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Strong cosmic censorship near two-ended Kerr data

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves local strong cosmic censorship near each fixed rotating subextremal Kerr bridge. A dense Gδ subset of a weighted smooth neighborhood of smooth complete two-ended asymptotically flat vacuum data has full maximal globally hyperbolic developments with no future continuous nondegenerate extension whose weak connection is locally square-integrable. No symmetry is imposed; extensions need not satisfy the vacuum equations.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

The paper states

Theorem 1.1 (Local C^2 strong cosmic censorship)

In Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility, 23 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The other possible matches are Theorem 1.2 in Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime () and Theorem 2.3 in Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes ().

For every fixed M>0 and 0<\mathfrak a<M, there exists \epsilon>0 such that the data in

\mathcal U_\epsilon =\{d\in\mathcal D:p_{10}(d-(h_*,K_*))<\epsilon\}

whose full maximal globally hyperbolic vacuum developments admit a C^2 future extension form a meagre subset of \mathcal U_\epsilon. Equivalently, the future-inextendible data contain a countable intersection of relatively open dense subsets of \mathcal U_\epsilon.

Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Three current papers, each readable here as a page beside its PDF. Two explanations by others.

Papers

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Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime

23 September 2026, current edition.

Read it herePDF

For each fixed rotating subextremal Kerr background with mass M > 0 and rotation \(0\lt |\mathfrak a|\lt M\), we prove that the smooth vacuum data near its complete two-ended bridge whose full maximal globally hyperbolic development admits a future \(C^0\cap W^{1,2}_{\mathrm{loc}}\) extension form a meagre set in the weighted smooth topology. The extension metric is continuous and nondegenerate, and its weak connection is locally square-integrable. The neighborhood requires smallness of only the tenth weighted seminorm, while the topology tests every finite order.

Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes

23 September 2026, current edition.

Read it herePDF

For each fixed Kerr spacetime with mass M > 0 and rotation \(0\lt \mathfrak a\lt M\), we prove that a dense Gδ set of nearby smooth, complete two-ended vacuum data has a maximal globally hyperbolic development with no future C1 extension. The neighborhood and genericity are defined in a weighted smooth topology, and ambient extensions may be nonvacuum.

Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility

23 September 2026, current edition.

Read it herePDF

For every fixed rotating subextremal Kerr bridge, we prove that smooth vacuum data admitting a C2 future extension of their full maximal globally hyperbolic development form a meagre set in a neighborhood defined by one finite-order seminorm. The topology allows arbitrary symbol-bounded asymptotically flat tails and imposes no symmetry.

Lean statements

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Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

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Explanations by others

Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.

  • seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source

    Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.

  • Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source

    Chinese guided reading of this family that compares the manuscripts’ main theorems with their abstracts and flags boundary conditions and limit orders that change the physical reading; drafted with Codex and not a re-proof.

What these papers cite in the release, and what cites them

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Within this family

Papers elsewhere in the release that this family’s papers cite

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Papers elsewhere in the release that cite this family’s papers

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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Identify the theorem the catalogue describes

  • Family 264 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 264 The release has no Lean statement for this family. Open the form