The classification of closed Einstein four-manifolds with positive sectional curvature

A Riemannian metric is Einstein if $\mathrm{Ric}=\lambda g$ . The round $S^4$ , the round $\mathbb{RP}^4$ and the Fubini-Study $\mathbb{CP}^2$ are closed Einstein four-manifolds with strictly positive sectional curvature. Yang (2000, Conjecture 1) conjectured that these are the only ones. Earlier results needed extra hypotheses: Berger's strict quarter pinching (1961), quantitative pinching bounds of Yang, Costa and Cao-Tran, Gursky-LeBrun's nonzero positive-definite intersection form, and in 2026 topological conditions of Cheng ( $\chi\le3$ ) and Gursky-Malchiodi ( $2\chi-3|\tau|\le4$ ). Is every connected closed Einstein four-manifold with $K>0$ , after positive rescaling, isometric to round $S^4$ , round $\mathbb{RP}^4$ or Fubini-Study $\mathbb{CP}^2$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Riemannian geometry; Einstein metrics in dimension four
Posed by: DaGang Yang, Rigidity of Einstein 4-manifolds with positive curvature (Invent. Math., 2000), Conjecture 1
Year posed: 2000
Years open: 26y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 38 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: a connected smooth closed Einstein four-manifold with strictly positive sectional curvature, with no orientability assumption, is after positive rescaling isometric to round $S^4$ , Fubini-Study $\mathbb{CP}^2$ or round $\mathbb{RP}^4$ . The core step is half-conformal flatness ( $W^+\equiv0$ or $W^-\equiv0$ ) from a weighted Hessian estimate and moment balance of the two Weyl blocks. Companions: a closed Einstein four-manifold with $\mathrm{Ric}=3g$ , $K\ge0$ and one zero-curvature plane has universal cover $S^2(1/\sqrt3)\times S^2(1/\sqrt3)$ , giving the full nonnegative classification; and a simply connected closed four-manifold with $K\ge0$ and small scale-invariant $L^2$ trace-free Ricci energy is diffeomorphic to $S^4$ , $\mathbb{CP}^2$ or $S^2\times S^2$ . Not shown: anything in higher dimensions or for complete noncompact metrics.

What the AI did

The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has three manuscripts: the strictly positive classification (September 23, 2026), a zero-plane rigidity theorem extending it to nonnegative curvature (October 4) and an L2 topological gap that uses that classification as a stated premise (October 5). The principal manuscript ships an exact-arithmetic verifier for its finite polynomial sign certificates; its coverage note says it checks only those finite claims, not the geometric or analytic reductions.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of 'Positively curved Einstein four-manifolds' were read against Yang's Conjecture 1 as the manuscript cites it; they match. The Lean challenge lean/ComparatorChallenges/EinsteinFour.lean (solution module OAI.Geometry.EinsteinFour.Classification, which exists at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/348.md. Its statement OAI.PositiveEinsteinFour.classification was read here: for every connected compact Hausdorff smooth four-manifold with a smooth metric that is Einstein (Ricci computed from chart Christoffel symbols) and has positive sectional curvature, some positive rescaling is distance-isometric to the round 4-sphere, Fubini-Study CP2 or round RP4. That is the headline claim. The formalization was not rebuilt here. The proof uses finite polynomial certificates checked by a shipped script, which was not run here.

The classification of closed Einstein four-manifolds with positive sectional curvature

What was actually shown

What the AI did

Verification

Sources

Discussion