---
title: "Potential integral density on curve character varieties"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 027; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "number-theory"
  - "unformalized"
seed_rank: 1887
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Integral points on character varieties of curves"
    url: "https://github.com/openai/math/blob/main/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Potential integral density on curve character varieties

## One-sentence takeaway

OpenAI's result family 027 (Number theory) claims: Resolves the determinant-one curve case of Litt's integral-density question.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SLr character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.
- *Integral points on character varieties of curves*: We prove potential Zariski density of integral points on SLr-character varieties of smooth complex curves in every rank. The result allows prescribed quasi-unipotent boundary monodromy, including exact nonsemisimple conjugacy classes, and holds on every component over the full ring of integers of one finite extension. Here integrality is measured in the ambient character variety, while the exact boundary conditions are imposed on the complex representation.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Integral points on character varieties of curves](https://github.com/openai/math/blob/main/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math