OpenAI Says Its AI Disproved the Exponential Time Hypothesis

SAN FRANCISCO, Oct. 26 — OpenAIOpenAI said on Monday that an internal AI system had disproved the Exponential Time Hypothesis, a conjecture about a basic logic problem that researchers have used in hundreds of papers to argue that other problems cannot be solved quickly. In the same post it claimed two more results: a proof in every dimension of the local smoothing conjecture, a question about how waves focus, and a proof that quantum chromodynamics, which describes how quarks and gluons interact through the strong force, exists as a mathematically rigorous theory.

All three are claims under review. OpenAI says the disproof was checked in Lean and will be verified by a group led by Russell Impagliazzo and Ramamohan Paturi of the University of California, San Diego, and Ryan Williams of the Massachusetts Institute of Technology.LeanLean is “a programming language that allows mathematical proofs to be checked by a computer,” as OpenAI described it on Oct. 6. It checks each step against the statement it is given; whether that statement says what the paper claims is for people to judge, as The Quantum Insider and Anthropic’s own Lean files both note. In an open letter, the three computer scientists named to review it told colleagues that a result of this significance requires “careful, independent scrutiny,” and wrote: “Our involvement should not be taken as certification of the claim.”

If it holds, the disproof removes an assumption that, the reviewers wrote, “has helped organize decades of research” on which computing problems are hard: “If the claim is correct, it will change an important part of theoretical computer science.” OpenAI says that turning it into practical improvements “will require significant further research.”

Our involvement should not be taken as certification of the claim.

The hypothesis concerns 3-SAT: given a list of conditions, each on up to three true-or-false variables, can all be met at once? Checking a proposed answer is easy; the obvious way to find one is to try all 2ⁿ settings of n variables.The HypothesisThe Exponential Time Hypothesis holds that there is some fixed number δ above zero such that no algorithm solves 3-SAT on n variables in time on the order of 2 to the power δn. Both labs’ papers trace it to work by Mr. Impagliazzo and Mr. Paturi published in 2001 in the Journal of Computer and System Sciences. OpenAI’s manuscript describes an algorithm that counts the satisfying settings in less than exponential time.

It would not settle P versus NP, which asks whether every problem whose solutions are easy to check is also easy to solve. Scott Aaronson, a computer scientist at the University of Texas at Austin, wrote in comments on his blog that “refuting the ETH is still a long way from P=NP.” Mr. Williams wrote on Bluesky that, if true, it “should make every computer scientist put at least a little more weight on the possibility of P=NP.”

Anthropic made the same claim hours later: a change to its GitHub repository, timestamped 11:07 p.m. Eastern on Monday, added a paper credited to “Claude” and a Lean formalization that claim to refute the hypothesis, a result Anthropic says was “discovered by an internal research Claude model.” Anthropic’s paper describes the work as independent and concurrent. Its notes call the result “purely asymptotic,” saying it “says nothing about solving formulas in practice.” The reviewers’ letter now lists it among its resources.

The reviewers are working from what OpenAI publishes. “We do not have access to their internal models. We only got the PDF,” Mr. Williams wrote. They are unpaid and signed no nondisclosure agreements, he said. At their request, OpenAI says, it will make the raw data public.

The Physics

The physics claim is the one the public cannot yet read. OpenAI says its models produced an existence proof of certain four-dimensional quantum field theories, including QCD and “a unified completion of the Standard Model,” and, as a subproblem, of pure Yang–Mills theory, which it calls part of a Clay Mathematics Institute Millennium Prize problem.The Prize ProblemThe Clay problem, as Arthur Jaffe and Edward Witten stated it for the institute, asks for a proof that, for any compact simple gauge group, a non-trivial quantum Yang–Mills theory exists in four dimensions and has a “mass gap” greater than zero. The institute says physicists have found the mass gap by experiment and computer simulation, but that it “still has not been understood from a theoretical point of view.”

OpenAI’s post does not mention the mass gap. “An existence claim alone should therefore not be described as a solution to the full Millennium Prize problem,” The Quantum Insider wrote early Tuesday. A group led by David Gross, who shared the 2004 Nobel Prize in Physics, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, has decided to hold back the AI-produced artifacts and write the work up itself, OpenAI said.

On Monday the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study, whose nine members include Edward Witten, who framed the prize problem with Mr. Jaffe, called on OpenAI to release the natural-language and Lean outputs behind the physics claims “without delay”: by OpenAI’s own account the Lean formalization was complete, it wrote, “so there is no scientific reason for delaying the release any further.” A day earlier it had recommended that AI companies not designate particular mathematicians to receive and present their output. OpenAI’s own post acknowledges that choosing such shepherds “creates asymmetry of information.”

“From this press release, it’s hard to know exactly what they’re claiming to have proved,” Peter Woit, a senior lecturer in mathematics at Columbia University, wrote on his blog, Not Even Wrong. In an update he wrote that, “while (deceptively…) not mentioning the mass gap, OpenAI appears to be claiming a solution to the ‘existence’ part of the Millennium Prize problem.”

The mathematics result has no review group: on an expert’s advice, OpenAI released its models’ write-up and Lean proof for anyone to examine. OpenAI says local smoothing implies the Kakeya conjecture, that any set containing a unit line segment in every direction has full dimension; Hong Wang and Joshua Zahl proved that conjecture in three dimensions in 2025.

What comes next is the checking: the reviewers propose a virtual meeting before the end of 2026, OpenAI has agreed to fund a conference, and mathematicians are pressing it to release the physics artifacts. OpenAI says it seeks no academic credit for its models’ discoveries.

Daily Cortex is written with AI models, including Anthropic’s Claude, and fact-checked with OpenAI’s GPT models.