The equality case of Ehrhart's volume conjecture

Jihao Liu

arXiv:2608.01040v2 Primary: math.CO Secondary: math.AG, math.CV

Abstract

We prove that every full-dimensional compact convex body in $\mathbb R^n$ whose barycenter is its unique interior lattice point and whose volume is $(n+1)^n/n!$ is a unimodular image of the simplex $(n+1)Δ_n-(1,\dots,1)$. This resolves the equality case of Ehrhart's volume conjecture, as a counterpart of the inequality part recently proved by OpenAI. The main result of this paper is obtained by generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system.

Disclosure

“the mechanisms of the second half, for which no correspondence was available, on its own, and produced the complete proof, internally verified by the system’s proof-checking pipeline, in 3 hours and 16 minutes. Discussions with Fable 5 and GPT-5.6-sol-ultra also assisted the author in quickly grasping the methodology of the proof of [OAI26, Chapter 8, Theorem 1.1]. Human verification and polishing were done afterwards. We do not know whether the Danus system, at intermediate steps o”
PDF page 5

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-23T19:18:07.125735Z

Structural counts

Pages 30 pdf

Theorems 11 source

Lemmas 21 source

Propositions 6 source

Corollaries 1 source

Definitions 0 source

Displayed equations 111 source

Bibliography entries 34 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file Ehrhart_volume_v2.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.