OpenAI reports integral density on curve character varieties

For every smooth connected complex algebraic curve, every rank r, and prescribed quasi-unipotent boundary conjugacy classes, including nonsemisimple classes, one finite extension of the number field generated by their eigenvalues makes ambient integral points Zariski dense in the exact-boundary locus of the SL_r character variety and in every irreducible component.

Sources for this record

What was already known

Esnault–Groechenig proved integrality for cohomologically rigid local systems, and de Jong–Esnault obtained prime-by-prime integral l-adic existence without full-ring density over one field. Coccia–Litt proved full-ring potential density in rank two under related hypotheses; Litt asked whether one number field suffices in general (manuscript introduction, History and relation to previous work, pp. 2–3).

AI involvement AI-led

OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.

OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.

Importance Major

Answers the determinant-one curve case of Litt’s potential integral-density question in every rank with exact quasi-unipotent boundary conditions.

Limitations

Integrality is measured in the ambient character variety; exact boundary classes are imposed on the complex representation and need not persist as strict classes after reduction at each finite prime. The theorem does not cover higher-dimensional varieties or general reductive groups.

Technical verification Reported

Reported is the default when evidence for a higher level is insufficient. The OpenAI-authored manuscript reports and argues the result, but the reviewed sources do not document a completed check of the full assessed claim.

Human understanding Few

Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.

Correction and publication history

2026-10-18 · publication

Primary manuscript, attribution, and prior-work review for non-notifying local inclusion.

Read this published revision