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Theta Bases for Rank-Two Quantum Cluster Algebras of Geometric Type

A mix of human-written and AI-generated textHuman understanding: no partsmath.QA — Quantum Algebramath.RA — Rings and Algebrasmath.RT — Representation Theory

Contributed by 琪越 唐

Version 2 / Oct 04, 2026 / CC BY 4.0

Abstract

We prove that quantum theta and greedy bases coincide in coefficient-free rank two for all positive exchange parameters. Their homogeneous lifts form integral theta bases for arbitrary geometric coefficients and compatible integral quantizations, with frozen variables inverted. For nonnegative initial coefficient blocks, we also establish the basis theorem without frozen inverses. 2026.10.15:This version underwent AI-assisted correctness review using Rethlas-OSV with GPT-6-Astra at xhigh reasoning effort. The review comprised 24 completed segment checks and a completed supplementary full-paper review. No confirmed mathematical errors were reported within the reviewed scope. Rethlas-OSV: https://github.com/OSVerify/Rethlas-OSV

Provenance statement

My motivation was to understand the relationship between quantum greedy bases and quantum theta bases. I initiated the project and chose this comparison as the research question. In searching the literature, I found Gao’s doctoral thesis, which claimed equality in the symmetric and finite-type rank-two cases and conjectured equality when one exchange parameter divides the other. Our subsequent audit identified gaps and errors in parts of the thesis’s arguments. The project developed proofs of the rank-two comparison, extended it to all positive exchange parameters, and established corresponding theta basis results with geometric coefficients. The resulting comparison theorem includes Gao’s Conjecture 5.2 as a special case. The mathematical development proceeded through several conversations with GPT-6-Astra. The manuscript subsequently underwent separate, full-manuscript adversarial reviews by GPT-6-Astra and several revisions using my configured mathematical-writing skill. Uploading this paper to this platform does not mean that I claim ownership or copyright over it. My sole intention is to let more people know that the mathematical problem addressed in this paper has been solved by AI. Once it has been reviewed and its correctness has been verified, it may be cited directly. I welcome anyone who would like to organize this work; if you do so, the credit will be yours. Additionally, if you need more related literature or detailed materials, you can contact me by email. Email: tangqy24@mails.tsinghua.edu.cn

References

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Version history

  1. v1Submitted by 琪越 唐Initial depositSupersededOct 04, 2026
  2. v2Submitted by 琪越 唐Submitted as a new versionCurrentOct 15, 2026