What exactly does OpenAI claim about Donaldson's hypersymplectic deformation conjecture?

Proves Donaldson's hypersymplectic deformation conjecture in a cohomology-preserving form. Every positive triple of smooth closed two-forms on a closed connected oriented four-manifold, normalized by ∫ωᵢ∧ωⱼ=δᵢⱼ, deforms through positive closed triples to a hyperkähler triple while preserving all three cohomology classes. Any positive triple can first be normalized by a constant linear change.

In OpenAI's preprint Deforming hypersymplectic four-manifolds to hyperkähler triples (PDF): https://github.com/openai/math/blob/main/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026/paper.pdf

Yes, Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More on YouTube (29:20, chapter 36): https://www.youtube.com/watch?v=lckiipi69OY&t=733s

Donaldson's hypersymplectic deformation conjecture | OpenAI Math, Explained

Donaldson's hypersymplectic deformation conjecture

One of the 372 results in OpenAI's public mathematics catalogue, filed under differential geometry: 1 preprint, the latest dated 7 October 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.

In one line: OpenAI reports a proof of Donaldson's hypersymplectic deformation conjecture (differential geometry) in a preprint dated 7 October 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 3 of 4: Operator Paving, Naimark's Problem and 82 More, is on YouTube.

What OpenAI claims

Proves Donaldson's hypersymplectic deformation conjecture in a cohomology-preserving form. Every positive triple of smooth closed two-forms on a closed connected oriented four-manifold, normalized by ∫ω_i∧ω_j=δ_ij, deforms through positive closed triples to a hyperkähler triple while preserving all three cohomology classes. Any positive triple can first be normalized by a constant linear change.
OpenAI, in its public catalogue of maths results

Paper

Deforming hypersymplectic four-manifolds to hyperkähler triples
7 October 2026 · Preprint, not peer reviewed · PDF
OpenAI's abstract Every smooth normalized positive triple of closed two-forms on a closed connected oriented four-manifold admits a smooth deformation, with each cohomology class fixed, to a hyperkähler triple. This resolves Donaldson's hypersymplectic deformation conjecture.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.

Quick answers

Is it proven?

Not independently yet. OpenAI posted a preprint on 7 October 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.

What exactly does OpenAI claim?

Proves Donaldson's hypersymplectic deformation conjecture in a cohomology-preserving form. Every positive triple of smooth closed two-forms on a closed connected oriented four-manifold, normalized by ∫ω_i∧ω_j=δ_ij, deforms through positive closed triples to a hyperkähler triple while preserving all three cohomology classes. Any positive triple can first be normalized by a constant linear change.

Where is the proof?

In the paper, Deforming hypersymplectic four-manifolds to hyperkähler triples (PDF).

Is there a video?

Yes, on YouTube (29:20, chapter 36).

Donaldson's hypersymplectic deformation conjecture

What OpenAI claims

Paper

Deforming hypersymplectic four-manifolds to hyperkähler triples

Status

Quick answers

Also in

The video

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