OpenAI Claims Three Scientific Breakthroughs: ETH Refuted, Kakeya Solved, Yang-Mills Existence Proved
OpenAI announced on October 26 that its models produced three major results across mathematics, computer science and physics — claims that, if independently confirmed, would mark a turning point in AI-assisted science. The math and computer-science results come with Lean formalizations published on GitHub; the physics materials are being withheld at the reviewing group's own decision, and the company is explicit that all three await independent verification.
Mathematics: local smoothing in every dimension
The company says its models proved the local smoothing conjecture, proposed by Sogge in 1991, in every dimension. The conjecture concerns how energy in wave equations concentrates over time, asserting that the most extreme focusing must be brief. Because of its foundational role in harmonic analysis, resolving local smoothing also settles the Kakeya conjecture — how much space a needle needs to point in every direction — and the Stein restriction conjecture in all dimensions. OpenAI dates the model output to October 18 and its Lean formalization to the following day, noting that the result builds on prior work by Guth, Wang and Zhang in two dimensions and on Wang and Zahl's 2025 proof of the three-dimensional Kakeya conjecture. On external advice, the AI-generated write-up is being released openly on GitHub for researchers to organize their own efforts.
Computer science: the Exponential Time Hypothesis is false
OpenAI says its system proved that the Exponential Time Hypothesis — a long-standing conjecture about the complexity of Boolean satisfiability that underpins hundreds of papers in graph theory, computational biology, databases and computational geometry — is false, with the result subsequently verified in Lean. The natural-language disproof dates to October 16 and the Lean formalization to October 17, both public on GitHub. A verification group led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego together with Ryan Williams of MIT will check and explain the work; the group asked for all raw data to be public and for OpenAI to fund a conference, and OpenAI agreed. If the refutation holds, it reopens the question of how much faster these computation tasks can actually be made.
Physics: existence proofs touching a Millennium Prize problem
The third claim is the most consequential: OpenAI says its models produced a mathematically rigorous existence proof for certain interacting quantum field theories in four dimensions, including quantum chromodynamics and a unified completion of the Standard Model, with a proof of the existence of pure Yang-Mills theory — a subproblem of a Clay Mathematics Institute Millennium Prize problem. The Yang-Mills proof dates to September 9, the QCD proof to September 18, and the Lean formalization of Yang-Mills existence to October 16. A review group led by Nobel laureate David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, has decided to hold off releasing the AI-produced artifacts and instead write up the work themselves.
The reaction from the mathematical community is split. The Advisory Group on Mathematics and AI — whose members include Timothy Gowers, Martin Hairer and Edward Witten — says OpenAI has told them the Lean formalization is complete and that there is no scientific reason for delaying release any further, and the group published a separate statement on October 25 criticizing the practice of routing AI output to designated mathematicians first. Times of AI notes that Lean verification confirms a formal proof follows from a formal statement, not that the formal statement faithfully captures the written conjecture — a caveat that proved relevant in an earlier OpenAI mathematics manuscript. For now, the three claims stand as the boldest demonstration yet that AI systems can do more than answer questions: they may be able to discover and prove new science.
Sources: OpenAI, The Paper, PANews, Times of AI, The Verge, Wallstreetcn, threerouter
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