🔍 Read the full analysis: Did AI Disprove A Major Hypothesis? What OpenAI’s Work Actually Shows on ThorstenMeyerAI.com
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TL;DR
OpenAI says an unreleased AI system produced a proof of a subexponential-time algorithm for exactly counting solutions to k-SAT, which would refute the Exponential Time Hypothesis. Lean checked a slightly weaker formal statement, while independent expert review is still pending. The result does not solve P vs. NP or show that practical computing will become faster.
OpenAI said October 26 that an unreleased internal 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). The proof has been checked by the Lean proof assistant, but that formalization covers a slightly weaker runtime bound than the preprint claims, and independent expert review has not yet been completed.
The preprint describes a deterministic algorithm that exactly counts satisfying assignments for any k-SAT formula when k is fixed, using subexponential time and polynomial memory. For 3-SAT, that would contradict ETH, which holds that solving 3-SAT requires time exponential in the number of variables. The preprint also says its result refutes the stronger Strong Exponential Time Hypothesis, or SETH, and three related hypotheses. Those conclusions depend on the proof being correct and the stated result surviving outside scrutiny.
OpenAI released the preprint and Lean code at the request of the outside group tasked with examining the computer science work. The source reports that its own scan counted 3,977 files and about 710,000 lines of Lean code; it also says the formal statement itself is short. Lean checks whether a proof follows from its formal definitions and rules. In this case, however, the formalized statement is weaker than the sharpest runtime bound in the paper, so the machine check should not be described as verifying every claim in the preprint.
OpenAI says its system found the ETH result on October 16 and that formalization followed the next day. The preprint is dated October 17 and was revised October 25. OpenAI announced it alongside two other research claims: a mathematics result concerning Sogge’s local smoothing conjecture and a physics result concerning quantum chromodynamics and a proposed completion of the Standard Model. Those projects have separate review arrangements; they do not independently establish the computer science claim.
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
ETH is widely used in theoretical computer science to argue that certain problems cannot have algorithms substantially faster than exponential time. If this result is validated, researchers would need to revisit hardness conclusions that rely on ETH, including results about the limits of algorithms for related problems. The impact would be on the field’s theoretical assumptions and proofs, not an immediate change to everyday computing.
The claimed algorithm is still slower than every polynomial-time algorithm. It would not make hard instances of SAT suddenly practical, and the supplied account says OpenAI itself expects significant further research before practical improvements. The result is also not a solution to P vs. NP: that question asks whether every problem whose answers can be checked efficiently can also be solved efficiently, and remains open. ETH is a stronger hardness hypothesis than P ≠ NP alone requires.
The other significance is methodological. OpenAI says an AI system generated the proof, followed by a formalization that a proof assistant could check. That combination offers a possible route for examining complex AI-produced mathematical work. But formal verification only supports the exact statement encoded and checked; here, the gap between that statement and the paper’s strongest bound makes independent mathematical review especially relevant.
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From SAT Algorithms to ETH
In k-SAT, each rule is a logical clause involving no more than k variables, and the task is to determine whether assignments of true or false values can satisfy all clauses. With n variables, trying every assignment takes 2^n possibilities. Researchers have developed algorithms faster than brute force by reducing the exponential base, but that still leaves exponential growth.
ETH, proposed by Russell Impagliazzo, Ramamohan Paturi and Francis Zane in 2001, formalizes the expectation that 3-SAT cannot be solved in subexponential time. The source material describes earlier algorithmic improvements, including Schöning’s 1999 method and later PPSZ refinements, as advances that reduced the base without removing the exponential character. The new preprint claims a different kind of improvement: a runtime roughly 2 raised to n divided by the square root of log n, which is subexponential but not polynomial.
The announcement follows a broader OpenAI release of AI-generated research manuscripts on October 6, according to the source. That repository later withdrew three manuscripts because of a sign error and revised 14 others. Those corrections do not resolve the ETH proof, but they illustrate why publication, formal checks and outside review are distinct steps rather than interchangeable assurances.
The Formalization and Review Gap
The key unresolved issue is whether independent specialists agree that the full argument establishes the preprint’s strongest claim. The source says a group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the work; Impagliazzo and Paturi were among ETH’s original authors. The review has not yet been reported as complete, and no independent sign-off is supplied here.
There is also a precise scope question: Lean checked a slightly weaker bound than the sharpest runtime stated in the paper. The available account does not explain whether reviewers will extend the formalization to that sharper bound, or how they will assess any gap between the formal statement and the paper’s argument. Until those questions are addressed, the careful description is that OpenAI announced a claimed refutation and released a machine-checked formal result—not that the hypothesis has been conclusively disproved.
Independent Review of the Proof
The next milestone is the external group’s examination of the preprint and its Lean formalization. Its members are expected to explain and verify the computer science work, according to the source account. The review may clarify whether the paper’s main theorem follows, whether the stronger runtime bound is supported, and what is covered by the formal check.
Readers should watch for a public assessment, any revisions to the paper or code, and whether the formalization is expanded to cover the sharpest stated result. OpenAI also said it agreed to fund a conference on the work. Until review findings are available, the claim remains under examination, and practical consequences remain speculative.
Key Questions
Did OpenAI prove that P equals NP?
No. The announced result concerns ETH, a stronger hypothesis about the difficulty of 3-SAT. P vs. NP remains open, and the claimed algorithm is still slower than polynomial time.
What does the claimed result say about 3-SAT?
The preprint claims a deterministic subexponential-time algorithm for exactly counting satisfying assignments to k-SAT when k is fixed. For 3-SAT, that would contradict ETH if the proof and its runtime bound are validated.
Did Lean verify the entire preprint?
No, according to the supplied account. Lean checked a formal statement with a slightly weaker runtime bound than the paper’s sharpest claim. Independent experts have not yet completed their review.
Will this make computers solve hard problems faster now?
There is no confirmed immediate practical improvement. The claimed algorithm remains superpolynomial, and OpenAI said practical gains would require significant further research.
Source: ThorstenMeyerAI.com
