OpenAI says its AI refuted ETH and proved Kakeya

What was announced?

OpenAI announced three new results in mathematics, computer science and physics on Monday (OpenAI). All three were produced by the company's internal models. Each has a Lean formalization, covering the Yang-Mills part in the physics result. The release process is new too: external experts organize how the community examines and verifies the results.

Mathematics: the Kakeya conjecture

OpenAI says its models proved Sogge's 1991 local smoothing conjecture in every dimension, which implies both the Kakeya conjecture and Stein's restriction conjecture. Human researchers had previously proved local smoothing in two spatial dimensions in 2019 and the Kakeya conjecture in three dimensions in 2025. The new proof is based on a novel auxiliary partial differential equation.

The AI output arrived on October 18 and was formalized in Lean the next day. The natural language writeup and the Lean proof are public in OpenAI's GitHub repository, and researchers are free to organize their own efforts to explain the result.

Computer science: ETH is false

The biggest surprise: OpenAI says its system refuted the Exponential Time Hypothesis (ETH), a conjecture about the complexity of Boolean formula satisfiability that is closely tied to P vs. NP. ETH was widely believed to be true, and hundreds of papers use it for conditional hardness results in fields from graph theory and computational biology to databases.

The AI found the disproof on October 16, and it was verified in Lean on October 17. Verification is led by Russell Impagliazzo and Ramamohan Paturi (UC San Diego), two of the original proposers of ETH, and Ryan Williams (MIT). The natural language proof and the Lean code are already public, and OpenAI has committed to funding a conference on the implications. The company adds that faster algorithms will not follow automatically and will need significant further research.

Physics: QCD and Yang-Mills

The third result is a mathematically rigorous existence proof for certain interacting four-dimensional quantum field theories, including quantum chromodynamics (QCD) and a unified completion of the Standard Model. As a subproblem, it also proves the existence of pure Yang-Mills theory, which is part of a Clay Mathematics Institute Millennium Prize problem.

The Yang-Mills proof was produced on September 9, the QCD proof on September 18, and the Yang-Mills Lean formalization was finished on October 16. The AI materials are not being released for now: a group led by Nobel laureate David Gross (with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov) will write up the result themselves.

What to keep in mind

For now these are OpenAI's claims. Lean formalization is a strong signal, but independent expert review is only starting, and the physics materials are not yet public. OpenAI is not seeking academic credit for the discoveries, and group members take part voluntarily, unpaid and without nondisclosure agreements.

Why it matters: if the ETH refutation holds, conditional lower bounds built on it need rethinking, which matters to algorithm designers and complexity researchers. It also shows that machine verification in Lean is becoming the entry ticket for AI-driven research claims.