OpenAI says its models disproved the Exponential Time Hypothesis and solved Kakeya and Yang-Mills problems

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OpenAI announced on October 26, 2026 that its internal models have produced three new results that, if they survive expert scrutiny, would rank among the most consequential claims ever attributed to an AI system. The company says one of its models found a disproof of the Exponential Time Hypothesis, the conjecture about Boolean satisfiability that underpins hundreds of conditional hardness results in theoretical computer science. A second result is a claimed proof of the local smoothing conjecture in every dimension, which would also settle the Kakeya conjecture and Stein's restriction conjecture in all dimensions. The third is a claimed rigorous existence proof for certain four-dimensional interacting quantum field theories with gauge fields and fermions, including Quantum Chromodynamics, a unified completion of the Standard Model and pure Yang-Mills theory, whose existence is one of the Clay Millennium Prize problems.

The announcement is as much about process as about the results themselves. OpenAI describes the AI-written manuscripts as research data that still require human interpretation, and it has handed each result to a group of outside experts, which it calls "shepherds", who decide how the material is released, checked and explained. Two of the three results come with Lean formalizations, meaning that a computer proof assistant has checked the formal statements, while the physics result has only a completed Lean formalization of the Yang-Mills part and its AI-produced artifacts are being withheld for now at the request of the physicists overseeing it.

None of the three claims has been peer-reviewed, and OpenAI's own post says independent verification is still under way. The company does not name the model that produced the work, referring only to internal models, and it has not made that system available to the public.

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WHAT OPENAI IS CLAIMING IN EACH FIELD.

The computer science result is the one with the most immediate reach. The Exponential Time Hypothesis, proposed by Russell Impagliazzo and Ramamohan Paturi around the turn of the century, states roughly that 3-SAT cannot be solved in time that is subexponential in the number of variables. Researchers have used it as an assumption to derive lower bounds for a very wide range of problems, so a disproof would turn many conditional statements of the form "this cannot be done faster unless ETH fails" into statements whose premise is false. OpenAI says the natural-language disproof was found on October 16, 2026 and formalized in Lean on October 17.

The documentation that OpenAI published alongside the Lean files gives the formal shape of the claim. For any fixed k of at least 2, a deterministic machine exactly counts the satisfying assignments of a canonically encoded k-CNF formula on n variables in time (L+1)^O_k(1) · 2^O_k(n(log log n)²/√(log n)), where L is the bit length of the encoding, while using only a polynomial number of tape cells. The accompanying manuscript, titled "The Exponential Time Hypothesis is False" and dated October 25, states a sharper bound of (n+m)^O_k(1) · 2^O_k(n log log n/√(log n)) for formulas with m clauses, which becomes 2^O(n/√(log n)) up to polynomial factors when the number of clauses grows at most linearly with the number of variables. Two details matter here: the claimed algorithm counts solutions rather than merely deciding satisfiability, which is a harder task, and it is deterministic rather than randomized.

The mathematics result concerns how waves focus. The local smoothing conjecture, proposed by Christopher Sogge in 1991, is known to imply both the Kakeya conjecture and Stein's restriction conjecture. Larry Guth, Hong Wang and Ruixiang Zhang proved local smoothing in two spatial dimensions in 2019, and Wang and Joshua Zahl proved the Kakeya conjecture in three dimensions in 2025. OpenAI says its model produced a proof of local smoothing in every dimension, built around a novel auxiliary partial differential equation, with the AI output obtained on October 18 and a Lean formalization completed the following day.

The physics result is the broadest in scope. OpenAI says the work gives a mathematically rigorous existence proof for certain interacting quantum field theories in four dimensions with gauge fields and fermions, building on the constructive approach developed by Tadeusz Balaban. According to the company's timeline, the pure Yang-Mills proof was produced on September 9, 2026, the QCD proof on September 18, and the Lean formalization of the Yang-Mills part was completed on October 16.

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Result	Field	AI output	Lean formalization	Public material
ETH is false (k-SAT counting)	Computer science	October 16, 2026	October 17, 2026	Manuscript and Lean proof on GitHub
Local smoothing in all dimensions	Mathematics	October 18, 2026	October 19, 2026	Manuscript and Lean proof on GitHub
Yang-Mills existence	Physics	September 9, 2026	October 16, 2026	Withheld by shepherding group
QCD and Standard Model completion	Physics	September 18, 2026	Not stated	Withheld by shepherding group

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HOW THE RESULTS ARE BEING RELEASED AND CHECKED.

OpenAI chose an initial shepherd for each result, and the shepherds then picked very different release strategies. For the local smoothing proof, the AI-written paper and its Lean proof were published as raw output in OpenAI's public mathematics repository on GitHub, with researchers left to organize their own follow-up work, which mirrors the approach the company used for a batch of results released on October 6. For the Exponential Time Hypothesis, a group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will verify and explain the result; at the group's request, OpenAI agreed to make all raw data public and to fund a conference on the implications. For the physics work, a group led by David Gross and including Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov decided not to release the AI-produced artifacts and to write up the work themselves, with communications routed through a designated email address.

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Result	Shepherds	Release strategy
Local smoothing and Kakeya	Not organized as a group	Raw AI output and Lean proof published
Exponential Time Hypothesis	Impagliazzo, Paturi, Williams	Collaboration plus public raw data and Lean proof
Yang-Mills, QCD, Standard Model	Gross, Douglas, Jaffe, Klebanov, Nekrasov	Artifacts withheld; group writes up the work

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The company lists four principles for these collaborations. Transparency means no non-disclosure agreements. OpenAI says it will claim no academic credit for discoveries made by its internal models, while the outside researchers receive recognition for their own write-ups and analysis. Participation is described as voluntary and unpaid, and neither side speaks on the other's behalf. OpenAI also acknowledges a structural weakness in the arrangement: choosing the first shepherds gives a small set of experts earlier access to the material than everyone else, and the company says it is asking for feedback on that trade-off.

Funding for the wider research community comes through a separate commitment. OpenAI points to a $5 million contribution to a Simons Foundation initiative, first announced in an October 15 update to its earlier mathematics post, which will pay for workshops, conferences and programs aimed at understanding AI-produced results.

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WHAT LEAN VERIFICATION DOES AND DOES NOT ESTABLISH.

A Lean formalization is a strong form of evidence, because the proof assistant mechanically checks every logical step of a formal proof against a small trusted kernel. It is not the same as confirming the headline claim, however. Human experts still have to check that the formal statement says what the informal theorem says, that the definitions used in the formalization, such as the canonical encoding of formulas or the machine model, match the standard ones, and that the proof does not rely on extra axioms or unproven assumptions. OpenAI's scope document for the ETH formalization describes the statement and the comparator file but does not, on its own, settle those questions, which is precisely the job assigned to the Impagliazzo, Paturi and Williams group.

The company's own record shows why this review step matters. The history file of the same GitHub repository records that on October 7, 2026, one day after the first large release, a sign error in one AI-written manuscript invalidated an argument and a construction relied on by two dependent papers, and all three manuscripts were withdrawn, while fourteen others received fixes. The same entry reports that 300 of 719 top-line results in the repository had been formalized at that point, roughly 42%. A formalized result is far less likely to contain that kind of error, which is why the presence of Lean proofs for the ETH and local smoothing claims carries weight, but formalization of the statement-to-meaning link remains a human task.

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WHAT CHANGES FOR RESEARCHERS AND FOR CHATGPT USERS.

For researchers in complexity theory, the practical consequence of a confirmed ETH disproof would be a large-scale audit of conditional lower bounds. Many fine-grained hardness results for graph problems, parameterized algorithms and approximation schemes are stated relative to ETH, and each would need to be reconsidered or re-based on a different assumption. The claimed algorithm does not imply that P equals NP: its running time is still superpolynomial, only faster than any exponential of the form 2^(cn). For working programmers and SAT-solver users, nothing changes immediately, because asymptotic bounds of this type say little about performance on practical instance sizes and OpenAI has published no implementation benchmarks.

For harmonic analysts, the local smoothing material is available now in the public repository, and the open release means anyone can start checking the argument and the novel auxiliary equation it relies on. Physicists have no artifacts to examine yet, since the Gross group has chosen to produce its own account before anything is published.

For ChatGPT users, the announcement does not come with a product. The system behind these results is an internal model that OpenAI has not released. For the October 6 batch of results, the company described the average compute per result as roughly equivalent to three hours of ChatGPT Pro reasoning, and said it is working to release the underlying model responsibly, but it has given no date, plan tier or pricing for that release, and it has published no compute figures for the three new results.

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WHAT IS ESTABLISHED AND WHAT REMAINS UNCERTAIN.

What can be verified today is that OpenAI has made the three claims in an official post dated October 26, 2026, that the ETH and local smoothing manuscripts and their Lean proofs are publicly available, and that named senior researchers in each field have agreed to oversee verification and interpretation. The formal statement attached to the ETH claim is specific: deterministic exact counting of k-SAT solutions in time 2^O(n(log log n)²/√(log n)) times a polynomial factor, with polynomial space.

Everything else is still open. No independent expert has publicly confirmed any of the three results, none has passed peer review, and the physics claims, including the Yang-Mills existence proof, cannot be examined by outsiders until the shepherding group publishes its write-up. The correspondence between the Lean statements and the classical conjectures needs human checking, and OpenAI's own withdrawal of three manuscripts earlier this month shows that AI-produced mathematics can contain errors that survive initial review. If the ETH and local smoothing claims hold, they would resolve conjectures that have shaped their fields for decades, with an AI system credited as the primary source of the proofs; until the expert groups report, they remain company claims backed by machine-checked formal proofs whose interpretation is not yet settled.

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DATA STUDIOS

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