Result 264, Mathematical physics

Strong cosmic censorship near two-ended Kerr data

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.

Proof

The bigger picture

Why it matters

Can the evolution predicted by Einstein's equations continue beyond the region fixed by initial data? Near certain rotating black holes, the manuscript reports that generic small perturbations block even a relatively weak kind of continuation.

What changes?

For each fixed Kerr black hole with positive mass and nonzero rotation magnitude smaller than its mass, the claim concerns nearby smooth vacuum initial data: no matter, two complete asymptotically flat ends, and no imposed symmetry. A dense G-delta set, a topological notion of genericity, yields maximal globally hyperbolic developments, the largest spacetimes determined by those data, with no future extension having a continuous, nondegenerate metric and locally square-integrable weak connection. Candidate extensions need not obey the vacuum equations.

What does that help mathematicians do?

Smallness is required only in the tenth weighted seminorm, a measure of derivative size and behavior at infinity, while genericity uses the weighted smooth topology testing every finite order. The connection describes how directions change across spacetime; square-integrability permits considerable irregularity while bounding its squared size over local regions. Ruling out extensions at this weak regularity strengthens the companion manuscripts' C1 and C2 barriers, narrowing the possible ways predictability could break down near these Kerr backgrounds.

Are there practical applications?

Its immediate value is foundational for general relativity: the claim supports a local form of strong cosmic censorship by excluding continuations with a specified, relatively low regularity for generic nearby data. It does not exclude every continuous extension or settle censorship for all black holes. Its scope is a neighborhood of each fixed rotating subextremal Kerr bridge.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

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

September 23, 2026 59 pages

For each fixed rotating subextremal Kerr background with mass M > 0 and rotation 0<∣a∣<M0\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 C0∩Wloc1,2C^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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026,
  author = {{OpenAI}},
  title = {{Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026/paper.pdf}{OAI:Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026}},
  year = {2026}
}

Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes

September 23, 2026 68 pages

For each fixed Kerr spacetime with mass M > 0 and rotation 0<a<M0\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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026,
  author = {{OpenAI}},
  title = {{Generic $C^1$ Future Inextendibility Near Rotating Subextremal Kerr Spacetimes}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026/paper.pdf}{OAI:Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026}},
  year = {2026}
}

Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility

September 23, 2026 151 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026,
  author = {{OpenAI}},
  title = {{Quantitative Near-Kerr Evolution and Generic $C^2$ Future Inextendibility}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026/paper.pdf}{OAI:Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.