Donaldson's hypersymplectic deformation conjecture

Donaldson's hypersymplectic deformation conjecture concerns hypersymplectic structures: triples of closed two-forms on an oriented four-manifold whose pointwise span is three-dimensional and positive definite for the wedge product. After a constant normalization of the intersection matrix, it asks whether such a triple can be deformed through hypersymplectic triples, with all three cohomology classes fixed, to a hyperkähler triple. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), in editions dated 23 September and 7 October 2026 and listed as its result 341, claims the conjecture in full. The family has no Lean formalization in the repository's catalogue. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.

Background

• As recounted in the introduction of the manuscript:
  • Donaldson's 2006 paper on two-forms on four-manifolds asks, as Question 3 of its Section 5.3, whether a compact oriented four-manifold carrying a positive closed triple admits a hyperkähler structure. In the same section he proposes a prescribed-volume continuity argument in the simply connected case, conditional on a priori estimates and an auxiliary involution.
  • Fine and Yao formulated the normalized deformation problem, keeping all three cohomology classes, as Conjecture 1.1 of their 2017 paper, and introduced the hypersymplectic flow by reducing the G₂-Laplacian flow on the product with a three-torus. They proved that a solution extends while the associated torsion stays bounded, and later gave an integral torsion criterion, global existence near pointwise orthogonality and subsequential convergence under further bounds.
  • Complete convergence results were known in restricted classes: Huang, Wang and Yao for triples of simple type on the four-torus; Picard and Suan for G₂ flows from Kähler Calabi–Yau data; Fine, He and Yao for T³-invariant triples on the four-torus in a symmetric normal form, and, in later work, a cohomologous linear isotopy to a hyperkähler triple when an effective circle action preserves the initial triple.
  • The topology already restricts such a triple: Bauer's bound for symplectic four-manifolds with torsion first Chern class gives b⁺ = 3, so the intersection lattice is the K3 lattice or three hyperbolic planes.
  • The manuscript uses the current estimates of the collection's companion paper on Donaldson's tamed-to-compatible conjecture, with the needed proofs included.

The original result

• Claim. Theorem 1.1 of Deforming hypersymplectic four-manifolds to hyperkähler triples states that for every smooth hypersymplectic structure ω on a closed connected oriented smooth four-manifold, normalized so that the integral of ω_i ∧ ω_j is δ_ij, there is a smooth family of hypersymplectic structures ω(t), 0 ≤ t ≤ 1, starting at ω, with each cohomology class [ω_i(t)] fixed, whose endpoint forms are parallel and self-dual for a hyperkähler metric. Any positive closed triple can first be normalized by a constant linear change.
• Corollaries, as stated. Every nonzero constant combination of the forms of a positive closed triple is a Kähler form for a parallel complex structure of some hyperkähler metric, chosen separately for each combination; and a closed connected four-manifold admitting such a triple is diffeomorphic to the standard K3 manifold or to the four-torus.
• Method, as described: the triple is handled as a pair of forms and a third form that must keep a tamer in its original class. A positive-current argument gives a criterion for a prescribed taming class. Families Seiberg–Witten wall-crossing of Li and Liu and Taubes's curve correspondence, applied over the sphere of almost complex structures defined by the triple, give a torus fibration whose singular fibres are nodal rational curves. Gibbons–Hawking and Ooguri–Vafa local models near the nodal fibres then give an auxiliary closed form of small fibre area. Two Kähler steps in the original classes, using the Kähler-cone criteria of Buchdahl, Lamari and Demailly–Păun and Yau's volume theorem, reach the hyperkähler endpoint. The manuscript states that the path it constructs is a separate question from convergence or uniqueness for the hypersymplectic flow.
• Editions. The repository's version note states that the 7 October edition clarifies the prescribed cohomology-class criterion used in the deformation argument, removes an unnecessary cone-comparison citation and updates the citation of the companion paper.
• Formalization. The repository's catalogue lists no Lean formalization for this family; the companion tamed-to-compatible result has one.
• Sources (openai/math repository, commit 7ef5c0c): Deforming hypersymplectic four-manifolds to hyperkähler triples (7 October 2026); the earlier edition dated 23 September remains in the repository.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 30 October 2026)
2026-10-07	Deforming positive triples to hyperkähler geometry, an article of the Latent Seminar blog at lowrank.github.io	Reading (exposition)	Explains the manuscript's claim that every normalized positive triple of closed two-forms on a closed connected oriented four-manifold deforms, with its three cohomology classes fixed, to a hyperkähler triple. The article covers definite pairs and the third-form problem, the prescribed-class criterion, the torus fibration obtained from the coefficient sphere, the auxiliary equal-square form chosen once for every later curve, the integrability of closed equal-square pairs, the two volume equations in the original classes, and the scope of the conclusion for individual forms.	Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs.
2026-10-16	J. Jardine, Lattice24, referee review rank_208 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	Reviews the 7 October edition and reports no gap and no minor note. The reviewer states that it traced the whole logical chain and lists explicit checks section by section, among them the topological arithmetic, an angular identity, the families Seiberg–Witten dimension and wall-degree computations, the Ooguri–Vafa mean constant and both Kähler steps. It states that it did not independently re-verify whether the cited external theorems apply as used, naming Taubes's compactness and incidence results, the Li–Liu families wall-crossing, Rivière–Tian regularity, Preiss's rectifiability theorem, Eichler's criterion and the Gross–Wilson and Ooguri–Vafa exponential estimates.	Part of an audit of the collection's manuscripts that the audit describes as complete. The review is published under CC BY 4.0.
2026-10-19	AI Proved What?, episode 341 Every triple becomes perfect?, remade version of 23 October 2026	Reading (exposition)	Video explainer of the claim that every normalized hypersymplectic triple on a closed connected oriented four-manifold deforms, with all three cohomology classes fixed, to a hyperkähler triple, with the corollary that the manifold is diffeomorphic to a K3 surface or the four-torus. The episode illustrates with its own computations that a straight-line interpolation between cohomologous symplectic forms on the four-torus can fail to stay symplectic. It states that convergence of the hypersymplectic flow, a classification of symplectic Calabi–Yau surfaces and a machine-checked proof are not claimed, that the argument relies on deep inputs including Taubes's work relating Seiberg–Witten invariants to curves and Li and Liu's wall crossing, and that the name of the conjecture follows the paper, Donaldson's original question having asked only about existence.	Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 72 pages, dated September 23, 2026, revised October 7, introduction only; under expert review: not yet; peer reviewed: not yet; checked in Lean: not for this paper, whose first step is result 342's theorem, which has Lean. The page states that the rest of the argument has no Lean proof and that it found no specialist comment and no reported error.

Notes

• The background summarizes the manuscript's own account of earlier work.
• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.