---
title: "A characteristic-zero counterexample to Lipman–Zariski"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 048; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebraic-and-complex-geometry"
  - "unformalized"
seed_rank: 1907
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "A singular normal affine surface with free tangent sheaf"
    url: "https://github.com/openai/math/blob/main/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# A characteristic-zero counterexample to Lipman–Zariski

## One-sentence takeaway

OpenAI's result family 048 (Algebraic and complex geometry) claims: Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.

## 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

- *A singular normal affine surface with free tangent sheaf*: We construct a singular normal affine complex surface whose tangent sheaf is free of rank two. This disproves the Lipman–Zariski conjecture in characteristic zero.

## 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: [A singular normal affine surface with free tangent sheaf](https://github.com/openai/math/blob/main/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/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