Symmetric edge polytopes are not gamma-positive

Luis Ferroni

arxiv: 2607.02424 · v1 · pith:K7UBNROCnew · submitted 2026-07-02 · 🧮 math.CO

Symmetric edge polytopes are not gamma-positive

Luis Ferroni This is my paper

Pith reviewed 2026-07-03 10:15 UTC · model grok-4.3

classification 🧮 math.CO

keywords symmetric edge polytopesEhrhart h*-polynomialsgamma-positivitycounterexamplesEhrhart theorypolytope combinatorics

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The pith

The Ehrhart h*-polynomials of symmetric edge polytopes are not gamma-positive.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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.

What carries the argument

Symmetric edge polytopes together with explicit computation of the coefficients in their Ehrhart h*-polynomials to test the gamma-positivity condition.

If this is right

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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

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.

Load-bearing premise

The constructed objects qualify as symmetric edge polytopes and the computed h*-polynomials correctly display negative coefficients in their gamma expansions.

What would settle it

Direct computation of the h*-polynomial for the 36-dimensional example that shows at least one gamma coefficient is negative.

Figures

Figures reproduced from arXiv: 2607.02424 by Luis Ferroni.

**Figure 1.** Figure 1: The graph G(8, 8, 8, 8, 8). The natural question is how to compute the h ∗ -polynomials of these graphs. Although reasonably compact formulas should be possible for arbitrary vectors a ∈ Z≥1, for the purposes of the counterexample it suffices to focus on the case in which all entries of a are equal. The main computational tool we employ is a formula of K´alman and T´othm´er´esz [KT23], which writes the h … view at source ↗

read the original abstract

A conjecture posed by Ohsugi and Tsuchiya (2019) postulates that the Ehrhart $h^*$-polynomials of symmetric edge polytopes are $\gamma$-positive. We disprove this conjecture by exhibiting an infinite family of counterexamples. The smallest example provided by our construction is a $36$-dimensional symmetric edge polytope.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

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.

read the 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.

What 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.

The 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.

This 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.

Referee Report

2 major / 1 minor

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.

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 (2)

[§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.
[§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.

minor comments (1)

[Abstract] The abstract states the dimension but does not indicate the order of the underlying graph; adding this datum would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments on reproducibility. We address each major comment below.

read point-by-point responses

Referee: [§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.

Authors: 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: yes
Referee: [§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.

Authors: 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: yes

Circularity Check

0 steps flagged

Direct counterexample construction with no circularity

full rationale

The paper disproves the Ohsugi-Tsuchiya conjecture by exhibiting an explicit infinite family of symmetric edge polytopes whose h*-polynomials fail gamma-positivity, with the smallest member 36-dimensional. This is a direct falsification via construction and computation; no positive result is derived from fitted parameters, self-definitions, or self-citation chains. The central claim rests on verifiable object definitions and explicit polynomial computation rather than any reduction to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The disproof rests on standard definitions and theorems from Ehrhart theory and polytope combinatorics; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard definitions and properties of Ehrhart polynomials, h*-polynomials, and gamma-positivity in the theory of lattice polytopes.
The paper invokes the established framework of Ehrhart theory without additional assumptions beyond those in the cited 2019 conjecture.

pith-pipeline@v0.9.1-grok · 5560 in / 1136 out tokens · 47745 ms · 2026-07-03T10:15:30.807911+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 1 internal anchor

[1]

Arithmetic aspects of symmetric edge polytopes , JOURNAL =

Higashitani, Akihiro and Jochemko, Katharina and Micha. Arithmetic aspects of symmetric edge polytopes , JOURNAL =. 2019 , NUMBER =. doi:10.1112/s0025579319000147 , URL =

work page doi:10.1112/s0025579319000147 2019
[2]

On a generalization of symmetric edge polytopes to regular matroids , fjournal =

D'Al. On a generalization of symmetric edge polytopes to regular matroids , fjournal =. Int. Math. Res. Not. , issn =. 2024 , language =. doi:10.1093/imrn/rnae107 , keywords =

work page doi:10.1093/imrn/rnae107 2024
[3]

Interlacing

Higashitani, Akihiro and Kummer, Mario and Micha. Interlacing. Sel. Math., New Ser. , issn =. 2017 , language =. doi:10.1007/s00029-017-0350-6 , keywords =

work page doi:10.1007/s00029-017-0350-6 2017
[4]

Ehrhart theory of symmetric edge polytopes via ribbon structures , year =

Tam. Ehrhart theory of symmetric edge polytopes via ribbon structures , year =

work page
[5]

K. Algebr. Comb. , issn =. 2023 , language =. doi:10.5802/alco.318 , keywords =

work page doi:10.5802/alco.318 2023
[6]

arXiv e-prints , keywords =

The number of edges of a symmetric edge polytope. arXiv e-prints , keywords =. doi:10.48550/arXiv.2512.16572 , archivePrefix =. 2512.16572 , primaryClass =

work page doi:10.48550/arxiv.2512.16572
[7]

On the Ehrhart Theory of Generalized Symmetric Edge Polytopes

On the Ehrhart Theory of Generalized Symmetric Edge Polytopes. arXiv e-prints , keywords =. doi:10.48550/arXiv.2401.03383 , archivePrefix =. 2401.03383 , primaryClass =

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2401.03383
[8]

Matsui, Tetsushi and Higashitani, Akihiro and Nagazawa, Yuuki and Ohsugi, Hidefumi and Hibi, Takayuki , TITLE =. J. Algebraic Combin. , FJOURNAL =. 2011 , NUMBER =. doi:10.1007/s10801-011-0290-8 , URL =

work page doi:10.1007/s10801-011-0290-8 2011
[9]

Selecta Math

Ohsugi, Hidefumi and Tsuchiya, Akiyoshi , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s00029-020-00588-0 , URL =

work page doi:10.1007/s00029-020-00588-0 2020
[10]

On the gamma-vector of symmetric edge polytopes , JOURNAL =

D'Al\`. On the gamma-vector of symmetric edge polytopes , JOURNAL =. 2023 , NUMBER =. doi:10.1137/22M1492799 , URL =

work page doi:10.1137/22m1492799 2023
[11]

Bruns, Winfried and R\". J. Combin. Theory Ser. A , FJOURNAL =. 2007 , NUMBER =. doi:10.1016/j.jcta.2006.03.003 , URL =

work page doi:10.1016/j.jcta.2006.03.003 2007
[12]

Real root conjecture fails for five- and higher-dimensional spheres , JOURNAL =

Gal,. Real root conjecture fails for five- and higher-dimensional spheres , JOURNAL =. 2005 , NUMBER =. doi:10.1007/s00454-005-1171-5 , URL =

work page doi:10.1007/s00454-005-1171-5 2005
[13]

Discrete Comput

Ohsugi, Hidefumi and Tsuchiya, Akiyoshi , title =. Discrete Comput. Geom. , issn =. 2021 , language =. doi:10.1007/s00454-020-00236-6 , keywords =

work page doi:10.1007/s00454-020-00236-6 2021
[14]

Sign-graded posets, unimodality of

Br. Sign-graded posets, unimodality of. Electron. J. Combin. , FJOURNAL =. 2004/06 , NUMBER =. doi:10.37236/1866 , URL =

work page doi:10.37236/1866 2004
[15]

2026 , note =

Matt Larson , title =. 2026 , note =

work page 2026

[1] [1]

Arithmetic aspects of symmetric edge polytopes , JOURNAL =

Higashitani, Akihiro and Jochemko, Katharina and Micha. Arithmetic aspects of symmetric edge polytopes , JOURNAL =. 2019 , NUMBER =. doi:10.1112/s0025579319000147 , URL =

work page doi:10.1112/s0025579319000147 2019

[2] [2]

On a generalization of symmetric edge polytopes to regular matroids , fjournal =

D'Al. On a generalization of symmetric edge polytopes to regular matroids , fjournal =. Int. Math. Res. Not. , issn =. 2024 , language =. doi:10.1093/imrn/rnae107 , keywords =

work page doi:10.1093/imrn/rnae107 2024

[3] [3]

Interlacing

Higashitani, Akihiro and Kummer, Mario and Micha. Interlacing. Sel. Math., New Ser. , issn =. 2017 , language =. doi:10.1007/s00029-017-0350-6 , keywords =

work page doi:10.1007/s00029-017-0350-6 2017

[4] [4]

Ehrhart theory of symmetric edge polytopes via ribbon structures , year =

Tam. Ehrhart theory of symmetric edge polytopes via ribbon structures , year =

work page

[5] [5]

K. Algebr. Comb. , issn =. 2023 , language =. doi:10.5802/alco.318 , keywords =

work page doi:10.5802/alco.318 2023

[6] [6]

arXiv e-prints , keywords =

The number of edges of a symmetric edge polytope. arXiv e-prints , keywords =. doi:10.48550/arXiv.2512.16572 , archivePrefix =. 2512.16572 , primaryClass =

work page doi:10.48550/arxiv.2512.16572

[7] [7]

On the Ehrhart Theory of Generalized Symmetric Edge Polytopes

On the Ehrhart Theory of Generalized Symmetric Edge Polytopes. arXiv e-prints , keywords =. doi:10.48550/arXiv.2401.03383 , archivePrefix =. 2401.03383 , primaryClass =

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2401.03383

[8] [8]

Matsui, Tetsushi and Higashitani, Akihiro and Nagazawa, Yuuki and Ohsugi, Hidefumi and Hibi, Takayuki , TITLE =. J. Algebraic Combin. , FJOURNAL =. 2011 , NUMBER =. doi:10.1007/s10801-011-0290-8 , URL =

work page doi:10.1007/s10801-011-0290-8 2011

[9] [9]

Selecta Math

Ohsugi, Hidefumi and Tsuchiya, Akiyoshi , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s00029-020-00588-0 , URL =

work page doi:10.1007/s00029-020-00588-0 2020

[10] [10]

On the gamma-vector of symmetric edge polytopes , JOURNAL =

D'Al\`. On the gamma-vector of symmetric edge polytopes , JOURNAL =. 2023 , NUMBER =. doi:10.1137/22M1492799 , URL =

work page doi:10.1137/22m1492799 2023

[11] [11]

Bruns, Winfried and R\". J. Combin. Theory Ser. A , FJOURNAL =. 2007 , NUMBER =. doi:10.1016/j.jcta.2006.03.003 , URL =

work page doi:10.1016/j.jcta.2006.03.003 2007

[12] [12]

Real root conjecture fails for five- and higher-dimensional spheres , JOURNAL =

Gal,. Real root conjecture fails for five- and higher-dimensional spheres , JOURNAL =. 2005 , NUMBER =. doi:10.1007/s00454-005-1171-5 , URL =

work page doi:10.1007/s00454-005-1171-5 2005

[13] [13]

Discrete Comput

Ohsugi, Hidefumi and Tsuchiya, Akiyoshi , title =. Discrete Comput. Geom. , issn =. 2021 , language =. doi:10.1007/s00454-020-00236-6 , keywords =

work page doi:10.1007/s00454-020-00236-6 2021

[14] [14]

Sign-graded posets, unimodality of

Br. Sign-graded posets, unimodality of. Electron. J. Combin. , FJOURNAL =. 2004/06 , NUMBER =. doi:10.37236/1866 , URL =

work page doi:10.37236/1866 2004

[15] [15]

2026 , note =

Matt Larson , title =. 2026 , note =

work page 2026