{"id":"b7ab15d6-8cda-46ec-96b7-7f3b8dfab39c","arxiv_id":"2607.02424","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Disproves the conjecture that Ehrhart h*-polynomials of symmetric edge polytopes are gamma-positive by exhibiting an infinite family of counterexamples, with the smallest in dimension 36.","lead":"The paper disproves the 2019 Ohsugi-Tsuchiya conjecture by constructing an infinite family of symmetric edge polytopes whose Ehrhart h*-polynomials fail to be gamma-positive. A smart generalist might read it to see the boundaries of positivity properties in high-dimensional combinatorial objects.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"The load-bearing step is whether the 36-dimensional construction yields a valid symmetric edge polytope whose h*-polynomial (computed computationally) has a negative gamma coefficient.","rationale":"The reader's weakest assumption is precisely the validity of the objects and the correctness of the h*-computation; that is the single point on which the entire disproof rests. Because the dimension precludes hand verification, the concern is concrete and testable by the check above. If the recomputation confirms a negative gamma coefficient, the counterexample stands and the verdict can move to ACCEPT; otherwise the claim fails. This matches the reader's diagnosis exactly, so agreement is full and the verdict adjustment is from UNVERDICTED to CONDITIONAL pending the independent check.","tokens_in":1548,"tokens_out":398,"duration_ms":24919,"concrete_test":"Take the explicit graph (or other defining data) for the 36-dimensional example from the paper, construct the symmetric edge polytope in an independent system (e.g., Polymake or SageMath), recompute its h*-polynomial via the standard Ehrhart series method, convert to the gamma basis, and verify whether any gamma coefficient is negative.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The disproof is by explicit counterexample: an infinite family whose smallest member is 36-dimensional. For the claim to hold, two things must be true: (1) the graphs or objects in the family must satisfy the definition of symmetric edge polytopes, and (2) the h*-polynomial must be computed correctly and shown to lie outside the gamma-positive cone. At dimension 36 the second step is necessarily machine-assisted; any error in the input description, in the Ehrhart computation routine, or in the subsequent change-of-basis to the gamma basis would invalidate the counterexample. The paper must therefore supply an unambiguous, reproducible description of the smallest member together with the exact computational pipeline.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims to disprove the conjecture of Ohsugi and Tsuchiya (2019) that the Ehrhart h*-polynomials of symmetric edge polytopes are gamma-positive. The disproof proceeds by explicit construction of an infinite family of counterexamples whose smallest member is a 36-dimensional symmetric edge polytope.","tokens_in":1672,"tokens_out":378,"duration_ms":31733,"significance":"If the counterexamples are valid, the result is significant: it supplies a negative answer to an open conjecture together with an explicit infinite family, which is a strong form of evidence in Ehrhart theory. The work is a direct disproof by counterexample rather than a derived positive statement, and the provision of an infinite family strengthens the claim beyond a single sporadic example.","major_comments":[{"comment":"§3 (Construction of the family): the manuscript must supply an unambiguous, machine-readable description (e.g., adjacency list or edge set) of the smallest 36-dimensional graph so that independent verification that the resulting polytope is a symmetric edge polytope can be performed.","section":"§3"},{"comment":"§4 (Computation of the h*-polynomial): because the gamma-coefficient sign is obtained by a change-of-basis computation at dimension 36, the paper must document the exact software, input data, and algorithm used for the Ehrhart polynomial and the subsequent gamma-basis conversion; without this pipeline the sign claim cannot be reproduced.","section":"§4"}],"minor_comments":[{"comment":"The abstract states the dimension but does not indicate the order of the underlying graph; adding this datum would improve readability.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the detailed comments on reproducibility. We address each major comment below.","responses":[{"response":"We agree that an explicit machine-readable description will facilitate independent verification. The revised manuscript will include the full edge set (or adjacency list) of the smallest 36-dimensional graph together with a precise description of the infinite family generated from it.","revision_made":"yes","referee_comment":"[§3] §3 (Construction of the family): the manuscript must supply an unambiguous, machine-readable description (e.g., adjacency list or edge set) of the smallest 36-dimensional graph so that independent verification that the resulting polytope is a symmetric edge polytope can be performed."},{"response":"We will add a new subsection in the revised version that documents the full computational pipeline: the software employed, the precise input data or files, and the sequence of steps used to obtain the Ehrhart polynomial and perform the change-of-basis to the gamma basis. This will make the sign computation fully reproducible.","revision_made":"yes","referee_comment":"[§4] §4 (Computation of the h*-polynomial): because the gamma-coefficient sign is obtained by a change-of-basis computation at dimension 36, the paper must document the exact software, input data, and algorithm used for the Ehrhart polynomial and the subsequent gamma-basis conversion; without this pipeline the sign claim cannot be reproduced."}],"tokens_in":1152,"tokens_out":326,"duration_ms":23170,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper's core contribution is a direct counterexample to the Ohsugi-Tsuchiya conjecture: an infinite family of symmetric edge polytopes whose Ehrhart h*-polynomials fail gamma-positivity. The smallest member sits in dimension 36, and the construction appears to be new relative to the 2019 statement and the cited literature.\n\nWhat the work does cleanly is deliver a concrete disproof rather than a conditional or asymptotic argument. Once the graphs or defining data for the family are fixed, the claim reduces to checking that the objects meet the symmetric edge polytope definition and that their h*-polynomials have at least one negative coefficient in the gamma basis. That is a standard, falsifiable route.\n\nThe soft spot is the computational step at dimension 36. Ehrhart computations and the change-of-basis to the gamma basis are necessarily machine-assisted here, so the paper must give an unambiguous description of the smallest graph or polytope together with the exact pipeline used. Any ambiguity in the input or in the software verification would leave the counterexample open to doubt. If those details are supplied and reproducible, the argument holds; if they are only sketched, a referee will ask for them.\n\nThis is the sort of paper that belongs in the Ehrhart-theory and combinatorial-convex-geometry literature. Readers who track positivity conjectures for polytopes will want to see the construction and decide whether the family can be simplified or whether similar failures appear in lower dimensions. It deserves a serious referee because it settles a named conjecture with an explicit family rather than with fitting or non-constructive arguments.","headline":"Ferroni supplies an explicit infinite family of counterexamples showing that symmetric edge polytopes need not have gamma-positive h*-polynomials, with the smallest at dimension 36.","tokens_in":2124,"tokens_out":404,"would_cite":true,"duration_ms":17187,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The Ehrhart h*-polynomials of symmetric edge polytopes are not gamma-positive.","keywords":["symmetric edge polytopes","Ehrhart h*-polynomials","gamma-positivity","counterexamples","Ehrhart theory","polytope combinatorics"],"falsifier":"Direct computation of the h*-polynomial for the 36-dimensional example that shows at least one gamma coefficient is negative.","tokens_in":2417,"feed_emoji":"","tokens_out":583,"duration_ms":38102,"temperature":0.7,"pith_summary":"The paper sets out to determine whether the Ehrhart h*-polynomials of symmetric edge polytopes are always gamma-positive. It constructs an infinite family of counterexamples where this property fails. A sympathetic reader would care because gamma-positivity encodes a refined form of positivity often tied to combinatorial structure in Ehrhart theory. The family produces counterexamples in arbitrarily high dimensions. The smallest member of the family is 36-dimensional.","feed_headline":"Symmetric edge polytopes fail gamma-positivity test","feed_subtitle":"An infinite family of counterexamples begins at dimension 36, showing their Ehrhart h*-polynomials are not always gamma-positive.","key_machinery":"Symmetric edge polytopes together with explicit computation of the coefficients in their Ehrhart h*-polynomials to test the gamma-positivity condition.","core_discovery":"By constructing an infinite family of symmetric edge polytopes, the paper shows that their Ehrhart h*-polynomials are not gamma-positive. This supplies explicit counterexamples to the general claim of gamma-positivity for this class, including polytopes of arbitrarily high dimension whose smallest instance is 36-dimensional.","pith_inferences":["Constructions of this type could be adapted to produce counterexamples for gamma-positivity conjectures on other families of polytopes.","Cases of dimension less than 36 may still satisfy gamma-positivity, pointing to a possible threshold effect.","It would be natural to classify which symmetric edge polytopes do satisfy gamma-positivity and which do not."],"forward_implications":["The claim that Ehrhart h*-polynomials of symmetric edge polytopes are always gamma-positive does not hold.","There exist symmetric edge polytopes whose h*-polynomials have at least one negative gamma coefficient.","Gamma-positivity fails for this class of polytopes in every dimension at or above 36.","The property cannot be assumed without additional restrictions on the underlying graph or signed graph."],"fun_headline_variants":["Symmetric edge polytopes disprove gamma-positivity conjecture","Infinite counterexamples to gamma-positivity in symmetric edge polytopes","Ohsugi-Tsuchiya conjecture disproved by symmetric edge polytopes","Symmetric edge polytopes yield infinite gamma-positivity counterexamples"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The constructed objects qualify as symmetric edge polytopes and the computed h*-polynomials correctly display negative coefficients in their gamma expansions.","fun_headline_variants_meta":{"raw":{"variants":["Symmetric edge polytopes disprove gamma-positivity conjecture","Infinite counterexamples to gamma-positivity in symmetric edge polytopes","Ohsugi-Tsuchiya conjecture disproved by symmetric edge polytopes","Symmetric edge polytopes yield infinite gamma-positivity counterexamples"]},"model":"grok-4.3","cost_usd":0.010243,"raw_usage":{"total_tokens":4362,"prompt_tokens":476,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":102428000,"prompt_tokens_details":{"text_tokens":476,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3818,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":476,"tokens_out":68,"duration_ms":39995,"temperature":1.0,"reasoning_tokens":3818,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T10:15:30.807911+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Direct computation of the h*-polynomial for the 36-dimensional example that shows at least one gamma coefficient is negative.","supporting_citations":[],"review_version":1}