Claimed
On the twenty-sixth of October, OpenAI announced three new discoveries. One in mathematics. One in computer science. One in physics. The biggest: its system, in OpenAI's words, "has recently proved that the ETH is false."
The claim
OpenAI says its AI has refuted the Exponential Time Hypothesis, an assumption computer science has built on since 1999. The three experts it asked to organise the examination say their involvement "should not be taken as certification of the claim."
On 26 October 2026 OpenAI announced three discoveries: one in mathematics, one in computer science and one in physics. The biggest is the computer science. In OpenAI's words, its system "has recently proved that the ETH is false." Its page dates the find: "Our AI found its natural language disproof for the ETH on October 16th, 2026".
“has recently proved that the ETH is false.”
OpenAI, "Working with academic communities to build scientific understanding", 26 October 2026 · read 27 October 2026
ETH here is the Exponential Time Hypothesis, not the university and not the currency. It is a working assumption that some problems cannot be solved without exponential time: the time doubles, and doubles again, as the problem grows. Impagliazzo and Paturi stated it for 3-SAT (CCC 1999; JCSS 2001): 3-SAT needs time 2^Ω(n). In the reviewers' words, "ETH has helped organize decades of research". This piece treats the announcement as a claim under examination. It says neither proven nor refuted.
What anyone can check
The computer science paper is public. So is a formal proof of its main result, written in Lean, a language a machine can verify. The Lean proof establishes a slightly weaker bound than the paper, still enough, if correct, to refute the hypothesis. The reviewers have not certified it. The mathematics paper is public too.
In the repository: "The Exponential Time Hypothesis is False" (25 October 2026) with its Lean formalisation (lean/docs/378.md), and "Local smoothing for the Euclidean wave equation" with its own (lean/docs/079.md). Whether the Lean statement is exactly the textbook hypothesis is what the reviewers are examining. On the mathematics, the announcement says Stein's restriction conjecture is resolved "in all dimensions"; the repository's history says "Stein's spherical restriction estimates throughout the interior of the conjectured range". Both are printed here; neither is judged.
What is withheld
The physics is not public. OpenAI says the physicists it works with decided to hold off on releasing the AI's output and to write it up themselves. A group at the Institute for Advanced Study has called on OpenAI to release it "without delay."
OpenAI claims existence proofs for Yang–Mills theory, quantum chromodynamics and a completion of the Standard Model, not the mass gap. OpenAI's own sentence: "the group has decided to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves." The group at the Institute, AGMAI, the Advisory Group on Mathematics and Artificial Intelligence, answered the same day: "We call on OpenAI to make public without delay the natural-language and Lean outputs behind its announced existence proofs for Yang–Mills theory, quantum chromodynamics, and a completion of the Standard Model."
The checkers
OpenAI asked three computer scientists to organise the examination. Two of them first proposed the hypothesis. All three say their role is not a verdict: "Our involvement should not be taken as certification of the claim."
They are Russell Impagliazzo and Ramamohan Paturi, the hypothesis's original authors, and Ryan Williams, writing to the theoretical computer science community on 26 October 2026. Their list of resources puts Anthropic's refutation of the hypothesis beside OpenAI's.
The repository's count
There is also a count. On 7 October OpenAI's repository withdrew three papers after a sign error and revised 14 more. It said about 42% of its main results, 300 of 719, were machine-checked. The other 419 were not.
Source: OpenAI, github.com/openai/math, history.md and README.md, read at source 27 October 2026
The repository's history of 7 October: "a sign error invalidates a stabilization-trace cancellation argument" in one paper, withdrawn with the two that depended on it. The three were "Algebraicity of Weil classes on split abelian eightfolds", "Algebraicity of Kuga–Satake Correspondences for K3 Surfaces" and "The rational Hodge conjecture for products of K3 surfaces". Each now carries a notice explaining the gap.
Source: OpenAI, github.com/openai/math, history.md and README.md, read at source 27 October 2026
After the 26 October additions the README still says "~42%" with no new count, so the figures are dated 7 October. The 419 without a formal proof is our arithmetic: 719 less 300. The catalogue on 26 October: "721 manuscripts organized into 373 families".
The second claim
The next day Anthropic posted its own paper claiming the same result, found, it says, on the same day: 16 October. Anthropic makes Claude. A disclosure: Token Runner is built with Claude.
Anthropic's paper on Hexagon, 27 October 2026, is titled "A 2^{o(n)} Time Algorithm for 3-SAT": an algorithm faster than exponential time, which is what refuting the hypothesis would take. The paper says the core algorithm "was found on October 16 by an internal Claude research model with 14M output tokens and no human input", and that the result is formalised in Lean 4. And it names its rival: "Independently and concurrently, an internal model at OpenAI claims to have proven a similar result." OpenAI's page gives the same date. Who was first is not said here.
Certified by no one yet
Two companies, one claim, both machine-checked, they say.
Certified by no one yet. The people who will decide are the ones now reading the proof.
What to watch: the examination the three reviewers are organising, any release of the physics outputs the Institute's group asked for, and the repository's next count of how many of its results carry a machine-checked proof.
Every quote from OpenAI's announcement, its repository, the reviewers' own statement, the Institute for Advanced Study group and Anthropic's paper.
Update
28 October 2026: we added that the Lean proof establishes a slightly weaker bound than the paper (source: github.com/openai/math, commit 42907c9).
Sources
- OpenAI, "Working with academic communities to build scientific understanding", 26 October 2026, read at source by Token Runner on 27 October 2026: "Our AI found its natural language disproof for the ETH on October 16th, 2026"; its system "has recently proved that the ETH is false"; on the physics, "the group has decided to hold off on releasing the AI-produced artifacts and instead work on writing it up themselves."
- OpenAI, github.com/openai/math, history.md and README.md, read at source by Token Runner on 27 October 2026: the 26 October additions ("The Exponential Time Hypothesis is False", with its Lean formalisation at lean/docs/378.md; "Local smoothing for the Euclidean wave equation", lean/docs/079.md); the 7 October entry: three papers withdrawn after "a sign error invalidates a stabilization-trace cancellation argument", 14 manuscripts revised, and "the total percentage of top-line results formalized to 300 / 719 = ~42%"; the README's "721 manuscripts organized into 373 families". Token Runner's own table of the repository's counts.
- Russell Impagliazzo, Ramamohan Paturi and Ryan Williams, "Message to the TCS Community on ETH", 26 October 2026, read at source by Token Runner on 27 October 2026: "Our involvement should not be taken as certification of the claim."; "ETH has helped organize decades of research". The Exponential Time Hypothesis: Impagliazzo and Paturi, CCC 1999 and JCSS 2001 (the original papers, checked by Token Runner).
- AGMAI, the Advisory Group on Mathematics and Artificial Intelligence (Institute for Advanced Study), 26 October 2026: "We call on OpenAI to make public without delay the natural-language and Lean outputs behind its announced existence proofs for Yang–Mills theory, quantum chromodynamics, and a completion of the Standard Model."
- Anthropic, "A 2^{o(n)} Time Algorithm for 3-SAT", Hexagon, 27 October 2026, read at source by Token Runner: "The core algorithm was found on October 16 by an internal Claude research model with 14M output tokens and no human input."; "Independently and concurrently, an internal model at OpenAI claims to have proven a similar result." Anthropic makes Claude. A disclosure: Token Runner is built with Claude.
- Token Runner's own table of the repository's counts, as a data file to download (CC BY 4.0).