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Algebraicity of the G2\mathrm{G}_2-invariant tensors on Yun's exceptional motives

Primarily AI-generated textHuman understanding: some partsmath.AG — Algebraic Geometrymath.NT — Number Theorymath.RT — Representation Theory

Contributed by Benjamin Severn

Version 3 / Oct 20, 2026 / CC BY 4.0

Abstract

Zhiwei Yun constructed, from a rigid Hecke eigensheaf for G2\mathrm{G}_2 on P1\mathbb{P}^1 minus three points, a family of rank-seven motives whose motivic Galois groups are dense in G2\mathrm{G}_2. We show that on this family the invariant cubic form is the class of an algebraic cycle: at every Hodge-generic parameter, every rational Hodge class on every tensor construction of the rank-seven piece is algebraic, and at every parameter the transported G2\mathrm{G}_2-invariant tensors are algebraic. The cycles are top components of Satake convolution fibres transported by Hecke correspondences. The projector that cuts out the rank-seven piece is made algebraic by a canonical weighted resolution of Yun's boundary divisor, built on the moduli stack of bundles from the stabiliser tori of its points, together with a characteristic-zero census of those points and a cycle-class comparison on varieties with quotient singularities. Yun announced that the cubic is motivated in Andr\'e's sense; the step from motivated to algebraic is new. This version, with its computational certificates and review bundle, is archived on Zenodo: https://doi.org/10.5281/zenodo.23478625

Provenance statement

This paper was produced in a research programme directed by the author, in which AI systems did most of the mathematical derivation, the computer-algebra code and the writing. Claude models (Anthropic) derived the arguments, wrote the SageMath certificates, and drafted and revised the text. Steps were checked by independent adversarial audits from a different model family (OpenAI GPT-6 Astra), ten of them between review rounds. The manuscript then went through six rounds of adversarial review by ChatGPT (OpenAI), on drafts v0.1 to v0.6; the sixth report accepted it with minor revisions, which this version (v0.7) makes. Every objection raised in the six rounds was answered in the text or the certificates. The author set the research direction, chose which results to pursue, relayed the outside reviews and decided each revision. The computations (the characteristic-zero census of Cartan classes and the finite stabiliser data) are exact computations in SageMath 10.7, rerun from a pinned commit with SHA-256 hashes of every input and output, and the census script gives byte-identical output on consecutive runs; the outside reviewers inspected these deposited runs but did not rerun them. The reviews were source-level: no human expert other than the author has checked the paper. The same version is archived on Zenodo with the certificates and the full review record: https://doi.org/10.5281/zenodo.23478625

Tools used

Anthropic
Claude OpusVersion 5.5
Anthropic
Claude SonnetVersion 5.5
OpenAI
CodexVersion GPT-6 Astra
SageMath
SageMathVersion 10.7

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Version history

  1. v1Submitted by Benjamin SevernInitial depositSupersededOct 19, 2026
  2. v2Submitted by Benjamin SevernSubmitted as a new versionSupersededOct 20, 2026
  3. v3Submitted by Benjamin SevernSubmitted as a new versionCurrentOct 22, 2026