OpenAI claims its models have disproven the Exponential Time Hypothesis and proven the Kakeya conjecture

OpenAI announced on October 26 three scientific results produced by its internal models: a disproof of the Exponential Time Hypothesis (ETH), a pillar of theoretical computer science, a proof in all dimensions of the local smoothing conjecture, which implies that of Kakeya, and a proof of existence of quantum field theories, including quantum chromodynamics. These are claims, not yet established facts: the scientific community is only just beginning to examine them.

Three results, three fields

In theoretical computer science. ETH postulates, to put it simply, that the 3-SAT problem (deciding whether a logical formula can be satisfied) cannot be solved in sub-exponential time. It is widely accepted and serves as a starting assumption for hundreds of papers, which deduce from it that a problem is hard. OpenAI’s preprint, dated October 25, describes a deterministic algorithm that exactly counts the solutions of fixed-width formulas in sub-exponential time and polynomial space; the 3-SAT case would disprove ETH. According to OpenAI, the model found this proof on October 16 and it was formalized in Lean, a proof assistant that mechanically verifies each step, on October 17.

In mathematics. The local smoothing conjecture, proposed by Sogge in 1991, concerns the focusing of waves. It had been proven in dimension 2 in 2019 (Guth, Wang, and Zhang), and the Kakeya conjecture in dimension 3 in 2025 (Wang and Zahl). OpenAI’s models propose a proof in all dimensions, based on an auxiliary partial differential equation. The repository log is somewhat more cautious than the announcement about one of the consequences: it speaks of Stein restriction estimates “in the interior” of the conjectured domain.

In physics. The third result is a mathematically rigorous proof of existence of certain quantum field theories in four dimensions, including QCD, which describes the strong interaction between quarks and gluons. It includes the existence of pure Yang-Mills theory, which is part of one of the Clay Institute’s Millennium Problems. Neither the text nor the formalization are published: the group of physicists chose to write its own presentation.

A new publication method

OpenAI presents these manuscripts as “raw research data” and has asked specialists in each field to organize their review. For ETH, the group is led by Russell Impagliazzo and Ramamohan Paturi, who formulated the hypothesis, and by Ryan Williams (MIT). For physics, by Nobel laureate David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov, and Nikita Nekrasov. Participants sign no confidentiality agreement, are not paid, and speak in their own name. OpenAI says it claims no academic credit and has committed, at the group’s request, to funding a conference on the consequences of the result.

Ryan Williams announced the review speaking of a “claimed” disproof of ETH, and specifies that the group has requested publication of all raw data, PDFs and Lean code included.

Why caution is warranted

The Lean formalization is a strong argument, but it only guarantees what was formalized: it remains to verify that the statements verified by the machine correspond to the statements researchers have in mind. OpenAI’s history also reminds us that error remains possible: on October 7, the company withdrew three manuscripts from a previous publication, invalidated by a sign error, and corrected fourteen others.