OpenAI Says AI Refuted the Exponential Time Hypothesis

OpenAI says an internal model has produced three major results: a refutation of the Exponential Time Hypothesis (ETH) in computer science, a proof of the local smoothing conjecture in every dimension in mathematics, and existence proofs for quantum chromodynamics and related theories in physics. The company announced them in a post on 26 October, which describes the AI-written manuscripts as “a form of raw research data” rather than finished science.

None of the three has been peer reviewed. The computer science claim is being examined by a group of outside researchers, one of whom, MIT’s Ryan Williams, wrote on X that an internal OpenAI model “has produced a claimed refutation of ETH”. Hours later he added: “we haven’t fully vetted it yet!”

In brief

OpenAI says its models refuted the Exponential Time Hypothesis, proved local smoothing in all dimensions and gave existence proofs for QCD and pure Yang–Mills theory.
The ETH paper and a Lean formalisation are public; a group led by Impagliazzo, Paturi and Williams will examine them.
The physics artefacts are being held back while a group led by David Gross writes them up.
OpenAI’s own maths advisers had argued a day earlier against companies picking experts to handle results first.

The three claims

The ETH paper, published on OpenAI’s GitHub, runs to 67 pages and is credited to “OpenAI”. Its abstract describes a deterministic algorithm that exactly counts the satisfying assignments of k-CNF formulas for any fixed k of two or more, using polynomial space and running in time of roughly 2 to the power n/(log n)^(1/2+o(1)). Because ETH says 3-SAT needs 2^Ω(n) time in the worst case, such an algorithm would refute it. OpenAI says the model found the argument on 16 October and that it was formalised in the Lean proof assistant the next day.

The maths result concerns Sogge’s 1991 local smoothing conjecture about how focused waves behave. OpenAI says its proof works in every dimension and therefore also settles the Kakeya conjecture and Stein’s restriction conjecture in all dimensions. Its release log says this one is being released “directly”, as raw model output, as with an earlier batch on 6 October.

The physics claim is a rigorous existence proof for certain four-dimensional quantum field theories, including QCD and a unified completion of the Standard Model, with pure Yang–Mills theory as a sub-problem. That sub-problem is part of a Clay Millennium Prize problem. OpenAI has not released these materials.

Who is checking the work

OpenAI says it asked outside experts to act as “shepherds”, and that each advised a different release plan. For ETH, a group led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego and Williams will “verify and shepherd” the result. Impagliazzo and Paturi first proposed ETH, with Francis Zane. Williams said the group signed no NDAs, will receive no pay, and asked OpenAI to release all raw data and to sponsor a conference for the field; “They agreed to both.” He also wrote that one of the first things he asked OpenAI to check was that existing cryptosystems were not compromised, adding: “Apparently they are safe…”

For physics, a group led by Nobel laureate David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, has chosen to hold back the AI-produced artefacts and write the work up itself. OpenAI says it seeks no academic credit for the results.

The advisers’ objection

The release plan cuts against advice published a day earlier by the Advisory Group on Mathematics and Artificial Intelligence (AGMAI), an independent panel whose members include Timothy Gowers, Martin Hairer and Edward Witten. The group says it formed after OpenAI approached some of its members about an advisory board.

In a 25 October statement, AGMAI said that letting a company designate mathematicians to receive output first causes “delayed or unequal access”, “denied agency”, “biased selection” and “borrowed reputation”, and recommended that companies drop the practice for future releases. It stressed this was not criticism of the people chosen, who “have acted in good faith”. On 26 October it went further and called on OpenAI to release the Yang–Mills and QCD outputs “without delay”, arguing that once the Lean work was finished “there is no scientific reason” to wait. The panel’s advice is aimed at mathematics, and OpenAI has followed the open route for the maths result.

OpenAI, for its part, concedes that choosing shepherds “necessarily shapes who gets involved first and creates asymmetry of information”, and says it does not yet know which approach works best.

Lean proofs are not the end of checking

A Lean formalisation is strong evidence, but only for the statement that was formalised. Experts still need to confirm that the formal statement matches the claim, which is why outside review matters. OpenAI’s own record shows the risk. On 7 October its release log withdrew three manuscripts after a sign error broke a key argument, and revised 14 others.

ETH matters because hundreds of papers have used it to argue that certain problems cannot be solved quickly. If the claim holds, those results lose their footing, though the stated running time is only just below exponential and offers no practical speed-up yet. Williams told one follower he did not know whether practical SAT solving would change within a year.

The release follows a run of AI-assisted maths claims, from Microsoft’s KLS conjecture work to DeepMind’s AlphaProof Nexus and a preprint crediting Shanghai AI Lab’s Intern-S2. OpenAI also points to its $5m contribution to a Simons Foundation programme to help researchers check AI-produced results.