv.org/abs/2607.06447v2}. % exists: yes — arXiv:0708.0565v2. \bibitem[LiuXu09]{liu2009} K.~Liu and H.~Xu, \emph{Recursion formulae of higher Weil--Petersson volumes}, arXiv:0708.0565v2 (2009). % Numbered statements checked in the July 1970 notes: Proposition 11.1, % p. 27, and Corollary 10.3, p. 26. Publication metadata is supplied too; % the locators used here are explicitly for the notes, not an unchecked % assertion about the pagination of the 2012 edition. \bibitem[Mat70]{mather1970} J.~Mather, \emph{Notes on topological stability}, Harvard University, July 1970, Proposition~11.1 and Corollary~10.3. \url{https://webhomes.maths.ed.ac.uk/~v1ranick/surgery/mather.pdf}. Published in Bull. Amer. Math. Soc. (N.S.) \textbf{49} (2012), no.~4, 475--506, \url{https://doi.org/10.1090/S0273-0979-2012-01383-6}. % exists: yes -- arXiv:2402.00354v4, primary source checked. % used by: thm:cited-symplectic-borel-range in the genus-three correction. % matches: Theorem 4.6, degree