OpenAI Says Its Models Produced Three Proofs in Math, CS, and Physics

OpenAI said in an announcement on October 26, 2026, that its internal models produced three research results in mathematics, computer science, and physics, including a disproof of the Exponential Time Hypothesis, a longstanding conjecture in computational complexity theory.

In the company’s view, the AI-generated manuscripts amount to raw research data that must still be converted into scientific understanding, and it is experimenting with a new release approach toward that end. OpenAI asked external experts to act as shepherds, organizing groups to understand the new results, and said the shepherds advised three different release strategies: release the mathematics AI output directly, with researchers free to organize their own explanatory work; form an external group of collaborators for the computer science result while publishing the AI output for anyone to examine and build on; and form such a group for the physics result while deferring to it on the release of materials. The company said it does not 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 operate under four principles: transparency, with no nondisclosure agreements and participants free to discuss the work, share work in progress, and publish their conclusions; no academic credit, meaning OpenAI seeks none for discoveries produced by its internal models and that inviting a researcher to steward a collaboration does not imply that the researcher made the original discovery; recognition, under which each group receives credit for the understanding, writeups, and other materials it produces and helps ensure appropriate credit for the researchers, techniques, and ideas the work draws on; and independence, with participation voluntary and unpaid, members’ conclusions their own, and neither the members nor OpenAI speaking for the other. OpenAI said it will turn over materials and artifacts related to the results to the collaborations. The company acknowledged that the approaches involve tradeoffs — the choice of initial shepherds inevitably shapes who becomes involved first and creates an asymmetry of information — and said it welcomes feedback on how to broaden participation.

Local Smoothing and the Kakeya Conjecture

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

OpenAI said its models produced a proof of the local smoothing conjecture in every dimension, relying on an auxiliary partial differential equation of a novel kind, and in doing so also resolved the Kakeya conjecture and Stein’s restriction conjecture in all dimensions. The company obtained the AI output on October 18, 2026, formalized it in Lean the following day, and then contacted an expert to discuss forming a collaboration around the result. Following that expert’s recommendation, OpenAI published both an AI natural-language writeup produced by its models and a Lean proof in its public GitHub repository, the same strategy it used for its October 6, 2026 mathematics release.

Exponential Time Hypothesis Disproof

The computer science result concerns the Exponential Time Hypothesis, a conjecture about the complexity of determining whether a Boolean formula can be satisfied that is closely tied to the P versus NP problem. The announcement described the hypothesis as widely believed and said it has been used in hundreds of papers to derive hardness results conditional on it across fields including graph theory, computational biology, databases, and computational geometry. OpenAI said its system proved the hypothesis false and that the result was subsequently verified in Lean. The AI produced its natural-language disproof on October 16, 2026, and the result was formalized in Lean on October 17, 2026.

According to the formalization’s documentation, hosted in OpenAI’s public math repository with a latest commit dated October 26, 2026, the Lean proof establishes that for fixed k of at least 2, a deterministic machine exactly counts the satisfying assignments of a canonically encoded k-CNF formula on n variables in subexponential time and polynomial space, with sharper time bounds for instances with m clauses given in the accompanying October 25, 2026 manuscript.

The work will be verified and shepherded by a group 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 noted has refuted strong versions of it and connected algorithms and lower bounds. OpenAI said it agreed to the group’s requests that all raw data behind 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 field. The group also invited the community to help organize an open examination of the claim and to discuss the broader questions it said AI-generated breakthroughs raise about verification, access, credit, and the future of research. OpenAI said the refutation reopens questions about how much faster these tasks could become, and that turning theoretical advances into practical improvements will require significant further research.

QCD and Standard Model Existence Proof

OpenAI said the physics result supplies a mathematically rigorous existence proof of certain 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 Quantum Yang–Mills Theory Millennium Prize problem. The existence proof applies to the simplest grand unified theory incorporating the Standard Model’s gauge fields and fermions, the announcement said, 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 but 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 Yang–Mills existence on September 9, 2026, followed by a proof for QCD on September 18, 2026. Given the work’s length and complexity, the company postponed release until the Lean formalization was complete, finishing the formalization of Yang–Mills existence on October 16, 2026 — a milestone the company said led it to develop the 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 the theory of the strong interactions and earned him a share of the 2004 Nobel Prize in Physics. Jaffe formulated the Quantum Yang–Mills Theory Millennium Prize problem with Witten and co-founded the Clay Mathematics Institute. Douglas reported on the problem’s status in 2004 and provided a comprehensive update on progress 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 has developed exact methods for generalizations of Yang–Mills theory and won the Dannie Heineman Prize for Mathematical Physics in 2023. The physics group decided to hold off on releasing the AI-produced artifacts and instead write the result up itself, and asked that all communications with its members about the effort be directed to YANGMILLS4D@gmail.com.

Supporting 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 build on results produced by AI. The company said it is committed to ensuring funding for researchers to independently examine, explain, and build on the three results, and that it hopes humans working with AI can provide new understanding of computational complexity and shed light on the inherent capabilities and limits of efficient computation. OpenAI also said it continues to invite feedback on the best way to disclose results and engage with scientists.