OpenAI says AI models disproved the Exponential Time Hypothesis and proved a maths conjecture

What OpenAI announced

On Monday, 26 October, OpenAI said its AI models had produced three scientific results, in harmonic analysis, complexity theory and quantum field theory. The announcement followed a much larger release on 6 October, when the company published 722 maths papers, and the earlier manuscripts covered 372 math problems across more than 700 pages. This time OpenAI used three release methods. The math output went out so researchers could work through the explanations themselves, while the computer science and physics work went to outside expert groups the company calls "shepherds". The company says it will not seek academic credit for the models' discoveries.

A 1999 hypothesis under review

The Exponential Time Hypothesis (ETH) says some computational problems cannot be solved quickly, and it concerns Boolean satisfiability, a problem closely tied to P versus NP. Hundreds of computer science papers rest on it, and researchers have used it in work on graph theory, computational biology and databases. OpenAI says its model found the disproof on 16 October and checked it in Lean, a language that verifies proofs by machine, the following day. The write-up and the Lean proof have been published on GitHub. Russell Impagliazzo and Ramamohan Paturi of UC San Diego, with Ryan Williams of MIT, will check and explain the result, and Impagliazzo and Paturi were part of the group that originally proposed the hypothesis in 1999. In an open letter, the three wrote that taking part does not amount to endorsing the claim, and Williams said on X that the group signed no non-disclosure agreements and will not be paid. The reviewers asked OpenAI to publish its underlying data and finance a meeting for the field, and according to Williams and OpenAI the company agreed to both.

Local smoothing and the Kakeya link

OpenAI says its models proved the local smoothing conjecture, posed by Christopher Sogge in 1991, in every spatial dimension. According to the company, the same proof also resolves the Kakeya conjecture and Stein's restriction conjecture in all dimensions. Accounts differ on how far the result had reached before, with one saying it had been proved only in three dimensions and another saying two dimensions in 2019. The Kakeya conjecture had been proved in three dimensions in 2025. The company dates the math proof to 18 October and says it was later formalized in Lean. Alex Iosevich, a University of Rochester professor who works in harmonic analysis, said OpenAI appeared to have "cracked" the conjecture.

“The conjecture was formulated by my PhD adviser, Chris Sogge, so it is near and dear to my heart, and the news was a shock at first.”

Iosevich said theoretical mathematics was likely to "change drastically" because of AI, and added that many problems in pure mathematics came out of physical questions that the new tools can now explore quickly.

Physics claims and the review panel

OpenAI says its models produced a rigorous existence proof for certain interacting quantum field theories with gauge fields and fermions in four dimensions, including quantum chromodynamics, which studies the interaction of quarks and gluons. The company also says the work includes a unified completion of the Standard Model of particle physics. The group that will guide the physics work was named in the company's own statement.

“David Gross will lead a group including Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov that will shepherd this work.”

The physics material has not been released in full, and the group reviewing it has not yet published anything.

Review concerns and credit

On 25 October, the AGMAI panel of math advisers that OpenAI assembled said AI companies should not adopt the practice of naming certain people to receive output first. The panel cited unequal access, borrowed reputation and biased selection as problems. It recommended that labs release output on scholarly repositories with Lean-formalized proofs, publish information on how problems were chosen, and fund people working to understand the results. OpenAI itself acknowledged that its staged approach could have its own issues. Credit for the writeups will go to the people in each group, who are also responsible for making sure AI-generated results credit the researchers, techniques and ideas behind them. OpenAI has invited feedback on how to broaden participation in the process.