OpenAI said on October 26 that one of its internal models disproved the Exponential Time Hypothesis, a conjecture about how hard Boolean satisfiability problems are to solve and one that many results in computer science depend on. According to OpenAI, the model found the disproof on October 16, and the result was formalized in the Lean proof assistant the next day. The AI-written paper and the Lean proof are posted publicly on GitHub.
The ETH claim is one of three results OpenAI announced together. The company says its models also produced a proof of the local smoothing conjecture in every dimension, which would resolve the Kakeya conjecture and Stein's restriction conjecture in all dimensions as well. The third is a rigorous proof that certain quantum field theories exist in four dimensions, including quantum chromodynamics, with a proof of the existence of pure Yang-Mills theory as a subproblem.
Russell Impagliazzo and Ramamohan Paturi first conjectured the Exponential Time Hypothesis in 1999. Both of UC San Diego, they will now lead an examination of OpenAI's claim alongside Ryan Williams of MIT. In an open letter, the three researchers said their participation is not an endorsement of the result.
“Our involvement should not be taken as certification of the claim.”— Russell Impagliazzo, Ramamohan Paturi and Ryan Williams, Leads of the ETH review group, UC San Diego and MIT
Key facts
- 01OpenAI says an internal model disproved the Exponential Time Hypothesis on October 16, 2026, verified in Lean on October 17.
- 02OpenAI also claims a proof of the local smoothing conjecture in every dimension, which would settle the Kakeya conjecture.
- 03A third claim covers the existence of quantum chromodynamics and pure Yang-Mills theory in four dimensions.
- 04Russell Impagliazzo, Ramamohan Paturi (UC San Diego) and Ryan Williams (MIT) will lead the ETH review, unpaid and with no NDAs.
- 05OpenAI gave $5 million to the Simons Foundation for workshops on AI-produced results and seeks no academic credit.
Williams said the group signed no non-disclosure agreements and will not be paid. OpenAI agreed to the group's requests to make all raw data public and to fund a conference on the work's implications, and the group has invited the research community to help organize an open examination of the claim. In a post on X, Williams summarized the claim as 3SAT on n variables running in roughly exp(n/(log n)^(1/2+o(1))) time.
The math result builds on a line of work that began when Christopher Sogge proposed the local smoothing conjecture in 1991. 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 model produced the all-dimensions proof on October 18, and it was formalized in Lean on October 19. That paper and Lean proof are also on GitHub.
The physics result is being handled differently. A group led by Nobel laureate David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, will examine it. OpenAI says the Yang-Mills proof was produced on September 9 and the QCD proof on September 18, with the Yang-Mills Lean formalization finished on October 16. Citing the length and complexity of the work, the Gross group has chosen not to release the AI-produced material for now and will write up the work itself.
The pure Yang-Mills proof relates to one of the Clay Mathematics Institute's Millennium Prize problems, but OpenAI's claim covers existence only. The Clay problem also requires a mass gap, which OpenAI does not claim. A Lean formalization checks a proof against its formal statement, so reviewers will also need to confirm that each formal statement matches the original conjecture.
OpenAI did not name the model behind the results. The company described the manuscripts as a form of raw research data and set out four principles for the collaborations: they are publicly acknowledged with no NDAs, OpenAI claims no academic credit, the working groups get credit for their own understanding and writeups, and participation is voluntary and unpaid.
OpenAI also said it has contributed $5 million to the Simons Foundation to fund workshops, conferences and special programs on understanding AI-produced results. On the ETH disproof specifically, OpenAI said turning it into practical algorithms will require significant further research.
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