VibeMathedMath problems solved with AI
Sign in

The spacetime Penrose inequality: invariant ADM mass bounds the minimal area enclosing a trapped boundary, for general asymptotically flat initial data

Penrose (1973), as a test of weak cosmic censorship, argued that a black hole of horizon area AA requires total mass at least A/16π\sqrt{A/16\pi}. The time-symmetric (Riemannian) case was proved in dimension three by Huisken-Ilmanen and Bray, and in higher dimensions by Bray-Lee and Bi-Zhu. For initial data (Mn,g,K)(M^n,g,K) with nonzero second fundamental form the bound with the apparent horizon's own area fails (Ben-Dov; Carrasco-Mars), and the accepted formulation, following Bray and Khuri, uses the least area Amin⁡A_{\min} of surfaces enclosing the trapped boundary. It was known in spherical symmetry, for some cohomogeneity-one data, and with a non-sharp constant (Allen-Bryden-Kazaras-Khuri, 2025). Does every asymptotically flat initial-data set obeying the dominant energy condition, with weakly trapped boundary, satisfy m≥12(Amin⁡/ωn−1)(n−2)/(n−1)m\ge\frac12(A_{\min}/\omega_{n-1})^{(n-2)/(n-1)} for the invariant ADM mass mm, with equality only for Schwarzschild?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Mathematical general relativity
Posed by
Roger Penrose (heuristic inequality); enclosing-area initial-data formulation following Hubert Bray and Marcus Khuri
Year posed
1973
Years open
53y
Solved
2026-09-27
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
64 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.3: for smooth one-ended initial-data exteriors in every dimension n≥3n\ge3 with integrable constraint densities, dominant energy condition, weakly future trapped boundary (θ+≤0\theta^+\le0, nothing on θ−\theta^-), the stated decay, E>∣P∣E>|P| and Amin⁡>0A_{\min}>0: m=(E2−∣P∣2)1/2≥12(Amin⁡/ωn−1)(n−2)/(n−1)m=(E^2-|P|^2)^{1/2}\ge\frac12(A_{\min}/\omega_{n-1})^{(n-2)/(n-1)}, sharp; outermostness and outer area minimization are not assumed. Corollary 1.4 derives E>∣P∣E>|P|; Theorem 1.5 (n=3,4n=3,4) needs only two metric derivatives of decay. Companions: rigidity (equality only for Schwarzschild-Tangherlini slices, under horizon hypotheses), a conformal-flow Riemannian inequality in all dimensions, the boundary deformations, and a Bondi-mass version for CKS hyperboloidal data. Not shown: evolution or cosmic censorship. Priority: Allen-Bryden-Kazaras-Khuri (2025) proved the bound with a non-sharp constant.

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, Theorems 1.3 and 1.5 and Corollary 1.4 of the principal manuscript were read against the conjecture as the manuscript formulates it, and the companion abstracts were read. The proof (a coupled generalized-Jang and conformal deformation through a penalized scalar system, then a Riemannian Penrose inequality by conformal flow) was not refereed. Scope stated by the paper: smooth one-ended exteriors (designated-end versions for finitely many ends in dimensions three and four), weakly future trapped boundary, dominant energy condition, decay g−δ=O6(r−q)g-\delta=O_6(r^{-q}), K=O5(r−1−q)K=O_5(r^{-1-q}) with (n−2)/2<q<n−2(n-2)/2<q<n-2; for n≥5n\ge5 positive enclosing area is assumed, while in dimensions three and four weaker decay suffices and positivity and timelikeness are conclusions. Equality is a separate companion with extra horizon hypotheses. The family's only Lean challenge (CKSBondiPenrose) covers the Cha-Khuri-Sakovich end replacement and Schwarzschild equality examples, not this inequality, so the entry is not lean-checked.

Sources

Changelog1 change

Discussion