OpenAI says its models have produced three proofs in mathematics, computer science and physics – Unite.AI

OpenAI said in an October 26, 2026 announcement that its internal models produced three research results in mathematics, computer science and physics, including a refutation of the exponential time hypothesis, a long-standing conjecture in computational complexity theory.

From the company’s perspective, AI-generated manuscripts amount to raw research data that has yet to be converted into scientific understanding, and it is pioneering a new release approach to that end. OpenAI asked outside experts to act as shepherds, organizing groups to understand the new results, and said the shepherds recommended three different release strategies: releasing the output of the mathematical AI directly, with researchers free to organize their own explanatory work; train an external group of collaborators for the computing output while publishing the AI ​​output for anyone to review and build upon; and form such a group for the physical result by putting the release of the materials back to him. The company said it doesn’t yet know which of the three approaches will prove most useful and invited feedback on what works and what can be improved.

According to the announcement, the collaborations will work according to four principles: transparency, without non-disclosure agreements and participants free to discuss the work, share work in progress and publish their findings; no academic credit, meaning that OpenAI does not require any for discoveries produced by its internal models and that inviting a researcher to lead a collaboration does not imply that the researcher made the original discovery; recognition, whereby each group receives credit for the understanding, reports, and other materials it produces and helps ensure appropriate credit for the researchers, techniques, and ideas on which the work is based; and independence, with voluntary and unpaid participation, members’ own conclusions and neither members nor OpenAI speak for each other. OpenAI said it will deliver materials and artifacts related to the findings to the collaborations. The company acknowledged that the approaches involve trade-offs – choosing initial shepherds inevitably shapes who gets involved first and creates an information asymmetry – and said it welcomes feedback on how to broaden participation.

Local smoothing and Kakeya conjecture

In 1991, Sogge proposed the local smoothing conjecture, which concerns how waves behave when focused and argues that the most extreme focusing can only be short-lived, so the focusing at a typical time is much weaker than at the worst time. The announcement described the hypothesized phenomenon as fundamental, with wide-ranging implications throughout mathematical analysis; the conjecture involves the Kakeya conjecture and the Stein restriction conjecture. Guth, Wang and Zhang proved the conjecture for two spatial dimensions in 2019, and Wang and Zahl solved the three-dimensional Kakeya conjecture in 2025.

OpenAI said its models produced a proof of the local smoothing conjecture in every dimension, based on a novel auxiliary partial differential equation, and in doing so also solved the Kakeya conjecture and the Stein restriction conjecture in all dimensions. The company obtained the AI ​​output on October 18, 2026, formalized it into Lean the next day, and then reached out to an expert to discuss the possibility of collaborating around the outcome. Following that expert’s recommendation, OpenAI published both a natural language AI report produced by its models and a Lean proof to its public GitHub repository, the same strategy used for its October 6, 2026 math release.

Refutation of the exponential time hypothesis

The computing result concerns the exponential time hypothesis, a conjecture about the complexity of determining whether a Boolean formula can be satisfied, closely related to the P versus NP problem. The announcement described the hypothesis as widely believed and stated that it had been used in hundreds of papers to derive hardness results conditional on it in fields including graph theory, computational biology, databases, and computational geometry. OpenAI said its system proved the hypothesis false and that the result was later verified in Lean. The AI ​​produced its natural language refutation on October 16, 2026, and the result was formalized in Lean on October 17, 2026.

According to the formalization documentation, hosted in the OpenAI public mathematics repository with a final commitment dated October 26, 2026, the Lean proof establishes that for a fixed k of at least 2, a deterministic machine exactly counts satisfying assignments of a canonically encoded k-CNF formula on n variables in subexponential time and polynomial space, with more precise time bounds for instances with m-clauses provided in the accompanying October 25 document, Manuscript of 2026.

The work will be verified and led by a team led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego, who with Francis Zane originally proposed the hypothesis, and Ryan Williams of MIT, who OpenAI says has refuted strong versions of it and linked algorithms and lower bounds. OpenAI said it agreed to the group’s requests that all the raw data underlying the result be made public to facilitate independent verification by the community, and that OpenAI commit to funding a conference on the work’s implications for the industry. The group also invited the community to help organize an open review of the request and to discuss the broader questions it believes the AI-generated findings raise about verification, access, credit and the future of research. OpenAI said the rebuttal reopens questions about how much faster these tasks could become, and that turning theoretical advances into practical improvements will require significant further research.

Proof of the existence of QCD and the standard model

OpenAI said that the physics result provides a mathematically rigorous proof of the existence of some interacting quantum field theories containing gauge fields and fermions in four dimensions, including quantum chromodynamics (QCD) and a unified completion of the standard model of particle physics, and that the work builds on Balaban’s approach in constructive quantum field theory. As a subproblem, it includes a proof of the existence of pure Yang-Mills theory, part of the Clay Mathematics Institute’s Millennium Prize problem on quantum Yang-Mills theory. The existence proof applies to the simplest grand unification theory that incorporates the gauge fields and fermions of the Standard Model, the announcement says, because the Standard Model itself cannot be extrapolated to arbitrarily short distances. The Standard Model describes how fundamental subatomic particles interact under three of the four fundamental forces, all except gravity; QCD describes how particles such as quarks and gluons interact via the strong force; and pure Yang-Mills theory describes the self-interactions of gluons, the force carriers, rather than matter such as quarks.

OpenAI said its models produced a natural language proof of the existence of Yang-Mills on September 9, 2026, followed by a proof for QCD on September 18, 2026. Given the length and complexity of the work, the company postponed the release until the Lean formalization was complete, finishing the formalization of the existence of Yang-Mills on October 16, 2026 — a milestone the company says led it to develop collaboration.

David Gross will lead a group that includes Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. Gross’s 1973 work on asymptotic freedom in Yang-Mills theories helped establish QCD as a theory of strong interactions and earned him a share of the Nobel Prize in Physics in 2004. Jaffe formulated the Millennium Prize problem on quantum Yang-Mills theory with Witten and co-founded the Clay Mathematics Institute. Douglas reported on the state of the problem in 2004 and provided a full progress update in early 2026. Klebanov was among the pioneers in applying gauge string duality to confinement, a key property of QCD, and now directs the Simons Collaboration on Confinement and QCD Strings. Nekrasov developed exact methods for generalizations of Yang-Mills theory and won the Dannie Heineman Prize in Mathematical Physics in 2023. The group of physicists decided to hold off on releasing the artifacts produced by the AI ​​and instead write down the result itself, and asked that all communications with its members about the effort be directed to [email protected].

Support scientists

OpenAI said the collaborations complement its previously announced $5 million contribution to the Simons Foundation, which the foundation manages and awards to support workshops, conferences and special programs that help researchers understand and develop results produced by artificial intelligence. The company said it is committed to securing funding for researchers to independently examine, explain and develop the three findings, and that it hopes humans working with artificial intelligence can provide a new understanding of computational complexity and shed light on the inherent capabilities and limitations of efficient computation. OpenAI also said it continues to solicit feedback on how best to disseminate findings and engage scientists.