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Strong cosmic censorship near Kerr in Christodoulou's square-integrable-connection formulation

Penrose's strong cosmic censorship conjecture asserts that for generic asymptotically flat vacuum initial data the maximal globally hyperbolic development is inextendible, so that general relativity is deterministic. Kerr's smooth inner Cauchy horizon makes the regularity of extensions essential: Dafermos and Luk proved that near Kerr the development does extend continuously (C0C^0) across part of the Cauchy horizon, so the C0C^0 formulation fails. Christodoulou proposed the formulation at the threshold of locally square-integrable connection coefficients. For generic smooth vacuum data near the two-ended data of a rotating subextremal Kerr black hole (0<∣a∣<M0<|a|<M), is the maximal globally hyperbolic development future inextendible as a spacetime with continuous metric whose connection is locally square-integrable?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
General relativity; Einstein vacuum equations; Cauchy horizons
Posed by
Roger Penrose (strong cosmic censorship, 1979); Demetrios Christodoulou, The Formation of Black Holes in General Relativity (2009), p. 9 (square-integrable connection formulation)
Year posed
1979
Years open
47y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
58 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For each fixed M>0M>0, 0<∣a∣<M0<|\mathfrak a|<M there is ε>0\varepsilon>0 such that, among smooth complete two-ended vacuum data dd with tenth weighted seminorm p10(d−d∗)<εp_{10}(d-d_*)<\varepsilon from the Kerr bridge data, those whose full maximal globally hyperbolic development has a future extension with continuous nondegenerate metric in Wloc1,2W^{1,2}_{\mathrm{loc}} form a meagre set in the weighted smooth topology. No symmetry is imposed and extensions need not be vacuum. Companions prove the weaker C2C^2 and C1C^1 versions. This is strong cosmic censorship in Christodoulou's formulation near Kerr only; it is not the global conjecture, and it does not treat one-ended or nonrotating data.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has three manuscripts dated the same day; the principal one relies on intermediate results of the C1 and C2 companions but not on their headline theorems. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction, Definition 1.1 and Theorem 1.2 of the principal manuscript were read against strong cosmic censorship as Penrose and Christodoulou formulate it, as the manuscript cites them; the companions' abstracts and introductions were skimmed. The proofs were not refereed. No Lean formalization: none of the three manuscripts is in lean/formalization.yaml and lean/docs/264.md does not exist at the pinned commit. Scope the paper itself states: genericity is local, inside a neighbourhood of each fixed Kerr bridge whose size depends on M and a; no extremal or nonrotating case, no uniformity as a tends to 0 or M, and no genericity outside these neighbourhoods; inextendibility is not asserted to be open. Only two-ended data are treated, not one-ended collapse data.

Sources

Changelog1 change

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