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The spacetime Penrose inequality and enclosing area

OpenAI · September 27, 2026

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Abstract

We prove the sharp spacetime Penrose inequality for smooth one-ended initial-data exteriors in every spatial dimension n ≥ 3, under the dominant energy condition, weak future trapping, the stated differentiated decay and positive enclosing area. These hypotheses force the ADM energy-momentum to be future timelike. Area is the infimum over full enclosing cuts in the original metric. In dimensions three and four, two metric derivatives and one tensor derivative suffice, and both positive enclosing area and future timelikeness follow from the hypotheses.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-23) and not checked by a person. May be wrong or incomplete. Report an error.

Follow-up works

  • H. Bray, T. Dolmen, M. Khuri, D. Schmidt, Proof of the Penrose Conjecture for Maximally Parameterized Initial Data Sets (arXiv:2610.20559, 14 Oct 2026). Proves the spacetime Penrose inequality in all dimensions n ≥ 3 for "maximally parameterized" data via generalized Schoen–Yau identities, cites this paper as [27], and says "we believe this is what the OpenAI paper [27] posted last week on October 6, 2026 claims to have done." Its Appendix C ("AI statement") says this paper starts from regularized versions of their equations with little motivation and that "the mode of release concerns us"; it does not say they checked the proof.

Comments and discussion

  • Steve McCormick (@Quasilocal), post on X (2026-10-10 to 2026-10-12, thread). Says that, judging from an AI summary rather than a careful reading, the proof is "this same Jang equation approach" close to Bray and Khuri's idea for controlling horizon area, and that another AI condensed the main result from 171 to about 32 pages.

Formalization

Digests and expositions

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Elsewhere

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Cite

@misc{OAI:The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026,
  author = {{OpenAI}},
  title = {{The spacetime Penrose inequality and enclosing area}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026/paper.pdf}{OAI:The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/260.md. The Lean statements have not been checked by people against the paper's claims.

Family 260: Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions

Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions. It bounds invariant ADM mass below using minimum enclosing area, with equality rigidity under additional horizon hypotheses. Charged upper-area bounds treat dyonic three-dimensional data; the higher-dimensional purely electric extension uses the matched neutral theorem.

Other papers in this family: