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Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII

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

Proves that every complex K3 surface is Oka, including nonprojective surfaces. More generally, every connected minimal compact complex surface of Kodaira dimension zero is Oka; a connected minimal compact complex surface of class VII is Oka exactly when it is a Hopf or Enoki surface.

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

Theorem 1.1

In Every complex K3 surface is Oka, 23 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Let X be a complex K3 surface, with any smooth Hermitian distance d_X. For every integer m\geq1, every nonempty compact convex set K\subset\C^m, every open neighborhood U of K, every holomorphic map f:U\to X, and every \epsilon>0, there exists a holomorphic map F:\C^m\to X such that

\sup_{z\in K}d_X(F(z),f(z))<\epsilon.

Corollary 1.2

In Every complex K3 surface is Oka, 23 September 2026

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

Let S be a complex K3 surface. For every x\in S and every nonzero vector v\in T_xS, there is a holomorphic immersion f:\C\to S such that

f(0)=x,\qquad f'(0)=v,\qquad \overline{f(\C)}=S,

where closure is taken in the ordinary complex topology. If S is projective, then f(\C) is Zariski dense.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.

Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
One current paper, readable here as a page beside its PDF.

Papers

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Every complex K3 surface is Oka

23 September 2026, current edition.

Read it herePDF

We prove that every complex K3 surface X is Oka, resolving the K3 Oka conjecture. Equivalently, for every m ≥ 1, every holomorphic map to X from a neighborhood of a compact convex set in ℂm can be approximated uniformly on that set by entire maps \(\mathbb C^m\to X\).

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

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

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

Follow-up work by others

  • Yun-Heng Du, Bin Guo, Peng-Chao Wang, Song-Yan Xie, A Strong Dominability Criterion and an Oka Union Theorem, 7 October 2026. Source

    Du, Guo, Wang and Xie extend Theorem 3.1 of OpenAI’s K3-surfaces-are-Oka paper to arbitrary-dimensional targets and report a companion result that rationally connected manifolds are Oka (unpublished).

References outside the release

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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 proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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Identify the theorem the catalogue describes

  • Family 042 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 042 The release has no Lean statement for this family. Open the form