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TL;DR
OpenAI says an unreleased internal model produced a proof that refutes the Exponential Time Hypothesis, a long-standing conjecture about the limits of solving 3-SAT quickly. Lean has checked a formalized version, but it covers a weaker runtime bound than the preprint’s sharpest claim, and independent expert review is still pending. The result does not solve P versus NP or make practical computing problems immediately faster.
The preprint claims a deterministic algorithm that exactly counts solutions to any fixed-width k-SAT formula in subexponential time, using polynomial memory. For 3-SAT, that would contradict ETH, which says no algorithm can solve the problem in subexponential time. The paper also says the result refutes the stronger SETH and three related hypotheses.
OpenAI’s account says the model found the disproof on October 16 and the Lean formalization followed a day later. The preprint is dated October 17 and was revised on October 25. OpenAI published the paper and code at the request of an outside group that will examine the result. The source account reports that its scan counted 3,977 files and about 710,000 lines of Lean code; those figures are the author’s count, not a stated official total.
The formalization matters, but it does not check the paper’s sharpest stated runtime bound. According to the source account’s review of the formal statement and Lean scope notes, the machine-checked result establishes a slightly weaker bound. Lean can verify that a formal proof follows from its definitions and rules; that does not, on its own, establish that the result has survived independent scrutiny of the mathematical framing or the relationship between the paper and the formalized theorem.
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 New Route to Checking AI Proofs
If the result survives expert review, it would overturn a widely used assumption in theoretical computer science. ETH has supported conditional hardness results: researchers use it to argue that particular algorithms cannot run substantially faster unless ETH is false. A refutation would require those arguments to be reconsidered, although it would not automatically invalidate every result that has relied on the hypothesis.
The immediate significance may be as much about how the proof was produced and checked as about its conclusion. An AI system generated the argument, a proof assistant checked a formal version, and outside researchers are being asked to assess the work. That sequence offers a possible model for evaluating computer-generated mathematics while leaving a distinct role for human review.
There is no stated near-term practical speed-up for software or industry. The claimed algorithm is still slower than any polynomial-time algorithm, and OpenAI says practical improvements would require significant further research. The source account’s arithmetic also describes the speed-up as growing very slowly with problem size, so a theoretical change in asymptotic complexity should not be mistaken for a faster solver available to users now.
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What ETH Says About 3-SAT
In k-SAT, each rule places a limit of k on the number of variables, or their negations, in a clause. The task is to determine whether assignments of true or false can satisfy all clauses. With n variables, trying every assignment takes up to 2ⁿ combinations. Researchers have found faster approaches than brute force, but these still take exponential time.
ETH, formulated by Russell Impagliazzo, Ramamohan Paturi and Francis Zane in 2001, asserts that 3-SAT cannot be solved in subexponential time. It is stronger than the claim that P is not equal to NP. Refuting ETH would weaken that particular assumption about computational hardness; it would not resolve P versus NP, which remains open.
OpenAI’s October 26 announcement also described results in mathematics and physics. The source material says a mathematics manuscript concerns Sogge’s local smoothing conjecture, while a physics manuscript concerns existence results related to quantum chromodynamics and a grand unified completion of the Standard Model. Those are separate claims and do not serve as independent confirmation of the ETH result.
Expert Review and Bound Still Pending
The central open question is whether independent experts agree that the proof establishes what the preprint claims. A group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is expected to examine it; Impagliazzo and Paturi were two of ETH’s three original authors. The supplied material says that review has not yet produced a public sign-off.
There is also a specific scope gap: the Lean formalization checks a slightly weaker runtime bound than the sharpest bound in the paper. The available account does not provide enough detail to reconcile that difference or establish whether the stronger bound follows from the same argument. The distinction is material when describing precisely what a machine has verified.
Earlier releases make the checking process relevant. The source says OpenAI withdrew three manuscripts from a repository on October 7 after a sign error and revised 14 others; it also says about 42% of the repository’s top-line results were formalized in Lean. These details do not show that the ETH proof is wrong, but they underline why publication, machine checking and independent validation should not be treated as interchangeable.
Reviewers’ Assessment Comes Next
The next milestone is the external group’s examination of the preprint and its relationship to the Lean code. Readers will need to see whether reviewers endorse the theorem as stated, identify revisions, or find that the sharpest runtime claim is not established. OpenAI has also agreed to fund a conference on the result, according to the source material.
Until that review is public, the most precise description is that OpenAI has announced an AI-produced proof claiming to refute ETH, and that Lean has checked a formalized version with a narrower bound. Any change to computer science textbooks, conditional lower bounds or practical algorithms depends on what scrutiny establishes next.
Key Questions
What does OpenAI say its AI proved?
OpenAI says an internal model produced a proof of a deterministic subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT formulas. For 3-SAT, the preprint says this contradicts the Exponential Time Hypothesis.
Has the proof been independently confirmed?
Not in the supplied material. Lean has checked a formalized version, and an outside group is expected to review the work, but the source account says experts have not yet signed off.
Does this solve P versus NP?
No. The claimed result challenges ETH, a stronger hypothesis about the difficulty of 3-SAT. P versus NP remains an open problem.
Does the result make computers faster at solving these problems now?
No practical improvement is established. The claimed algorithm remains slower than polynomial time, and OpenAI says practical gains would require significant further research.
What has Lean actually checked?
The formalization checks a version of the result with a slightly weaker runtime bound than the preprint’s sharpest claim, according to the source account. The gap and the independent review remain important to understanding exactly what has been established.
Source: ThorstenMeyerAI.com
