{"id":"cc7a3dee-17a5-42aa-a9d6-bca21d03cc2a","arxiv_id":"2507.08995","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The weight-13 cohomology of M_{g,n} for 3g+2n=28 is computed via the Getzler-Kapranov graph complex, with explicit Sn-representations in two degrees.","lead":"This paper computes the weight-13 piece of the compactly supported cohomology of moduli spaces of curves for all pairs (g,n) with 3g+2n=28. The computation extends a program that uses graph complexes to extract motivic information, and it gives explicit symmetric group representations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The excess-28 reduction to six graphs rests on manually checked relation groups and differential elimination whose matrices are not printed; an independent rerun of the linear algebra is needed before Theorem 1.3 can be taken as verified.","rationale":"The reader's weakest_assumption identifies the same load-bearing point: the excess-28 computation depends on the exhaustive list of virtual blown-up representations and on the manually checked relation resolutions and Gaussian elimination in Lemma 3.2 and Theorem 3.3. I agree, with slightly more emphasis on the differential elimination than on the enumeration itself. The concern is load-bearing because Theorem 1.3 is exactly the cohomology of the six-graph complex obtained after elimination; if any elimination step is wrong, the S_n-representations change and the theorem fails. The paper deserves credit for supplying the code and for explicitly warning about computational reliability, so this is a verification concern rather than a demonstrated error or an internal inconsistency. No independent mathematical contradiction emerged from the manuscript text. Since the reader already made the verdict CONDITIONAL, my read does not move that verdict; it reinforces it.","tokens_in":18152,"tokens_out":4225,"duration_ms":53166,"concrete_test":"Run an independent script that reads the GitHub notebook's blown-up graph list, recomputes the incidence matrices for Tables 2-4 with signs, and performs Gaussian elimination over Q for each (g,n) = (2,11), (4,8), (6,5), (8,2). Verify (i) after removing the ten graphs listed in Lemma 3.2, each of the six relation groups has the claimed rank; (ii) the images of IDs 80 and 81 span the subspace generated by IDs 116 and 117; and (iii) the only surviving graphs after elimination are IDs 70, 97, 98, 106, 125, 126 with exactly the differential shown in Table 5, reproducing the Specht decompositions of Corollary 3.4. If any rank or survivor set differs, Theorem 1.3 does not follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive step is Theorem 3.3, which reduces the 106 virtual blown-up representations to six survivors by Gaussian elimination on the differential. The argument does not display the actual matrix entries: Tables 2, 3, and 4 are only incidence grids, and the text asserts triangularity with 'one checks'. The two places where triangularity fails, rows/columns 116 and 117, are resolved by the assertion that the images of IDs 80 and 81 are independent and span that subspace. Lemma 3.2 similarly asserts, without displayed coefficients, that ten graphs (IDs 71, 69, 75, 72, 79, 96, 108, 105, 111, 107) are redundant in six relation groups. The author explicitly cautions that 'computer calculations are hard and allow room for many oversights'. A single omitted nonzero differential image, a wrong sign in a relation, or an incorrect redundancy claim would change the six-graph quotient, hence the Specht decompositions in Corollary 3.4 and the S_n-isomorphisms in Theorem 1.3. This is not an internal inconsistency; it is an unverified computational dependency at the exact point where the paper's new content is produced.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the weight-graded compactly supported cohomology of moduli spaces of curves using the Getzler-Kapranov graph complex. It extends previous weight-13 computations to the excess class 3g + 2n = 28, covering (g,n) = (2,11), (4,8), (6,5), (8,2). The main result, Theorem 1.3, states that gr^W_13 H^*_c(M_g,n) vanishes outside two degrees and gives Sn-equivariant isomorphisms to explicit Specht-module tensors with LS12. The proof is computational: the author lists 106 virtual blown-up representations, resolves weight-11 and weight-13 relations, performs Gaussian elimination on the differential using incidence tables, and obtains six surviving graphs from which the cohomology is read off. The paper relies on prior work [5] for the simplified graph complex and on [3,4] for generator presentations.","tokens_in":18590,"tokens_out":3499,"duration_ms":42037,"significance":"If the computation is correct, the paper is a useful extension of the weight-13 cohomology computations to the next excess class and provides concrete, falsifiable predictions with explicit Specht decompositions. The paper is accompanied by a GitHub repository, the appendix lists the generated graphs, and the reduction to six graphs is stated in a form that can in principle be checked independently. The main significance is therefore tied to the reliability of the computational steps, which are the paper's genuinely new content.","major_comments":[{"comment":"The decisive Gaussian elimination step is not fully reproducible from the manuscript. Tables 2, 3, and 4 record only zero/nonzero incidence patterns, not the actual coefficients or signs of the differential images. The places where triangularity fails, rows 116 and 117 of Table 4, are resolved by the assertion that the images of graphs 80 and 81 are independent and span that subspace. Since a single omitted nonzero coefficient, wrong sign, or incorrect independence statement would change the six-graph quotient and hence Corollary 3.4 and Theorem 1.3, the manuscript should provide the explicit differential matrices, including coefficients and signs, or a fully scripted deterministic computation whose output is included.","section":"Section 3.3, Theorem 3.3, Tables 2-4"},{"comment":"The redundancy of the ten underlined virtual blown-up representations in the six listed relation groups is asserted with 'One checks' but no coefficients or computation are shown. This reduction determines the input to Theorem 3.3 and is therefore load-bearing. The manuscript should display the actual linear relations among the graphs in each of the six groups, or provide code output verifying that the ten graphs are redundant and that the remaining graphs are independent for all four (g,n) pairs in the excess class.","section":"Section 3.2, Lemma 3.2"},{"comment":"The generation of the B1 family depends on the statement that for higher group sizes or higher valence the author manually checks each weight-2 relation group and hardcodes a basis into the script. These relation groups and bases are not listed in the paper, and the exact set of manually imposed relations is not documented. Since correctness of the entire enumeration relies on this step, the manuscript should either list all weight-2 relation groups with their bases or provide the relevant portion of the code and its output so that the relation resolution can be audited.","section":"Section 4.3"},{"comment":"Proposition 3.1 and the proof of Theorem 3.3 rest on the assertion that the enumeration algorithm generates all virtual blown-up representations of excess 28. The paper describes the algorithm but does not give a completeness proof or an independent verification that the component list is exhaustive. Given that the main theorem is a negative statement about vanishing, the completeness claim is essential. A second implementation or a systematic parametrization of all possible components would make the enumeration checkable.","section":"Section 4, Proposition 3.1"}],"minor_comments":[{"comment":"There appears to be a missing plus sign after the second displayed sum in (2.12); the five displayed summands are not clearly separated.","section":"Section 2.3, equation (2.12)"},{"comment":"There are several typos: 'traslations' in Section 2.1, 'indipendently' and 'indipendent' in Lemma 3.2, 'withing' in Section 2.5, 'ren' in Section 4.5, and 'column operations' and 'con be' in the proof of Corollary 3.4.","section":"Throughout"},{"comment":"The sentence 'the relations in 2.1 look as follows' refers to relations 1-6, 7', 7'' from [5], but those relations are not restated in the paper; please clarify the numbering and restate the relations being quoted.","section":"Section 2.4"},{"comment":"The phrase 'as in excess 2' appears to be a typo; the surrounding discussion concerns excess 28, so the intended comparison should be clarified.","section":"Corollary 3.4"}],"recommendation":"major_revision","confidential_remarks":"The paper's main new content is computational, and the weakest point is the reproducibility of the relation resolution and Gaussian elimination. The author's own caution that 'computer calculations are hard and allow room for many oversights' is honest, but it underscores the need for the explicit data. The dependence on [5], which shares authors with the author's supervisor, is not itself a problem, but it makes the lack of self-contained verification of the new steps more significant. I would encourage the editor to ask for the computational artifacts (matrices, hardcoded relation bases, and the full list of components) before final acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Marco,\n\nThe genuinely new thing here is the weight-13 computation at 3g+2n=28. That case was not covered in Canning–Larson–Payne–Willwacher, and the paper finds that cohomology is no longer concentrated in top degree: a second degree appears with nonzero Specht modules. If correct, it extends the known pattern and gives explicit S_n-representations for (2,11), (4,8), (6,5), (8,2). The result is new, and the author ships the Python/Sage code plus an honest warning that computer calculations have room for oversight. That transparency is real credit.\n\nThe soft spot is exactly where the new content is produced. Theorem 3.3 reduces 106 virtual blown-up representations to six survivors by Gaussian elimination, but the matrices are not printed, only incidence grids. The two places where triangularity fails, rows 116 and 117, are dismissed with ‘one checks’ that the images of IDs 80 and 81 are independent. Lemma 3.2 similarly asserts redundancy of ten graphs without displayed coefficients. The dependency on earlier quasi-isomorphism results from [5] is normal, though it makes the paper non-self-contained. The real issue is that the final linear algebra is not reproducible from the text alone; you have to rerun the code or trust the assertions. I did not rerun the code, so I cannot say the computation is wrong, but I can say the paper would be much stronger with the actual matrices or a machine-checked certificate.\n\nIf the computation is correct, this is a small but honest step in an active program. The right audience is specialists in moduli space cohomology and graph complexes. I would send it to a referee who can actually run the code and independently verify the elimination. My recommendation: accept conditional on verification, and ask the author to provide the differential matrices explicitly—or at least a script that prints them. That is not a desk-reject; the new case and the structural observation deserve referee time.","headline":"A genuine but thinly documented extension of the weight-13 computation to excess 28; the new cohomology is plausible, but the decisive Gaussian elimination is asserted rather than shown.","tokens_in":18934,"tokens_out":2501,"would_cite":false,"duration_ms":30397,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14H10","14C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Using the Getzler–Kapranov graph complex, the paper computes the weight-13 compactly supported cohomology of the moduli spaces of curves for every pair (g,n) with 3g+2n=28, showing it lives in exactly two degrees and is given by explicit…","keywords":["Getzler-Kapranov graph complex","weight filtration","compactly supported cohomology","moduli spaces of curves","weight 13","excess 28","Specht modules","symmetric group representations"],"falsifier":"Compute the $\\mathcal{S}_n$-equivariant Euler characteristic of the claimed weight-13 cohomology for, say, $(g,n)=(4,8)$ and compare it with the weight-13 term of the polynomial point count of $\\mathcal{M}_{4,8}$ from the paper's reference [2]; any mismatch of the character values would disprove Theorem 1.3.","tokens_in":17978,"feed_emoji":"📐","tokens_out":13309,"duration_ms":119647,"temperature":0.7,"pith_summary":"This paper computes the weight-13 piece of the compactly supported cohomology of the moduli spaces $\\mathcal{M}_{g,n}$ for every pair of genus $g$ and marked points $n$ satisfying $3g+2n=28$. Using the Getzler–Kapranov graph complex, it proves that this cohomology is nonzero only in two consecutive degrees, $3g+n-2$ and $3g+n-3$, and that in those degrees it is isomorphic to explicit symmetric-group representations, each tensored with the same motivic factor $LS_{12}$. The result extends the prior computation for $3g+2n\\leq 27$ and shows that the pattern changes at this excess: cohomology is no longer concentrated in the top degree, and the largest-genus pair $(8,2)$ vanishes in top degree. A fair reader should care because these are the first complete weight-13 computations beyond the previously known range, and they give explicit data about how the weight filtration on the cohomology of moduli spaces behaves.","feed_headline":"Weight-13 cohomology of curve moduli pinned down at 3g+2n=28","feed_subtitle":"Fills the last open case of the Getzler–Kapranov computation, showing cohomology sits in two degrees.","key_machinery":"The Getzler–Kapranov graph complex $GK^{\\kappa}_{g,n}$ is the central object: its cohomology is identified with the weight-$\\kappa$ graded piece $\\mathrm{gr}^{W}_{\\kappa}H^*_c(\\mathcal{M}_{g,n})$. The paper works in a quasi-isomorphic simplified subcomplex $GK^{12,1}_{g,n}$ generated by decorated graphs whose vertices carry classes in $H^{12,1}$, $H^{11,0}$ and $H^{1,1}$ of moduli spaces of curves, depicted by explicit graphical symbols. Generators are re-encoded as 'blown-up representations', combinatorial graphs whose hair labels record the decoration data, and are classified by the excess $E(g,n)=3g+2n$, which is additive over blown-up components. The argument enumerates all virtual blown-up representations of excess 28, groups them by weight-11 and weight-13 relations, and uses Gaussian elimination on the differential matrices to reduce the complex to six graphs whose cohomology gives the theorem.","core_discovery":"The paper's central claim is Theorem 1.3: for $(g,n) = (2,11), (4,8), (6,5), (8,2)$, the associated graded piece $\\mathrm{gr}^{W}_{13}H^*_c(\\mathcal{M}_{g,n})$ vanishes outside degrees $k_1 = 3g+n-2$ and $k_2 = 3g+n-3$, and there are $\\mathcal{S}_n$-equivariant isomorphisms $\\mathrm{gr}^{W}_{13}H^{k_1}_c(\\mathcal{M}_{g,n}) \\cong Z_{g,n}\\otimes LS_{12}$ and $\\mathrm{gr}^{W}_{13}H^{k_2}_c(\\mathcal{M}_{g,n}) \\cong W_{g,n}\\otimes LS_{12}$, where $Z_{g,n}$ and $W_{g,n}$ are the specific representations listed in the theorem. The computation proceeds by resolving the weight-13 Getzler–Kapranov graph complex on the list of all 106 virtual blown-up representations of excess 28, eliminating redundant generators through weight-11 and weight-13 relations, and then running Gaussian elimination on the differential matrices to cut the complex down to six graphs whose cohomology is read off directly.","pith_inferences":["The same enumeration pipeline could plausibly be pushed to excess 29 and beyond, but the paper notes that the excess-29 list is unverified; certifying that list would be a natural next step.","If $LS_{12}$ appears at every excess, weight-13 cohomology may be governed by a single universal motivic factor, with all remaining complexity encoded in the symmetric-group representation.","The shift from single-degree concentration at excess 26 and 27 to two-degree concentration at excess 28 suggests that the width of the cohomology window may grow with excess, a pattern worth testing in higher excesses.","An independent implementation of the graph complex or a direct algebro-geometric computation of the $\\mathcal{S}_n$-character would provide a check on the 'one checks' steps that the paper leaves to the reader."],"forward_implications":["With excess 28 settled, the weight-13 cohomology groups for all four pairs $(2,11),(4,8),(6,5),(8,2)$ are now known completely, both as vector spaces and as symmetric-group representations.","The motivic factor $LS_{12}$ again appears as a tensor factor, so the pattern seen for excess $\\leq 27$ persists rather than breaking at 28.","Cohomology is concentrated in two adjacent degrees instead of one, and the top-degree part for $(8,2)$ vanishes, mirroring the behavior observed in excess 2.","The explicit representations determine the $\\mathcal{S}_n$-equivariant Euler characteristics of these graded pieces, giving concrete numbers that can be compared with point-count data."],"supporting_citations":[{"why":"Supplies the identification of weight-graded cohomology with the Getzler–Kapranov graph complex, the quasi-isomorphic simplified complex, and the excess-26/27 results that this paper extends.","marker":"[5]"},{"why":"Gives the vanishing of odd cohomology below degree 11, which restricts the admissible vertex decorations in weight 13.","marker":"[2]"},{"why":"Provides the presentation of the weight-11 cohomology classes $\\omega_B$ used to draw and resolve weight-11 decorations.","marker":"[3]"},{"why":"Provides the presentation of the weight-13 classes $Z_{B\\subseteq A}$ used to draw and resolve weight-13 decorations.","marker":"[4]"},{"why":"Provides the presentation of weight-2 classes and the pullback maps that define the action of the differential.","marker":"[6]"},{"why":"Provides the code that generates the exhaustive list of virtual blown-up representations and the differential matrices used in the computation.","marker":"[1]"}],"fun_headline_variants":["Weight-13 cohomology of curve moduli computed at 3g+2n=28","Getzler-Kapranov weight 13: all cases resolved","Cohomology in weight 13: only two degrees survive","Curve moduli weight 13 cohomology: vanishing outside two degrees","Weight-13 cohomology for (g,n) with 3g+2n=28 determined"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole result rests on the completeness of the computer-generated catalogue of 106 building blocks and on the correctness of the hand-checked relation resolutions and Gaussian elimination steps; if one graph is missing from the list or one sign in the differential matrices is wrong, the theorem does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Weight-13 cohomology of curve moduli computed at 3g+2n=28","Getzler-Kapranov weight 13: all cases resolved","Cohomology in weight 13: only two degrees survive","Curve moduli weight 13 cohomology: vanishing outside two degrees","Weight-13 cohomology for (g,n) with 3g+2n=28 determined"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000275,"raw_usage":{"total_tokens":1609,"prompt_tokens":877,"completion_tokens":732,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":623}},"tokens_in":493,"tokens_out":732,"duration_ms":7670,"temperature":1.0,"reasoning_tokens":623,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:06:42.176967+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the $\\mathcal{S}_n$-equivariant Euler characteristic of the claimed weight-13 cohomology for, say, $(g,n)=(4,8)$ and compare it with the weight-13 term of the polynomial point count of $\\mathcal{M}_{4,8}$ from the paper's reference [2]; any mismatch of the character values would disprove Theorem 1.3.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the vanishing of odd cohomology below degree 11, which restricts the admissible vertex decorations in weight 13."},{"cited_title":"Sigma 11 (2023), no","cited_arxiv_id":null,"evidence_quote":"Provides the presentation of the weight-11 cohomology classes $\\omega_B$ used to draw and resolve weight-11 decorations."},{"cited_title":"Weight two compactly supported cohomology of moduli spaces of curves","cited_arxiv_id":"2110.05711","evidence_quote":"Provides the presentation of weight-2 classes and the pullback maps that define the action of the differential."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the code that generates the exhaustive list of virtual blown-up representations and the differential matrices used in the computation."}],"review_version":1}