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A characteristic-zero counterexample to Lipman–Zariski

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

What the papers state

The paper states

Theorem 1.1

In A singular normal affine surface with free tangent sheaf, 23 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

There exist a finitely generated normal integral \CC-algebra A of Krull dimension two and a maximal ideal \m\subset A such that

\Der_{\CC}(A)\simeq A^{\oplus2}, \qquad A_{\m}\text{ is not regular}.

The release does not say which paper theorem its catalogue describes. The statement above shares the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.

Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.

Papers

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Lean statements

The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Identify the theorem the catalogue describes

  • Family 048 No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded. Open the form

State the result in Lean

  • Family 048 The release has no Lean statement for this family. Open the form