Result 342, Differential geometry

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.

Lean formalization Proof

The bigger picture

Why it matters

Two ways of fitting an area-measuring structure to a four-dimensional space appear to impose different demands. The manuscript claims that satisfying the weaker demand always allows the stronger one, without changing the space's almost complex structure.

What changes?

The manuscript reports this for every smooth almost complex structure on a closed four-manifold, meaning a compact four-dimensional space without boundary. An almost complex structure acts like a quarter-turn on tangent vectors. A symplectic form is a closed, nondegenerate two-form measuring oriented areas. Taming requires positive area for each nonzero vector paired with its quarter-turn; compatibility additionally requires invariance under that rotation. The result keeps the almost complex structure fixed, but the new form's cohomology class may differ.

What does that help mathematicians do?

The claimed result rules out a specific obstruction: an almost complex structure in this setting cannot admit a taming symplectic form yet admit no compatible one. A researcher seeking compatibility can therefore focus on establishing the weaker taming condition. This is an existence reduction, not an algorithm, and it does not say that the original taming form is itself compatible.

Are there practical applications?

Its immediate value is foundational, clarifying when symplectic and almost complex structures can fit together on closed four-manifolds. The cohomology qualification matters: integrals of the form over closed surfaces need not be preserved. Questions requiring those global quantities to remain fixed therefore need more than the existence statement reported here.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Taming implies compatibility on four-manifolds

September 23, 2026 18 pages Main result formalized in Lean

We prove that every smooth almost complex structure on a closed four-manifold which is tamed by a symplectic form is compatible with a symplectic form. This gives a positive solution to Donaldson's tamed-to-compatible conjecture.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/342.md.

Donaldson's tamed-to-compatible conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Donaldson's tamed-to-compatible conjecture asks whether an almost-complex structure tamed by a symplectic form also admits a compatible symplectic form. The formalized result establishes this for every closed connected smooth four-manifold, keeping the almost-complex structure fixed. No integrability assumption or prescribed cohomology class for the compatible form is required.

Comparator links

Result Comparator statement
Taming implies compatibility TamingCompatibility.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.