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Positive polytopes and the full Fock–Goncharov conjecture

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.RT — Representation Theory

Contributed by Qiyue Tang ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

We prove the origin-interior form of the polytope criterion of Gross, Hacking, Keel and Kontsevich: over a field of characteristic zero, full Fock–Goncharov for a principal coefficient cluster variety is equivalent to the existence of a bounded full-dimensional positive polytope containing the origin in its interior. Replacing the interior condition by membership of the origin gives a false criterion.

Provenance statement

The initial research manuscript was developed using ChatGPT-6-Pro during a research session lasting approximately 270 minutes. A further ChatGPT-6-Pro session of approximately 14 minutes examined the role and necessity of the origin-interior condition. Codex reviewed the arguments, checked the literature, and revised the exposition. A preceding version of the manuscript passed an AI review using Rethlas-OSV, which incorporates the OSVerify framework of Nie, Ye, Liu, Wang, Ermon and Zou.

References

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  9. OSVerify contributors, Rethlas-OSV , software repository (2026). https://github.com/OSVerify/Rethlas-OSV.link
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Version history

  1. v1Submitted by Qiyue TangInitial depositCurrentNov 01, 2026