Strong cosmic censorship near Kerr black holes

Strong cosmic censorship is Penrose's conjecture that, for generic initial data, the maximal globally hyperbolic development of the Einstein equations cannot be continued, so that the data determine the spacetime. In the exact Kerr solution of a rotating black hole, the development has a smooth inner Cauchy horizon beyond which it can be continued; the conjecture asserts that this disappears under generic perturbations. The regularity allowed for a continuation is part of the question, since continuous extensions are known to exist in some settings. Three manuscripts in OpenAI's public mathematics collection (github.com/openai/math), all dated 23 September 2026, claim local, generic forms of the conjecture near each fixed rotating subextremal Kerr black hole, for extensions of class C², C¹ and continuous with locally square-integrable connection. The family has no Lean formalization in the repository's catalogue. This page summarizes the published claims 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 introductions of the manuscripts:
- Penrose's cosmic-censorship programme connects gravitational collapse with predictability. The mass-inflation analysis of Poisson and Israel and Ori's charged and rotating models explain how blueshift at the inner horizon can produce a singularity with comparatively weak tidal effects.
- For the spherically symmetric Einstein–Maxwell–scalar-field system, Dafermos proved stability and instability results, including a continuous metric together with an unbounded Hawking mass, and Luk and Oh proved generic C² future inextendibility for two-ended asymptotically flat data.
- Christodoulou identified square-integrable connection coefficients as a natural threshold for formulating strong cosmic censorship. Dafermos and Luk proved that suitably controlled interior data near Kerr have a development with a continuous extension across part of a Cauchy horizon. Sbierski developed holonomy obstructions to locally Lipschitz extensions; Gurriaran and Luk and Sbierski excluded locally Lipschitz extensions in vacuum interiors under smallness and asymptotic assumptions; Cameron and Sbierski obtained obstructions to extensions in C⁰ ∩ W^1,s_loc, s > 1, in spherical symmetry.
- For the black-hole exterior, Häfner, Hintz and Vasy proved linear stability of Kerr, nonlinear stability for slow rotation was established by Klainerman, Szeftel, Giorgi and Shen, and Hintz proved nonlinear exterior stability throughout the subextremal range for a class of data that, as the C² manuscript notes, does not include the slowly decaying asymptotic tails it allows.

The original results

Setting. Each manuscript fixes a Kerr solution with mass M > 0 and rotation 0 < |a| < M and considers smooth complete two-ended asymptotically flat vacuum data near its two-ended bridge data, in a weighted smooth topology in which only one finite-order seminorm must be small. No symmetry is imposed, and the extensions are not required to satisfy the vacuum equations. "Generic" means a residual set in the sense of Baire category.
Claim (C²). The main theorem of Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility states that, for every such Kerr solution, the data in a neighbourhood whose full maximal globally hyperbolic vacuum developments admit a C² future extension form a meagre set. The manuscript states that this is not a global assertion for all vacuum data and that no estimate uniform as the rotation tends to 0 or to M is claimed.
Claim (C¹). Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes states that a dense G_δ set of nearby data has developments with no future C¹ extension.
Claim (square-integrable connection). The main theorem of Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime states that the data whose development admits a future extension with a continuous nondegenerate metric in W^1,2_loc, that is, with locally square-integrable weak connection coefficients, form a meagre set in a neighbourhood of the Kerr data. Since C² ⊂ C¹ ⊂ C⁰ ∩ W^1,2_loc, this is the strongest of the three conclusions. The manuscript states that no nonrotating or extremal case, no genericity outside these neighbourhoods and no openness of the inextendible set is asserted.
Method, as described: each proof is a Baire category argument. A possible extension supplies a test from a countable family of closed conditions on finite observations; near any datum that passes a test, a small, late, high-frequency signal is placed so that blueshift near the Cauchy horizon makes the test fail. The C² manuscript uses a large tidal curvature measurement; the weaker regularities use holonomy around small loops and, at the square-integrable threshold, holonomy averaged over families of observers. The C² manuscript proves its own exterior and interior control for its data class, combining the stationary analysis of Häfner, Hintz and Vasy with energy estimates on the asymptotically flat ends, and states that a global nonlinear stability statement for this data class is not assumed. The square-integrable manuscript uses intermediate results of the two companions but not their headline theorems.
Sources (openai/math repository, commit 7ef5c0c): Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime; Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes; Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility; all dated 23 September 2026.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 30 October 2026)
2026-10-07	Kerr cosmic censorship: curvature, holonomy, and weak connections, an article of the Latent Seminar blog at lowrank.github.io	Reading (exposition)	Explains Cauchy evolution and extension regularity, the data neighbourhood and the three conclusions, why curvature alone does not suffice, the passage from a possible extension to a finite experiment and averaging at the square-integrable threshold, with proof outlines on the finite witnesses of the C² category argument, the boost powers in the C¹ argument and stopping before averaging for weak connection coefficients.	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 reviews rank_368 (square-integrable connection), rank_162 (C¹) and rank_223 (C²) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	Reports no gap in any of the three manuscripts. Its minor notes are limits on verification, not demonstrated errors: the C² review states that the exterior analysis amounts to global nonlinear stability of the full subextremal Kerr exterior, built on the linear input of Häfner, Hintz and Vasy, and is too compressed to verify line by line; the square-integrable review says the same of its quadratic energy estimate and records that it checked the averaged holonomy obstruction; the C¹ review notes that one pointwise decay step is imported from a companion manuscript.	Part of an audit that its README describes as complete for the collection's manuscripts without a Lean formalization. The reviews are published under CC BY 4.0.
2026-10-17	AI Proved What?, episode 264 Is the future determined?, remade version of 22 October 2026	Reading (exposition)	Video explainer of the claims and their history. The episode states that the papers do not claim cosmic censorship globally, for one-ended collapse data, for Schwarzschild or extremal black holes, for an open set of data or with a positive cosmological constant, and that genericity is in the sense of Baire category only. It identifies as the part most in need of expert checking the 151-page foundation paper's own exterior control at every subextremal spin, which relies on an unrefereed linear stability preprint by Häfner, Hintz and Vasy.	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 preprints: OpenAI, three papers, 59, 68 and 151 pages, dated September 23, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: no. The page states that the preprints have not been revised and that it found no comment from specialists in black hole interiors.

Notes

The background summarizes the manuscripts' 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.