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Taming implies compatibility on four-manifolds

OpenAI · September 23, 2026

Earlier edition Lean

This is an earlier edition (September 23, 2026). Current edition: Taming implies compatibility on four-manifolds, October 6, 2026.

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Cite

@misc{OAI:Taming-implies-compatibility-on-four-manifolds-September-23-2026,
  author = {{OpenAI}},
  title = {{Taming implies compatibility on four-manifolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/paper.pdf}{OAI:Taming-implies-compatibility-on-four-manifolds-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/342.md. The Lean statements have not been checked by people against the paper's claims.

Family 342: Donaldson's tamed-to-compatible conjecture

Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change.

This is the only paper in this family.