OpenAI said on 26 October 2026 that its system has proved the Exponential Time Hypothesis (ETH) false, a result it says was subsequently verified in Lean. The ETH is a long-standing conjecture about how hard it is to solve Boolean formula satisfiability, a problem OpenAI describes as deeply related to P vs. NP.
The ETH claim was one of three results OpenAI announced in its post, “Working with academic communities to build scientific understanding”: a proof in harmonic analysis, the ETH refutation, and an existence proof for quantum chromodynamics that OpenAI says includes pure Yang–Mills theory. Outside groups have been asked to examine the claims.
Business Pill · BENCHMARKS
A short Business Engineer explainer on benchmarks: why a test score is not your own result. It teaches the general idea only and says nothing about OpenAI or these proofs.
The key insight: As we read it, OpenAI is positioning its models as producers of research results, not only research tools: it says they produced three proofs, and it has handed the checking to outside experts, including two of the ETH’s original proposers.
OpenAI said it found its natural-language disproof of the ETH on 16 October 2026 and formalized it in Lean the next day. It said the ETH “has been used in hundreds of papers to derive conditional computational hardness results” in fields from graph theory to databases.
OpenAI said the refutation “reopens questions about how much faster these tasks could become”, and added that turning it into practical improvements “will require significant further research.” It has published the AI-written proof and the Lean proof in its public GitHub repository.
Field
What OpenAI says its models produced
Dates OpenAI gives
Who will examine it
Mathematics
A proof of the local smoothing conjecture in every dimension, which it says also resolves the Kakeya and Stein restriction conjectures
Output 18 Oct, Lean 19 Oct
Released publicly for any researcher
Computer science
A proof that the Exponential Time Hypothesis is false
Disproof 16 Oct, Lean 17 Oct
Impagliazzo, Paturi (UC San Diego), Williams (MIT)
Physics
An existence proof for QCD and a unified completion of the Standard Model, including pure Yang–Mills theory
Proofs 9 and 18 Sep, Lean (Yang–Mills only) 16 Oct
A group led by David Gross
Source: OpenAI, 26 October 2026. All entries are OpenAI’s own descriptions; none has yet been independently confirmed.
The dates OpenAI gives for each AI-produced proof and its Lean formalization, from OpenAI’s announcement of 26 October 2026. All three results are OpenAI’s claims.
Who Will Check the ETH Claim
OpenAI said the ETH work will be verified and shepherded by a group led by Russell Impagliazzo and Ramamohan Paturi of UC San Diego and Ryan Williams of MIT. It noted that Impagliazzo and Paturi, with Francis Zane, are the original proposers of the ETH.
Williams called it “a claimed refutation of ETH” on X, and said the group has “signed no NDAs and will receive no compensation.” OpenAI said it agreed to the group’s requests to make all raw data public and to fund a conference on the implications.
The Physics and Math Results
OpenAI said its models produced a natural-language proof of Yang–Mills existence on 9 September 2026 and a proof for QCD on 18 September, and finished the Lean formalization of Yang–Mills existence on 16 October. It said pure Yang–Mills existence is part of a Clay Mathematics Institute Millennium Prize problem.
A group led by Nobel laureate David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, will shepherd the physics work, OpenAI said, and has chosen to hold back the AI-produced artifacts and write the result up itself. For the math result, OpenAI released the AI output and a Lean proof publicly.
OpenAI said it does not seek academic credit for the discoveries produced by its internal models, and that working-group members participate voluntarily, unpaid and independently.
As we read it, the release design matters as much as the claims: OpenAI says each collaboration works without NDAs, unpaid and independently, and that it seeks no academic credit for the discoveries.
As we read it, Lean formalization is the part OpenAI leans on for trust: it says each math and ETH result was formalized within a day or so of the AI output, and the Yang–Mills formalization finished on 16 October.
As we read it, this lands on the same day Anthropic’s S-1 described using Claude to write over 90% of its own new code, so both labs are now describing their models as doing core work inside the lab.
OpenAI, Working with academic communities to build scientific understanding, 26 October 2026
“Surprisingly, our system has recently proved that the ETH is false, a result that was subsequently verified in Lean”
Three Implications
FOR RESEARCHERSOpenAI says the ETH proof and its Lean formalization are public in its GitHub repository, and the examining group invites the community to help organize an open examination.
FOR THE PHYSICS COMMUNITYOpenAI says the group led by David Gross has chosen to hold back the AI-produced artifacts and write the result up itself.
FOR FUNDERS AND INSTITUTIONSOpenAI says it agreed to fund a conference on the implications of the ETH work, alongside its $5 million contribution to the Simons Foundation.
The framework describes “The Verifier Moat” as one “built by RLVR on domain-specific reward signals.”
As we read it, a Lean check is that kind of verifier signal for mathematics, which is why OpenAI pairs each claim with a formal proof. The framework was published in July 2026 and does not discuss these results.
What Is Not Established
The three results are OpenAI’s claims. The math and ETH results were published with natural-language proofs and Lean formalizations that outside researchers have only begun to examine; the physics artifacts have not been released. A Lean check confirms a proof of the formal statement it is given; whether those statements match the conjectures is part of what the groups are examining. No peer-reviewed paper has been published, and OpenAI does not say which model produced the results.
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OpenAI said on 26 October 2026 that its system proved the Exponential Time Hypothesis false, a result verified in Lean, alongside a harmonic-analysis proof and a quantum field theory existence proof. A group including two of the ETH’s original proposers will now examine the claim in public.
A note on sourcing. This piece is based on OpenAI’s announcement and posts on X by OpenAI staff and Ryan Williams, all read on 26 October 2026, and on our own earlier reporting. We have not examined the proofs ourselves. We did not contact OpenAI.
Gennaro Cuofano is a CRO and tech executive who has worked in AI since late 2015, starting with bringing NLP, natural-language generation, voice and chatbot products to the marketing industry. His background is in law and finance: he holds a Master's degree in Law and an International MBA with an emphasis on corporate finance (LUISS Business School and the University of San Diego, 2012), and worked as a financial analyst at a real-estate investment firm in San Diego and as an assistant controller. His work focuses on business model strategy, business engineering and, more broadly, structural analysis: how companies make money, read from their own filings. He created FourWeekMBA and leads research there and at The Business Engineer, his newsletter on AI and business strategy, with over 95,000 subscribers and more than 1,000 published analyses. His writing has also appeared on Entrepreneur (for example, How to Digitize a Family Business in Europe) and HackerNoon. Find him on LinkedIn and Substack. See how we source.
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