AI Proved What?

Episode 260, October 2026

How heavy must a black hole be?

Thirteen AI preprints claim the spacetime Penrose inequality in its enclosing-area form, in every dimension

Covers result 260 in OpenAI’s catalog: all 13 papers. See its connections to other results and the work it cites.

Play Episode 260: How heavy must a black hole be? A dark disk with a dashed wavy curve inside, wrapped by a glowing gold circle.

4:54. English captions available. The video loads from YouTube when you press play.

Chapters

  1. 0:00The question
  2. 1:27The claims
  3. 2:49How the proof goes
  4. 3:49Where it stands

The claim, in plain words

In 1973 Roger Penrose argued that a black hole’s mass must be at least the square root of its horizon area divided by 16π, in units where the speed of light and Newton’s constant are 1. A counterexample would have been evidence against cosmic censorship, the idea that singularities stay hidden behind horizons.

As a theorem, the inequality starts from one moment in time: the geometry of space, together with how it is bending in time. The case where space is not bending in time was proved in 2001, by Huisken and Ilmanen for one horizon and by Bray for several. Bray and Lee later extended it up to dimension 7. The general case, called the spacetime Penrose inequality, stayed open; a 2025 result proved it in three dimensions, but only with a far worse constant.

Which area? The area of an apparent horizon itself can be too big, as Ben-Dov showed, and Carrasco and Mars refuted a version with a weaker kind of trapping. On a Schwarzschild slice, a sphere past the horizon’s neck still counts as trapped toward us, yet has about 1.3 times the horizon area, over 100 times in 10 dimensions, and without limit deeper in (our computation). So the right ruler is the least area of any surface enclosing the trapped region, an idea going back to Jang and Wald.

Result 260 is thirteen OpenAI preprints, 1,355 pages, dated September 27 and October 5, 2026. The main paper, 100 pages, claims the enclosing-area spacetime Penrose inequality in every space dimension 3 or more: the mass is at least the bound built from the least enclosing area. Its hypotheses are smooth initial data with one end, the dominant energy condition, a boundary that traps light weakly in the future, and fast enough decay; from five dimensions up it also assumes a positive enclosing area. Schwarzschild attains equality, and a companion paper claims it is the only case, under extra horizon conditions. Other companions claim versions with electric charge, with rotation for axisymmetric Kerr–Newman type data under further assumptions, and in three-dimensional anti-de Sitter space under extra conditions.

What it doesn’t claim

The version with the trapped surface’s own area, which is false in general; rough data or slower decay; or cosmic censorship itself. The decay condition matters: a standard slice of the Schwarzschild black hole is exactly flat as a metric, so its energy reads zero, yet it has a horizon. Its decay is exactly the borderline the theorem excludes (our check).

Where it stands

Posted as preprints
OpenAI, 13 papers, 1,355 pages, dated September 27 and October 5, 2026
Peer reviewed
Not yet
Checked in Lean
Only an auxiliary construction (docs)

Last checked October 19, 2026. The preprints have not been revised. Lean covers only an auxiliary construction from one paper and some Schwarzschild examples (about 59,000 lines); the main inequality is not formalized. OSVerify has not reviewed them. A week after the release, Bray, Dolmen, Khuri and Schmidt posted a paper proving a special case by their own method. They write that regularizing their equations to prove the general case is, essentially, what they believe the OpenAI paper claims to have done, and that it begins with regularized versions of their equations “with very little motivation, a bit like a magic trick”. They do not say whether it is correct. They also say they cannot know whether their privately circulated notes influenced it, and they disclose using Claude for algebra.

How the proof goes

  1. Lift, reshape, compare. The main proof follows a strategy proposed by Bray and Khuri around 2010. Lift the slice to a graph one dimension higher and change its shape by a conformal factor, chosen so that the new space has nonnegative scalar curvature, its mass barely grows, and enclosing areas barely shrink. What the papers present as new is a regularized coupled system for that graph and factor, with the analysis to solve it globally.
  2. The time-symmetric finish. The time-symmetric Penrose inequality then finishes the job. In dimensions above 7, a companion paper supplies that step, building on two human preprints from 2026.

The numbers in the video

  • Ours: two black holes side by side, each of mass 1. When they are close, the area bound uses all of their mass. Once their common horizon disappears, at a separation between 1.5 and 1.55, the ratio falls toward 0.71.

Read the source

The 13 papers in this result

OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.

Transcript

Read the full script

0:00The question

How heavy must a black hole be? In 1973 Roger Penrose argued that a black hole’s mass must be at least the square root of its horizon area divided by 16π, in units where the speed of light and Newton’s constant are one. A counterexample would have been evidence against cosmic censorship, the idea that singularities stay hidden behind horizons. As a theorem, the inequality starts from one moment in time: the geometry of space, together with how it is bending in time. When space is not bending in time, it was proved in 2001, by Huisken and Ilmanen for one horizon and by Bray for several, and later up to dimension 7 by Bray and Lee. The general case, called the spacetime Penrose inequality, stayed open; a 2025 result proved it in three dimensions, but only with a far worse constant. Which area? The area of an apparent horizon itself can be too big, as Ben-Dov showed, and Carrasco and Mars refuted a version with a weaker kind of trapping. Our computer: on a Schwarzschild slice, a sphere past the horizon’s neck still counts as trapped toward us, yet already has about 1.3 times the horizon area, in 10 dimensions over 100 times, and deeper in, without limit. So the right ruler is the least area of any surface enclosing the trapped region, an idea going back to Jang and Wald.

1:27The claims

Result 260 is 13 OpenAI preprints, 1,355 pages, dated September 27 and October 5, 2026. The main paper, 100 pages, claims the enclosing-area spacetime Penrose inequality in every space dimension 3 or more: the mass is at least the bound built from the least enclosing area. Schwarzschild attains equality, and a companion claims it is the only case, under extra horizon conditions. The hypotheses: smooth initial data with one end, the dominant energy condition, a boundary that traps light weakly in the future, fast enough decay, and from 5 dimensions up, a positive enclosing area. The decay matters. Our check: a standard slice of the Schwarzschild black hole is exactly flat as a metric, so its energy reads zero, yet it has a horizon; its decay is exactly the borderline the theorem excludes. Companion papers claim versions with electric charge, with rotation for axisymmetric Kerr-Newman type data under further assumptions, and in three-dimensional anti-de Sitter space under extra conditions. Not claimed: the version with the trapped surface’s own area, which is false in general; rough data or slower decay; or cosmic censorship itself.

2:49How the proof goes

The main proof follows a strategy proposed by Bray and Khuri around 2010: lift the slice to a graph one dimension higher and change its shape by a conformal factor, chosen so that the new space has nonnegative scalar curvature, its mass barely grows, and enclosing areas barely shrink. What the papers present as new is a regularized coupled system for that graph and factor, with the analysis to solve it globally. Then the time-symmetric Penrose inequality finishes the job. In dimensions above 7, a companion paper supplies that step, building on two human preprints from 2026. Our computer shows the time-symmetric case at work with two black holes side by side: when close, the area bound uses all of their mass; once their common horizon disappears, at a separation between 1.5 and 1.55 hole masses, the ratio falls toward 0.71.

3:49Where it stands

The preprints are unrefereed and unchanged since release. Lean covers only an auxiliary construction from one paper and some Schwarzschild examples; the main inequality is not formalized. OSVerify has not reviewed them. A week after the release, Bray, Dolmen, Khuri and Schmidt posted a paper proving a special case by their own method. They write that regularizing their equations to prove the general case is, essentially, what they believe the OpenAI paper claims to have done, and that it begins with regularized versions of their equations with very little motivation, a bit like a magic trick. They do not say whether it is correct, say they cannot know whether their privately circulated notes influenced it, and disclose using Claude for algebra. If they hold, Penrose’s 1973 bound, in its enclosing-area form, becomes a theorem for general initial data in every dimension: the mass of a black hole must pay for the smallest surface that can wrap it.

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.