OpenAI Claims ETH Disproof and Kakeya Proof

OpenAI said on October 26 that its internal models have produced three major results: a disproof of the Exponential Time Hypothesis (ETH) in computer science, a proof of the local smoothing conjecture that would settle the Kakeya conjecture in every dimension, and a rigorous existence proof for quantum chromodynamics (QCD). None of the claims has been independently confirmed, and the outside experts reviewing them say their involvement is not an endorsement.

Key takeaways

• OpenAI says its system found a disproof of the ETH on October 16 and formalized it in the Lean proof assistant on October 17; the write-up and Lean code are public on GitHub.
• Three computer scientists, including two of the hypothesis's originators, will lead an open examination and wrote that their role "should not be taken as certification of the claim."
• A physics group led by Nobel laureate David Gross is withholding the AI-produced QCD and Yang–Mills material while it writes up the work itself.

What is the Exponential Time Hypothesis?

The ETH, formulated in 1999 by Russell Impagliazzo and Ramamohan Paturi, holds that the 3-SAT problem of Boolean satisfiability cannot be solved in subexponential time. It is unproven, but OpenAI notes it underpins conditional hardness results in hundreds of papers.

The review group is Impagliazzo and Paturi of UC San Diego, who proposed the ETH with Francis Zane, and Ryan Williams of MIT, who has refuted strong versions of the hypothesis. They asked OpenAI to publish all raw data and fund a conference on the implications, and OpenAI agreed, according to its post and reporting by The Next Web. "If the claim is correct, it will change an important part of theoretical computer science," the three wrote in an open letter.

Kakeya and local smoothing claim

Christopher Sogge proposed the local smoothing conjecture, which concerns how waves focus, in 1991. It implies both the Kakeya conjecture and Stein's restriction conjecture. Larry Guth, Hong Wang and Ruixiang Zhang proved it in two spatial dimensions in 2019, and Wang and Joshua Zahl proved the Kakeya conjecture in three dimensions in 2025.

OpenAI says its models produced a proof in every dimension using "a novel auxiliary partial differential equation." The output arrived October 18 and was formalized in Lean the next day. No external reviewer is named for it.

QCD and Yang–Mills existence

The physics claim is a rigorous existence proof for certain interacting four-dimensional quantum field theories, including QCD. It covers pure Yang–Mills theory, part of a Clay Mathematics Institute Millennium Prize problem. OpenAI dates the Yang–Mills proof to September 9, the QCD proof to September 18, and a Lean formalization of Yang–Mills existence to October 16.

Gross leads the group, which includes Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov. The group has deferred release of the AI artifacts.

Context: who gets credit

OpenAI says reviewers sign no nondisclosure agreements and are unpaid, and that it seeks no academic credit for its models. It also cited a $5 million contribution to the Simons Foundation, part of a $15 million initiative with Anthropic announced in mid-October, to fund workshops on AI-produced results. The models used are not named.

OpenAI's public math repository lists 719 manuscripts in 372 families and says about 42% of top-line results are formalized in Lean. Its last headline claim, a Navier-Stokes result that landed with an asterisk, set off a credit dispute.

What's next

Lean checks that a proof follows from its stated premises, not that the formal statement matches the conjecture mathematicians mean. That gap is what the review groups must close. The ETH group has invited the community to help organize the examination, and the physics group will release its own write-up.

FAQ

Did OpenAI prove P does not equal NP?
No. The ETH is a stronger, separate assumption about 3-SAT's running time. Refuting it would not settle P versus NP.

Has anyone verified the proofs?
OpenAI says the ETH and Kakeya-related proofs were checked in Lean. Human review is under way, and the ETH reviewers say they are not certifying the claim.

References

• Working with academic communities to build scientific understanding — OpenAI (Oct. 26, 2026)
• openai/math — GitHub
• OpenAI says its AI model has disproved the Exponential Time Hypothesis — The Next Web (Oct. 26, 2026)
• OpenAI tries to solve mathematician backlash problem by pledging $5 million to math foundation — Euronews (Oct. 16, 2026)
• Exponential time hypothesis — Wikipedia