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TL;DR
OpenAI says an unreleased AI model produced a proof of a subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT, which would refute ETH for 3-SAT. Lean checked a formalized version, but it covers a weaker runtime bound than the preprint’s sharpest claim, and independent experts have not yet completed their review.
OpenAI said October 26 that an unreleased AI model produced a proof of a subexponential-time algorithm for exactly counting solutions to k-SAT, a result that would refute the Exponential Time Hypothesis (ETH) for 3-SAT. The company released a preprint and a Lean formalization, but the machine-checked statement is weaker than the preprint’s sharpest runtime claim, and independent expert review is still underway.
The result concerns exactly counting satisfying assignments for k-SAT formulas, where each clause contains at most a fixed number k of variables or their negations. OpenAI’s preprint claims a deterministic algorithm with subexponential running time and polynomial memory. For 3-SAT, such an algorithm conflicts with ETH, which says that no subexponential-time algorithm can solve 3-SAT.
OpenAI says its model found the disproof on October 16, and that a Lean formalization followed the next day. The preprint is dated October 17 and was revised October 25. The company published the paper and formalization after consultation with outside academic experts, whom it calls “shepherds.” According to the source report’s count, the Lean repository contains 3,977 files and about 710,000 lines; the formal statement itself is much shorter.
The distinction between the paper and the formal check matters. The Lean code verifies a slightly weaker runtime bound than the sharpest one claimed in the preprint; that strongest bound is not included in the formalization. Lean’s check supports the formalized statement, but it does not by itself establish that every claim in the paper has been independently validated. OpenAI’s preprint also says the result refutes the Strong Exponential Time Hypothesis and three related hypotheses.
THREE RESULTS, THREE RELEASE STRATEGIES
What OpenAI released, what Lean checked, who reviews it
Dates and the physics status are OpenAI's own account. Lean-checked means a machine verified the proof, not that experts have accepted it.
Local smoothing, so Kakeya too
- AI output
- 18 Oct 2026
- Lean
- 19 Oct 2026 · strict ranges
- Public
- AI write-up + Lean proof
- Review
- No group: researchers organize themselves
The Exponential Time Hypothesis is false
- AI output
- 16 Oct 2026
- Lean
- 17 Oct 2026 · weaker bound
- Public
- 67-page preprint + Lean proof
- Review
- Impagliazzo, Paturi (UC San Diego), Williams (MIT)
QCD exists; pure Yang–Mills as sub-result
- AI output
- 9 Sep (Yang–Mills), 18 Sep (QCD)
- Lean
- Yang–Mills existence, 16 Oct (OpenAI)
- Public
- No: the group holds the AI materials back
- Review
- Gross, Douglas, Jaffe, Klebanov, Nekrasov
Before this release
The repository the results sit in
- 6 Oct: more than 700 AI-generated manuscripts published
- 7 Oct: 3 withdrawals after a sign error; 14 other manuscripts revised
- About 42% of top-line results formalized in Lean (repository’s own figure)
- OpenAI calls the manuscripts “raw research data”, not polished papers
Sources: OpenAI, 26 Oct 2026 · Lean note 378 · Lean note 079 · repository history. Status chips: our assessment.
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A Challenge to ETH, Not P vs NP
If the preprint’s result survives review, it would overturn a widely used assumption about the difficulty of 3-SAT and related problems. ETH has helped researchers establish conditional lower bounds: results of the form “this problem cannot be solved much faster unless ETH is false.” A refutation would mean those arguments need to be revisited, or restated under other assumptions.
That does not mean difficult problems become practical overnight. The claimed algorithm runs in roughly 2^(n / √log n), according to the source material. That is subexponential but still much slower than polynomial time, and its advantage over exponential approaches grows slowly as problem size increases. OpenAI itself said practical improvements would require significant further research. The result is not a solution to P vs NP, which remains open.
The announcement also draws attention to a research process: an AI system generated a mathematical result, and proof-assistant software checked a formal version. That is meaningful evidence about the checked statement, but it is not a substitute for specialists examining the reasoning, assumptions and relationship between the formalization and the full paper. The review will help establish how far the result can be trusted and what consequences follow.
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How ETH Became a Benchmark
In k-SAT, a formula consists of clauses such as “A or not-B or C”; the question is whether truth values can be assigned to satisfy every clause. With n variables, exhaustive search considers 2^n assignments. Researchers have developed faster-than-brute-force algorithms, but ETH posits that 3-SAT still requires exponential time in the worst case.
Russell Impagliazzo, Ramamohan Paturi and Francis Zane formulated ETH in 2001. The source report says an external group led by Impagliazzo and Paturi, two of ETH’s original authors, will examine the new result. The report also notes that expert views on ETH have not been unanimous: computer scientist Scott Aaronson wrote in a 2017 survey that there was “by no means a consensus in the field that ETH is true.”
OpenAI’s October 26 post announced three research claims, not just the computer-science result: a proof concerning local smoothing in harmonic analysis and a physics result involving quantum chromodynamics and a grand unified completion of the Standard Model. The company said each project had outside academic stewards and different release plans. The ETH preprint and Lean code were made public; the physics materials were held back. OpenAI’s account of the research process and claims remains the company’s own account.
What the Formal Check Covers
The main open issue is whether independent researchers accept the proof and its implications. The external review is pending, and no completed assessment from the reviewing group is included in the source material. In particular, the sharpest runtime bound in the preprint is not formally checked in the released Lean code. The exact gap between that claim and the weaker formal statement, and whether it affects the paper’s central conclusion, will need close scrutiny.
It is also not yet clear how the claimed refutation of SETH and the three related hypotheses will be assessed, or which conditional hardness results would change if the work is validated. The source report describes the repository’s prior corrections: after a separate October 6 release of AI-generated manuscripts, OpenAI withdrew three papers over a sign error and revised 14 others. That history does not establish a problem with this proof, but it underscores why formal checks and independent review matter.
The announced algorithm is a theoretical result. The available material does not show that it has been implemented in a way that improves practical solving or counting on real-world instances. The result should not be described as proof that all hard problems are easy, or that P equals NP.
Independent Review and Follow-Up
The next major milestone is the external group’s examination of the preprint and formalization. Its review can clarify whether the proof is sound, whether the Lean development matches the paper’s claims, and how much of the claimed runtime result is established by the checked version. OpenAI said it will fund a conference on the result, creating another venue for researchers to question and discuss the work.
For now, the public record consists of OpenAI’s announcement, the preprint and the released Lean code, alongside the stated review plan. Readers should distinguish the company’s account from conclusions reached by independent experts. If the review confirms the result, researchers can then assess which ETH-based lower bounds need revision and whether any algorithmic techniques have practical use. Until then, the broadest claims remain open to scrutiny.
Key Questions
What does OpenAI say its model proved?
OpenAI says the model produced a proof of a deterministic subexponential-time algorithm for exactly counting satisfying assignments to fixed-width k-SAT formulas. For 3-SAT, that would contradict ETH.
Has the entire result been checked by Lean?
No. Lean checked a formalized statement with a slightly weaker runtime bound than the preprint’s sharpest claim. The sharpest bound is not covered by the released formalization.
Does this settle P vs NP?
No. The claimed algorithm is still slower than polynomial time. P vs NP remains an open problem, and a refutation of ETH does not by itself resolve it.
Has the proof been independently accepted?
Not yet, based on the source material. An external group led by Russell Impagliazzo and Ramamohan Paturi is expected to examine the work.
Will the result make SAT problems faster to solve now?
No immediate practical speed-up has been established. OpenAI said practical improvements would require significant further research, and the reported algorithm is a theoretical result.
Source: ThorstenMeyerAI.com
