🔍 Read the full analysis: OpenAI Says AI Disproved The Exponential Time Hypothesis—What Was Actually Proven? on ThorstenMeyerAI.com
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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 SAT formulas, which would contradict the Exponential Time Hypothesis. A Lean formalization checks a slightly weaker bound than the preprint’s sharpest result, and independent expert review has not yet been completed. The claim does not solve P versus NP or show that practical computing will speed up.
OpenAI said October 26 that an unreleased AI model produced a mathematical proof of an algorithm that would refute the Exponential Time Hypothesis (ETH), a longstanding conjecture about the difficulty of solving Boolean satisfiability problems. The company released a preprint and Lean proof-checking code, but the result has not yet received independent expert review, and the formalized statement is weaker than the paper’s sharpest runtime claim.
The preprint claims a deterministic algorithm that exactly counts satisfying assignments for any fixed-width k-SAT formula in subexponential time and polynomial memory. For 3-SAT, that would contradict ETH, which says no algorithm can solve the problem in subexponential time. The paper also says its result refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses.
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 source account reports that the public Lean repository contains 3,977 files and about 710,000 lines; that count is the account’s own, rather than an independently verified figure here.
Lean is software that checks whether a formal proof follows from its stated definitions and rules. The source account says the formalization verifies a slightly weaker runtime bound than the preprint’s sharpest claim, so it does not mechanically establish every detail of the strongest version. OpenAI says an outside group will examine the computer-science result, with researchers including Russell Impagliazzo and Ramamohan Paturi, two of ETH’s original authors, involved in the review.
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 paper’s central claim survives review, it would overturn a widely used assumption in theoretical computer science. ETH is often used to show that particular algorithms cannot run substantially faster than existing approaches. A valid refutation could require researchers to revisit results that rely on that assumption and could change how the limits of exact computation are understood.
The immediate significance is primarily theoretical, not practical. The claimed runtime is roughly 2 raised to n divided by the square root of log n, according to the source material. That is subexponential, but it remains far slower than polynomial time. OpenAI itself said practical improvements would require substantial further research; the announcement does not establish that current software or commercial computing tasks will get faster.
The process also matters: an AI-produced argument is being paired with a formal proof checker and scrutiny by specialists. That offers a possible route for evaluating machine-generated mathematical work, but a checker only verifies the statement encoded in its formalization. It does not, by itself, establish that the paper’s informal claims and the formal statement match in every important respect.
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How ETH Relates to SAT
In k-SAT, a formula is made up of clauses, each involving at most k variables. The question is whether the variables can be set true or false so that every clause is satisfied. With n variables, trying every assignment takes 2 to the n possibilities. Researchers have developed faster methods, but ETH asserts that for 3-SAT the exponential dependence cannot be eliminated: the running time must remain at least 2 raised to a positive constant times n.
ETH is stronger than the famous P versus NP question. P versus NP asks whether every problem whose proposed solution can be checked efficiently can also be solved efficiently. Refuting ETH would not prove P equals NP: the claimed algorithm remains slower than polynomial time, and P versus NP remains open. The source material also notes that expert opinion on ETH has not been unanimous, despite its extensive use in research.
OpenAI announced three research claims on October 26, according to its account: the ETH result, a mathematical result concerning local smoothing and related conjectures, and a physics result concerning quantum chromodynamics and a possible completion of the Standard Model. The physics materials are being held back at the request of their outside stewards, while the ETH preprint and code were released. The ETH claim is the one addressed here.
The Proof Still Needs Review
The central uncertainty is whether independent specialists will confirm the paper’s reasoning and whether the formal Lean statement fully supports the preprint’s strongest claim. The source material says the formalization checks a weaker bound, and that experts have not yet signed off on the result. The planned review is a step toward independent assessment, not a completed validation.
It is also not yet clear what revisions, if any, reviewers will request, when their assessment will be published, or whether the result will be accepted in a peer-reviewed venue. OpenAI’s statements about how the model found the proof and how the collaboration was organized are the company’s account; they do not replace an independent technical evaluation.
Separately, an ETH refutation would not settle P versus NP, establish polynomial-time algorithms for SAT, or show that practical applications will benefit. Those conclusions do not follow from the claim described in the preprint.
Independent Review and Formal Checks
The next milestone is the outside group’s examination of the preprint and its Lean formalization. Reviewers will need to assess the mathematical argument, clarify how the formalized bound relates to the paper’s sharper runtime, and report whether the result withstands scrutiny. The source material says OpenAI has agreed to fund a conference on the work, but gives no date for that meeting.
Until that review is complete, the result should be described as OpenAI’s announced claim, supported by a public formalization of a weaker statement, rather than as an independently confirmed overturning of ETH. Any changes to the manuscript or code, and the reviewers’ conclusions, will determine how the claim is assessed by the wider research community.
Key Questions
What does OpenAI say its AI proved?
OpenAI says its model produced a proof of a deterministic subexponential-time algorithm for exactly counting satisfying assignments to fixed-width k-SAT formulas. Applied to 3-SAT, that result would contradict ETH.
Has the result been independently confirmed?
No. The preprint and Lean code are public, but the source material says independent expert review is pending. The formalization checks a slightly weaker runtime bound than the paper’s sharpest claim.
Does this mean P equals NP?
No. The claimed algorithm is still slower than polynomial time. Refuting ETH would challenge a stronger hypothesis, but it would not resolve the open P-versus-NP problem.
Will this make SAT or other computing tasks faster now?
There is no reported immediate practical improvement. OpenAI said further research would be needed to translate the theoretical result into practical gains.
Why does the Lean proof matter?
Lean checks whether a formal proof follows from encoded definitions and rules, helping identify logical errors. Its check applies to the formalized statement, however, and reviewers still need to assess how that statement relates to the full preprint.
Source: ThorstenMeyerAI.com
