The asymptotically anti-de Sitter Penrose inequality for three-dimensional initial data with spherical conformal infinity
With cosmological constant , the Schwarzschild-anti-de Sitter relation and Penrose's 1973 argument suggest that asymptotically hyperbolic initial data with a trapped boundary of area have mass, in the sense of Wang and Chrusciel-Herzlich, at least with . Even the time-symmetric case was known only for special classes: graphs (Dahl-Gicquaud-Sakovich, de Lima-Girao), small perturbations of Schwarzschild-anti-de Sitter (Ambrozio; Khuri-Kopinski), and Neves showed inverse mean curvature flow cannot be used directly. Does every three-dimensional asymptotically anti-de Sitter initial-data set with spherical conformal infinity, the dominant energy condition and a weakly trapped boundary satisfy with the minimal enclosing area, with equality only for Schwarzschild-anti-de Sitter?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Mathematical general relativity
- Posed by
- Penrose's 1973 heuristic in the asymptotically anti-de Sitter setting; studied as a conjecture by Dahl, Gicquaud and Sakovich, Ambrozio, and Khuri and Kopinski, among others
- Year posed
- —
- Years open
- —
- Solved
- 2026-10-05
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 40 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1 (principal): under those hypotheses with the full enclosing-area infimum, sharp; no maximality, compact topology or connectedness assumption. Equality on the stated horizon subclass identifies the original data as a spacelike hypersurface in Schwarzschild-anti-de Sitter with the same mass. Companions: the maximal case with possibly disconnected boundary, and a local inequality for small maximal vacuum conformal perturbations of Schwarzschild-anti-de Sitter using the horizon's own area. Not shown: higher dimensions, toroidal or higher-genus conformal infinity (the Lee-Neves regime), or timelikeness of the mass as a conclusion.
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 manuscripts are authored 'OpenAI' and name no human author. The family has thirteen manuscripts (September 27 and October 5, 2026); the October papers cite the September neutral theorem as an input, consistent with the README's note that some results build on earlier model results.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the principal manuscript and the two asymptotically hyperbolic companion abstracts were read against the conjecture as the manuscript states it. Not refereed. Scope stated by the paper: dimension three, one spherical asymptotically anti-de Sitter end with stated decay and integrability, compact weakly future outer trapped boundary, and a future-timelike metric mass four-flux, which is assumed rather than derived; equality only on the connected, outermost, outer area-minimizing horizon subclass. The paper says it makes no evolution or cosmic-censorship claim. No Lean formalization.