OpenAI Says Its AI Refuted a 27-Year-Old Computer Science Bet. Hours Later, Anthropic's Claude Did Too.

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On October 26, 2026, OpenAI said three of its models' results were ready to publish: a proof of Sogge's local smoothing conjecture in every dimension, which would also settle the Kakeya and Stein restriction conjectures; a refutation of the Exponential Time Hypothesis; and a rigorous existence proof for four-dimensional gauge theories including quantum chromodynamics 1. OpenAI called the manuscripts "a form of raw research data" and, for the first time, named outside experts to shepherd each result 1.

The computer-science claim travels furthest. The hypothesis, proposed in 1999 by Russell Impagliazzo, Ramamohan Paturi and Francis Zane, holds that no algorithm solves 3-SAT meaningfully faster than testing assignments one at a time 2. It is the floor under hundreds of conditional hardness results in graph theory, computational biology, databases and computational geometry 1. OpenAI's manuscript gives a deterministic algorithm that exactly counts satisfying assignments of fixed-width formulas in sub-exponential time and polynomial space, and the 3-SAT case is what refutes the hypothesis 3. The model's natural-language disproof came on October 16, 2026, the Lean formalization the next day, and both were published on October 26 1.

Checking is the part OpenAI handed over. The collaboration is led by Impagliazzo and Paturi — the men who proposed the hypothesis — with Ryan Williams of MIT; at the group's request all raw data is public, and OpenAI agreed to fund a conference on what this does to the field 14. Participants are unpaid, no NDAs are signed, and OpenAI states it seeks no academic credit for its models 1.

The result carries its own ceiling. The improvement is asymptotic: the new bound beats brute-force search only past roughly 10^10,000 variables, while brute force itself gives out around 100, so the paper's own description is that the result "says nothing about solving formulas in practice" 2. What changes is the accounting. Claude's literature search, cited in Anthropic's paper, found more than 800 papers whose results depend on ETH or on SETH, which implies it 2.

Hours later came the twist. Anthropic published "A 2^o(n) Time Algorithm for 3-SAT", with the author line reading "Claude, Anthropic". Anthropic researcher Levent Alpöge said the core algorithm was found on October 16 by an internal Claude research model using 14 million output tokens with no human input in the discovery, and that Anthropic researchers then worked with Claude on the presentation 25. The paper acknowledges that an internal model at OpenAI "claims to have proven a similar result" 2. Its Lean development is 1,050 modules built from scratch on Mathlib, checked by Lean's kernel and by an independently built checker called nanoda; the same paper states the development was written and reviewed by AI models and that "at the time of writing no person is known to have checked it" 2.

The reason checking matters is in OpenAI's own release record. On October 7, 2026 it withdrew three manuscripts after a sign error invalidated a stabilization-trace argument and the construction two dependent papers had used 3.

The other two results have their own edges. The mathematics claim settles Kakeya in all dimensions a few months after Hong Wang won the Fields Medal for the three-dimensional case 6. The physics claim is an existence proof that includes pure Yang-Mills theory as a subproblem: the mass gap that the Clay Millennium Prize problem also asks for stays open, and that collaboration has held back the AI-produced artifacts to write the work up itself 1.

P versus NP stays open. OpenAI's Greg Brockman, posting the news, added: "p still probably doesn't equal np" 7.

Reporting cut off at October 27, 2026. If the verification group reports, that is the next episode.