🔍 Read the full analysis: The Exponential Time Hypothesis: What OpenAI Says Its AI Demonstrated on ThorstenMeyerAI.com
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TL;DR
OpenAI says an unreleased AI model produced a proof of a deterministic, subexponential-time algorithm for exactly counting solutions to k-SAT, which would contradict the Exponential Time Hypothesis for 3-SAT. Lean checked a formalization of a slightly weaker bound, while independent expert review is still pending; the result does not resolve P versus NP or make practical problems faster now.
OpenAI said on October 26 that an unreleased internal AI model had found a proof of a deterministic subexponential-time algorithm for exactly counting solutions to k-SAT, a result that would refute the Exponential Time Hypothesis for 3-SAT. The company published a preprint and Lean proof formalization, but independent experts have not yet verified the result, and the formalized statement uses a weaker runtime bound than the paper’s sharpest claim.
The preprint describes an algorithm that exactly counts satisfying assignments for any fixed clause width k using subexponential time and polynomial memory. For 3-SAT, that would contradict ETH, which says that no algorithm can solve 3-SAT in subexponential time. The paper also says the result refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses. Those are claims in the preprint, not conclusions independently endorsed by the field.
OpenAI says its model found the disproof on October 16 and that the Lean formalization followed the next day. The preprint is dated October 17 and was revised October 25. The source account reports that OpenAI released a Lean repository containing 3,977 files and about 710,000 lines, while describing the core formal statement as short enough to inspect separately. Lean is a proof assistant: it checks whether a formal proof follows from its stated definitions and rules. That check is not the same as independent review of the research argument or its significance.
A key qualification is that the Lean formalization covers a slightly weaker runtime bound than the sharpest one in the paper. The preprint’s most precise algorithmic claim is not fully covered by the published formalization, according to the source account. OpenAI has asked an outside group to examine the computer-science result; that review is not complete.
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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What an ETH Refutation Would Change
If the result survives review, it would overturn a major working assumption in theoretical computer science: that 3-SAT cannot be solved in subexponential time. ETH has been used to establish conditional lower bounds for many other problems, so a valid refutation could require researchers to revisit results whose difficulty depends on that hypothesis. The extent of those consequences would depend on the proof’s exact scope and on which reductions and assumptions other results use.
The stated algorithm would not make hard instances easy in practical terms. Its runtime is described as roughly 2 raised to n divided by the square root of log n—subexponential, but still much slower than any polynomial-time algorithm. OpenAI said practical improvements would require significant further research. The result is therefore more immediately important as a theoretical change and as a test of AI-assisted mathematics than as a tool for solving everyday computing problems.
The process also matters. An AI-produced argument has been paired with a machine-checked formalization and an outside review group. That offers a way to test complex claims, but the gap between the paper’s strongest bound and the formalized result shows why those checks must be described precisely. Formal verification can establish that a coded proof follows within a formal system; experts still need to judge whether the definitions capture the intended theorem and how the result fits the wider literature.
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ETH, k-SAT and the OpenAI Release
In k-SAT, a formula is made up of clauses, each containing at most k variables or their negations. The task is to find an assignment of true or false values that satisfies every clause, or, in the result described here, to count all satisfying assignments. A formula with n variables has 2ⁿ possible assignments, so checking them all takes exponential time. Researchers have developed faster-than-brute-force algorithms, but ETH predicts that the exponential dependence cannot be eliminated for 3-SAT.
Russell Impagliazzo, Ramamohan Paturi and Francis Zane proposed ETH in 2001. It is stronger than the unresolved P versus NP question: refuting ETH would not by itself show that P equals NP. The source account notes that ETH has been used in many conditional hardness results, while also citing a 2017 survey by Scott Aaronson that cautioned there was no consensus that ETH is true. The claim that the hypothesis is broadly believed should therefore be treated with care.
The ETH work was one of three research claims OpenAI announced October 26. The company also described a mathematics result concerning Sogge’s local smoothing conjecture and a physics result concerning quantum chromodynamics and a proposed completion of the Standard Model. According to OpenAI, outside academic groups are advising on how the manuscripts are handled. The source account says the physics materials are being held back, while the ETH preprint and Lean code were released at the reviewing group’s request.
Questions Awaiting Expert Review
The most immediate open question is whether independent reviewers will validate the proof and its implications. The source account says a group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine it; Impagliazzo and Paturi were among ETH’s original authors. Their review has not yet been reported as complete, and no independent conclusion is available in the supplied material.
It is also not clear from the announced materials whether the paper’s sharpest runtime bound can be formalized in Lean, or whether the weaker bound currently checked is sufficient to establish every headline claim. The source account distinguishes the paper’s result from the formalized statement, so they should not be treated as identical. The proof’s eventual standing will depend on the reviewers’ assessment of both the mathematics and the correspondence between the paper and the formal code.
More broadly, the supplied material does not establish what practical applications, if any, could follow, or when. It also does not show that the result changes the status of P versus NP. That problem remains open, and nothing in the described theorem supplies a polynomial-time algorithm for 3-SAT.
Review of the Proof and Formalization
The next milestone is the external group’s examination of the preprint and Lean code. Its findings could confirm the stated result, identify errors or gaps, or prompt revisions. OpenAI has also said it agreed to fund a conference on the result, which could give researchers a forum to discuss the proof and its consequences. No conference date or review timetable is specified in the supplied account.
Until that process produces a public assessment, the careful description is that OpenAI claims its AI found a proof, and that a Lean formalization checks a related, weaker bound. Readers should watch for a review that states exactly which theorem has been verified, whether the formalization matches the paper’s strongest claim, and which consequences follow if the proof holds.
Key Questions
What does OpenAI say its AI proved?
OpenAI says an internal AI model produced a proof of a deterministic, subexponential-time algorithm for exactly counting solutions to k-SAT when k is fixed. For 3-SAT, that would refute the Exponential Time Hypothesis.
Has the proof been independently verified?
Not yet, according to the supplied account. Lean checked a formalized statement, but an outside group is expected to review the research. The formalization covers a slightly weaker runtime bound than the paper’s sharpest claim.
Does this solve P versus NP?
No. Refuting ETH would not settle P versus NP. The source material says P versus NP remains open, and the claimed algorithm is still slower than any polynomial-time algorithm.
Will the result make SAT problems faster to solve now?
There is no evidence of an immediate practical speed-up. The runtime described is subexponential but still very large, and OpenAI said practical improvements would require significant further research.
What happens next?
An external group is expected to examine the preprint and Lean formalization. OpenAI has also agreed to fund a conference, but the supplied account gives no date for the review or event.
Source: ThorstenMeyerAI.com
