OpenAI says AI models refuted the Exponential Time Hypothesis

OpenAI has announced that its internal models produced a proof that refutes the Exponential Time Hypothesis (ETH), a longstanding conjecture in computational complexity that has underpinned hundreds of conditional hardness results across computer science. The company formalized the disproof in the Lean proof assistant on October 17th, 2026, one day after its models generated the natural language argument on October 16th.

The announcement bundles three claimed discoveries in mathematics, computer science, and physics, and pairs each with a newly designed external collaboration intended to convert what OpenAI calls "raw research data" into scientific understanding. The stakes are considerable: if verified, the results touch problems that have resisted mathematicians and physicists for decades, including one Clay Mathematics Institute Millennium Prize problem.

What did the models prove in mathematics?

The mathematics result addresses harmonic analysis. In 1991, Sogge proposed the local smoothing conjecture, which concerns the behavior of waves as they become focused. The conjecture states that the most extreme focusing must be short-lived: the amount of focusing at a typical random time is always much smaller than what can happen at the worst-case time. OpenAI notes this reflects a familiar pattern throughout mathematics, in which the worst case is more extreme than what is observed after averaging, and that the conjecture has wide-ranging implications because it implies the Kakeya conjecture and Stein's restriction conjecture.

The result extends prior human work. In 2019, Guth, Wang, and Zhang proved the local smoothing conjecture in two spatial dimensions. In 2025, Wang and Zahl proved the Kakeya conjecture in three dimensions. OpenAI states that its models produced a proof of the local smoothing conjecture in every dimension, which therefore also resolves the Kakeya conjecture and Stein's restriction conjecture in all dimensions. The company says the new proof rests on a novel auxiliary partial differential equation.

OpenAI obtained the AI output on October 18th, 2026, formalized it in Lean the following day, and then contacted an expert to discuss forming a collaboration. Following that expert's recommendation, the company released the AI output publicly, using the same strategy as its October 6th, 2026 release. Both a natural language writeup and a Lean proof are available in OpenAI's public GitHub repository.

Why does refuting the ETH matter?

The Exponential Time Hypothesis is a conjecture about the computational complexity of Boolean formula satisfiability, a problem deeply related to P versus NP. The ETH has been widely believed to be true, and researchers have used it in hundreds of papers to derive conditional hardness results in graph theory, computational biology, databases, computational geometry, and other fields.

OpenAI reports that its system proved the ETH is false, and that the result was subsequently verified in Lean. The company writes that the refutation "reopens questions about how much faster these tasks could become, and may help guide the search for better algorithms." It cautions that finding such algorithms, and turning theoretical advances into practical improvements, will require significant further research.

A group led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego, together with Ryan Williams of MIT, will verify and shepherd the work. All three are experts in algorithm design and complexity theory. Impagliazzo and Paturi, with Francis Zane, originally proposed the Exponential Time Hypothesis and have authored influential papers on it over the last 30 years. Williams has refuted strong versions of the ETH, used ETH for hardness results, and connected algorithms and lower bounds.

OpenAI has agreed to two requests from the group: all raw data of the results will be made public to allow independent verification by the community, and OpenAI will commit to funding a conference on the implications of the work for the field. The group invites the community to help organize an open examination of the ETH claim and to discuss the broader questions that AI-generated breakthroughs raise about verification, access, credit, and the future of research. As with the mathematics result, OpenAI has published a natural language writeup and a Lean proof on GitHub.

OpenAI frames the broader ambition plainly: computational complexity has many open questions whose proofs or refutations have so far eluded scientists, and the company hopes that humans working with AI can shed light on the inherent capabilities and limits of efficient computation.

Who is checking the physics claim?

The physics result is 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. 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, which forms part of the Quantum Yang–Mills Theory Clay Mathematics Institute Millennium Prize problem.

The Standard Model describes the interactions of fundamental subatomic particles for three of the four fundamental forces, excepting gravity. The existence proof covers the simplest grand unified theory that includes the gauge fields and fermions of the Standard Model, since the Standard Model itself cannot be extrapolated to arbitrarily short distances. QCD describes interactions of particles such as quarks and gluons via the strong force; pure Yang–Mills theory describes only the self-interactions of the gluons, the force carriers, rather than matter such as quarks.

David Gross will lead the group shepherding the work. Its members include 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. OpenAI notes asymptotic freedom is also the key mechanism underlying the existence of these theories as mathematically rigorous objects.
• Jaffe formulated the Quantum Yang–Mills Millennium Prize problem with Witten, envisioned the idea of Millennium Prize problems, and co-founded the Clay Mathematics Institute.
• Douglas has stewarded this problem for 20 years, reporting on its status in 2004 and providing a comprehensive update at the beginning of this year.
• Klebanov helped pioneer the application of gauge–string duality to confinement, an important property of QCD, and directs the Simons Collaboration on Confinement and QCD Strings.
• Nekrasov has done groundbreaking work on exact methods for generalizations of Yang–Mills theory and won the Dannie Heineman Prize for Mathematical Physics in 2023.

The timeline differs from the other two results. OpenAI's models produced a natural language proof for Yang–Mills existence on September 9th, 2026, and a proof for QCD on September 18th, 2026. The company postponed release until the Lean formalization completed; it finished formalizing Yang–Mills existence on October 16th, 2026, which led to the collaboration. The group has decided to hold off on releasing the AI-produced artifacts and instead write up the work itself. Communications to group members should go to [email protected].

OpenAI says it hopes the results mark the beginning of a transition in which quantum field theory, the most important framework in modern theoretical physics, becomes a mathematically rigorous field.

How will the collaborations work?

OpenAI asked external experts to act as shepherds, organizing groups to understand the new results. The shepherds advised three different release strategies: release the math output openly and let researchers organize their own explanation efforts; form an external group for the computer science result while making the AI output public for anyone to examine; and form an external group for the physics result while deferring to it on the release of materials. OpenAI says it does not yet know which approach will prove most useful.

The collaborations follow four stated principles:

• Transparency. OpenAI is transparent about the existence of its collaborations. Participants will not sign nondisclosure agreements and may discuss the work, share intermediate products, and publish their conclusions.
• No Academic Credit. OpenAI does not seek academic credit for discoveries produced by its internal models, and inviting a researcher to steward a collaboration does not imply that researcher or their working group made the original discovery.
• Recognition. The groups receive recognition for the understanding they develop, the writeups they author, and other materials they create, and will help ensure the work appropriately credits researchers, techniques, and ideas it draws on.
• Independence. Participation is voluntary and unpaid. Members' conclusions are their own; they do not speak for OpenAI, and OpenAI does not speak for them.

OpenAI acknowledges the tradeoffs. The collaborations empower experts to unpack and contextualize results, but choosing initial shepherds necessarily shapes who gets involved first and creates an asymmetry of information. The company invites feedback on broadening participation.

The effort sits alongside a recently announced $5 million contribution from OpenAI to the Simons Foundation, which will manage and award the funding for workshops, conferences, and special programs that help researchers understand and build on AI-produced results. OpenAI says it is committed to funding independent examination of the three results, and continues to invite feedback on how to disclose discoveries and engage scientists to accelerate research.