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The charged spacetime Penrose inequality in upper-area form for Einstein-Maxwell initial data

With electric and magnetic charge, Q2=QE2+QB2Q^2=Q_E^2+Q_B^2, Penrose's heuristic suggests that a black hole of given area cannot have less mass than the Reissner-Nordstrom black hole of that area and charge. For possibly disconnected horizons the lower form m≥(rA+Q2/rA)/2m\ge(r_A+Q^2/r_A)/2, rA=A/4πr_A=\sqrt{A/4\pi}, can fail, and the appropriate statement is the upper-area inequality m≥∣Q∣m\ge|Q|, rA≤m+m2−Q2r_A\le m+\sqrt{m^2-Q^2}; Khuri, Weinstein and Yamada proved it for time-symmetric data with multiple horizons. Does the upper-area inequality hold for general asymptotically flat initial data with arbitrary second fundamental form satisfying the charged dominant energy condition, with AA the least area enclosing a weakly trapped boundary, and is equality attained only by Reissner-Nordstrom?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Mathematical general relativity
Posed by
Charged extension of Penrose's 1973 heuristic; upper-area form for multiple horizons from Marcus Khuri, Gilbert Weinstein and Sumio Yamada
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
34 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Principal (Theorem 2.3): for one-ended three-dimensional data with arbitrary KK, nonzero ADM momentum, disconnected weakly future trapped boundary and the stated decay, m≥∣Q∣m\ge|Q| and rA≤m+m2−Q2r_A\le m+\sqrt{m^2-Q^2} with AA the minimal enclosing area; positive area and timelike ADM vector are conclusions. Equality (for m>∣Q∣m>|Q|, connected outermost horizon) identifies the data as a slice of dyonic Reissner-Nordstrom. Companions: the purely electric case in every dimension n≥4n\ge4; a Kerr-Newman Penrose inequality for axisymmetric electrovacuum exteriors with zero ADM momentum that assumes the area condition A≥4πQ4+4J2A\ge4\pi\sqrt{Q^4+4J^2}; and counterexamples to the Kerr-Newman bound when JJ is the bare gravitational angular momentum and fields decay only like r−2r^{-2}. Not shown: magnetic charge in dimensions four and up, or a general rotating inequality.

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 and introduction of the principal manuscript and the abstracts of the charged and Kerr-Newman companions were read against the conjecture as the manuscript states it. Not refereed. Scope stated by the paper: dimension three, one-ended data, source-free electric and magnetic fields plus uncharged matter with its own dominant energy condition, two metric and one second-form derivative of decay; the polynomial form is asserted only for rA>∣Q∣r_A>|Q|; equality needs m>∣Q∣m>|Q| and a connected, outermost, outer-area-minimizing future horizon. The numerical proof uses the family's neutral theorem (the spacetime Penrose entry) as an input. No Lean formalization.

Sources

Changelog1 change

Discussion