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OpenAI says its models refuted the ETH and proved local smoothing

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OpenAI says its models refuted the ETH and proved local smoothing

OpenAI announced three new discoveries in mathematics, computer science and physics, covering harmonic analysis, complexity theory and quantum field theory. It says its internal models produced all three. It does not name which models. OpenAI describes the manuscripts as "a form of raw research data" that must be turned into scientific understanding, so for this release it tried a new approach: it asked outside experts to act as shepherds and organise groups to understand each result. Those experts advised three different release strategies. For the math result, publish the AI output and let researchers organise their own efforts. For computer science, form an external group to explain the result while the AI output is public. For physics, form an external group and defer to it on releasing materials. OpenAI says it does not yet know which approach will work best.

Mathematics. In 1991 Sogge proposed the local smoothing conjecture, which concerns how waves behave as they become focused: the most extreme focusing must be short-lived, and the amount of focusing at a typical time is much smaller than at the worst-case time. The conjecture implies the Kakeya conjecture and Stein's restriction conjecture. Guth, Wang and Zhang proved it in two spatial dimensions in 2019, and Wang and Zahl proved the Kakeya conjecture in three dimensions in 2025. OpenAI says its models produced a proof of local smoothing in every dimension, which would also resolve the Kakeya and Stein restriction conjectures in all dimensions. The proof is based on a novel auxiliary partial differential equation. The AI output came on October 18, 2026 and was formalised in Lean the next day. OpenAI then consulted an expert, and on that advice published the AI-written natural language writeup and the Lean proof on its public GitHub repository, as it did for its October 6, 2026 release.

Computer science. The result concerns the Exponential Time Hypothesis (ETH), a longstanding conjecture about the computational complexity of Boolean formula satisfiability, which is closely tied to P vs. NP. The ETH has been widely believed and has been used in hundreds of papers to derive conditional hardness results in fields including graph theory, computational biology, databases and computational geometry. OpenAI says its system proved the ETH false, and that the result was then verified in Lean. The AI found its natural language disproof on October 16, 2026, and the Lean formalisation followed on October 17. OpenAI says the refutation reopens questions about how much faster these tasks could become and may help guide the search for better algorithms, but finding them and turning theory into practical gains will take significant further research. A group led by Russell Impagliazzo (UC San Diego), Ramamohan Paturi (UC San Diego) and Ryan Williams (MIT) will verify and shepherd the work. Impagliazzo and Paturi, with Francis Zane, originally proposed the ETH. Williams has refuted strong versions of it. The group asked for all raw data to be made public for independent verification and for OpenAI to fund a conference on the implications; OpenAI agreed to both. The group also invites the community to help organise an open examination of the ETH claim. The natural language proof and Lean formalisation are on OpenAI's GitHub.

Physics. The result is described as a mathematically rigorous existence proof of certain interacting quantum field theories with gauge fields and fermions in four dimensions, including QCD and a unified completion of the Standard Model. It 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 Millennium Prize problem on Quantum Yang-Mills theory. The proof pertains to the simplest grand unified theory that includes the Standard Model's gauge fields and fermions, because the Standard Model itself cannot be extrapolated to arbitrarily short distances. OpenAI says its models produced a natural language proof of Yang-Mills existence on September 9, 2026 and a proof for QCD on September 18. Because of the length and complexity, it held back release until the Lean formalisation of Yang-Mills existence finished on October 16. David Gross, who shared the 2004 Nobel Prize in Physics, will lead the shepherding group with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. The group decided to hold off releasing the AI-produced artifacts and to write the result up itself. It asks that all communications go to a single email account, YANGMILLS4D@gmail.com.

The collaborations follow four principles. Transparency: no nondisclosure agreements, and participants can discuss the work, share intermediate products and publish their conclusions. No academic credit: OpenAI does not seek credit for what its internal models produced, and inviting someone to steward a collaboration does not imply they made the discovery. Recognition: the groups get credit for the understanding and writeups they create. Independence: participation is voluntary and unpaid, members' conclusions are their own, and they do not speak for OpenAI. OpenAI also admits a tradeoff: choosing the first shepherds shapes who gets involved first and creates an asymmetry of information. It points to its recently announced $5 million contribution to the Simons Foundation, which the Foundation will manage and award for workshops, conferences and special programs to help researchers understand and build on AI-produced results.

Key facts

  • OpenAI says its models proved the local smoothing conjecture in every dimension, which would also resolve the Kakeya and Stein restriction conjectures in all dimensions, using a novel auxiliary partial differential equation. The AI output came on October 18, 2026 and a Lean formalisation the next day.
  • OpenAI says its system proved the Exponential Time Hypothesis false, verified in Lean; the ETH has been used in hundreds of papers. A group led by Impagliazzo, Paturi and Williams will verify and shepherd it, and the proof and Lean files are on OpenAI's GitHub.
  • The physics result is described as a rigorous existence proof of certain four-dimensional gauge-and-fermion quantum field theories, including QCD, with pure Yang-Mills existence as a subproblem. A group led by David Gross is holding back the AI-produced artifacts and writing it up itself.
  • The release uses outside experts as shepherds under four principles: transparency, no academic credit for OpenAI, recognition for the groups, and independence. Participation is voluntary and unpaid.
  • OpenAI links the collaborations to its recently announced $5 million contribution to the Simons Foundation, for workshops, conferences and special programs.

Why it matters

If the claims hold, they touch three central open problems. The ETH has underpinned conditional hardness results in many fields, so its refutation would reopen questions about how fast those tasks could become. The local smoothing proof would extend earlier work that reached two dimensions (2019) and, for Kakeya, three (2025), to every dimension. The physics result would make progress toward putting quantum field theory on a mathematically rigorous footing, which OpenAI says it hopes these results begin. The release is also an experiment in how an AI lab discloses results: external shepherds, public data in some cases, and a stated aim of avoiding academic credit for the lab.

Who it affects

Complexity theorists, whose field leans on the ETH, and the CS group of Impagliazzo, Paturi and Williams, who will verify the claim. Harmonic analysts and researchers working on the Kakeya and Stein restriction conjectures. Mathematical physicists and quantum field theorists, including the group led by David Gross with Douglas, Jaffe, Klebanov and Nekrasov. Researchers more broadly are invited to help organise an open examination of the ETH claim and to discuss verification, access, credit and the future of research. Applied fields that use ETH-based hardness results, such as graph theory, computational biology, databases and computational geometry, may eventually be affected.

How to use it

The math and computer science AI natural language writeups and Lean proofs are published on OpenAI's GitHub repository, so anyone can examine and build on them. The CS group has asked the community to help organise an open examination of the ETH claim. For the physics work, the AI-produced artifacts are not being released for now, and the group asks that all communications go to YANGMILLS4D@gmail.com. The funding from the Simons Foundation, managed and awarded by the Foundation, is meant to support workshops, conferences and special programs around AI-produced results.

How solid is it

These are OpenAI's own claims, and the manuscripts are framed as raw research data, not finished papers. No independent verification or external expert confirmation of any of the three results is reported. The CS group is described as verifying and shepherding the work, and the physics group as understanding and explaining it. All three results have Lean formalisations of at least part of the work: the math and ETH proofs were formalised in Lean, and the Yang-Mills existence proof was formalised by October 16. The source does not say whether the QCD proof has been formalised in Lean. The source does not name the models that produced the results.

Risks and caveats

OpenAI itself notes that picking the first shepherds shapes who gets involved first and creates an asymmetry of information, and it says it does not yet know which of the three release approaches is most useful. The ETH refutation does not by itself yield faster algorithms: the source says finding them requires significant further research, and it names no specific faster algorithm. The physics outputs are not released and no release date is given. The source does not say the Yang-Mills Millennium Prize has been awarded or claimed; pure Yang-Mills existence is described only as a subproblem of the work. Group members speak for themselves, not for OpenAI.

“In our view, these manuscripts are best understood as a form of raw research data.”

— OpenAI