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OpenAI's release of 372 result families in mathematics

OpenAI released 719 manuscripts in 372 result families produced by an unreleased model; about 42% are formalized in Lean, none is peer reviewed, and three were withdrawn a day later over a sign error.

問題
Claims include a negative answer to Hilbert's tenth problem over Q\mathbb{Q}, the irrationality exponent of π\pi, the Unique Games Conjecture and L=RL=BPL\mathrm{L} = \mathrm{RL} = \mathrm{BPL}
貢献
OpenAI internal model · About 4,000 problems posed; about 3 hours of compute per result, per OpenAI
検証
Lean formalizations for about 42% of top-line results
経緯
More than 100 problems announced 2026-09-21 → 719 manuscripts released 2026-10-06 → 3 withdrawn 2026-10-07
出典
OpenAI announcementResults repositoryEarlier announcement (2026-09-21)
未検証≈42% Lean-formalized · none peer reviewed

The problem

On 21 September 2026 OpenAI said a model in training since late August had "resolved more than 100 long-standing open problems", without giving a list or proofs, and set up an independent advisory group on mathematics and AI hosted at the Institute for Advanced Study.

What AI did

On 6 October OpenAI published the results: 719 manuscripts in 372 result families, from about 4,000 problems posed, at an average of about 3 hours of ChatGPT Pro compute per result. Among the claims are a negative answer to Hilbert's tenth problem over Q\mathbb{Q}, the irrationality of Catalan's constant, that the irrationality exponent of π\pi is 2, a result on the Unique Games Conjecture, L=RL=BPL\mathrm{L} = \mathrm{RL} = \mathrm{BPL}, that the chromatic number of the plane is at least 6, and an improved upper bound for the unit distance problem. Some of these have their own entries on this site.

How it was checked

OpenAI's repository says about 42% of top-line results are formalized in Lean (300 of 719 manuscripts), and adds that "some of the unformalized results could have issues". None of the results has been independently peer reviewed, and the model is not public, so they cannot be reproduced.

Limitations

On 7 October a sign error invalidated one manuscript (on algebraicity of Weil classes) and two that depended on it, and 14 others were revised. Andrew Sutherland told Scientific American to "expect some of the proofs to contain mistakes, possibly serious ones". We grade the release "unverified" as a whole; individual results will be upgraded as they are checked.

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