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TL;DR
OpenAI says an unreleased AI model produced a proof that a deterministic algorithm can exactly count solutions to fixed-width k-SAT in subexponential time, challenging the Exponential Time Hypothesis. Lean checked a formalized version with a weaker runtime bound than the preprint claims, and independent experts have not yet completed their review. The result does not resolve P vs. NP or promise immediate practical speed-ups.
OpenAI said on October 26 that an unreleased internal AI model had produced a proof challenging the Exponential Time Hypothesis (ETH), a long-standing conjecture about the difficulty of solving Boolean satisfiability problems. The proof has a public Lean formalization, but independent expert review is still pending, and the formalized result checks a weaker runtime bound than the paper’s sharpest claim.
The preprint claims a deterministic algorithm can exactly count the satisfying assignments of any k-SAT formula, for fixed k, in subexponential time and polynomial memory. For 3-SAT, such an algorithm would contradict ETH, which holds that solving 3-SAT requires time exponential in the number of variables. The paper also says the result refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses.
OpenAI says its system found the disproof on October 16, with the Lean formalization following the next day. The public codebase includes 3,977 files and about 710,000 lines, according to the source report’s count. Lean is a proof assistant that checks whether a formal proof follows from its stated definitions and rules. The source report says its review of the scope notes found that Lean checks a slightly weaker bound than the preprint’s sharpest runtime claim; that strongest bound is not included in the formalization.
An external group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the computer-science result. Impagliazzo and Paturi were among the original authors of ETH. OpenAI says the outside experts requested public release of the preprint and code, and that it will fund a conference on the result. The manuscript and formalization provide material for scrutiny, but neither OpenAI’s announcement nor automated checking substitutes for independent assessment of the mathematical argument and its scope.
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 claims withstand scrutiny, it would change the theoretical understanding of how quickly certain SAT problems can be solved. ETH has served as a basis for conditional hardness results across theoretical computer science: researchers use it to show that other problems are unlikely to admit particularly fast algorithms. A counterexample could require revisiting some of those conclusions, depending on the exact theorem and assumptions involved.
That theoretical impact should not be confused with an immediate computing breakthrough. The reported algorithm remains superpolynomial—its runtime grows faster than any polynomial in the input size—and OpenAI said practical improvements would require significant further research. The source report’s arithmetic also suggests the asymptotic speed-up grows very slowly with problem size. The result does not establish that ordinary SAT instances will become easier to solve in practice.
The process is also being watched as a test of AI-generated mathematical research. A formal proof can make particular logical steps checkable by software, while expert review can assess the definitions, assumptions and relationship between the formal statement and the paper. The distinction matters here because the verified bound is weaker than the headline version of the claim.
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ETH Is Not the P vs. NP Problem
In k-SAT, each rule is a logical clause involving at most k variables, and the task is to determine whether some assignment makes all clauses true. With n variables there are 2ⁿ possible assignments to check by brute force, although algorithms can do better than that. ETH, proposed by Impagliazzo, Paturi and Francis Zane in 2001, says that for 3-SAT the exponential dependence cannot be eliminated: solving it requires at least 2^(c·n) time for some fixed positive c.
That is distinct from P vs. NP, the open question of whether every problem whose solutions can be checked efficiently can also be solved efficiently. The source report cites the Clay Mathematics Institute’s listing of P vs. NP as unsolved and describes ETH as a stronger hardness assumption. Refuting ETH would not prove P equals NP, nor would it settle P vs. NP. Researchers have already improved on brute force for SAT, but those gains have reduced the exponential base rather than removed exponential growth.
OpenAI’s October 26 announcement covered three separate research claims: the ETH result, a mathematics result concerning local smoothing, and a physics result concerning quantum chromodynamics and a grand unified completion of the Standard Model. The company says the latter materials are being handled differently by outside groups. The computer-science preprint and Lean code are public; the physics materials are being held back at the request of its stewards, according to OpenAI.
Expert Review and Bound Gaps
The central open question is whether independent researchers will validate the paper’s proof and its claimed consequences. The external review is ongoing, and the source material does not report a completed verdict. Lean’s check establishes that the formalized statement follows within the system’s specified framework; it does not, by itself, establish that every claim in the preprint has been formalized or that the strongest claimed bound is covered.
It is also not clear from the supplied material which earlier conditional hardness results, if any, would need revision if the result is accepted. That depends on the proof’s precise scope and on follow-up analysis. The source report notes that experts have disagreed about how broadly ETH is believed, so the result should not be described as overturning a universal consensus.
Independent Checks and Follow-Up
The next milestone is the external group’s examination of the preprint and formalization, led by Impagliazzo, Paturi and Williams. The group’s review may clarify whether the proof establishes the stated theorem, how the Lean-checked version relates to the paper’s strongest runtime claim, and what implications follow for results that rely on ETH. OpenAI says a conference on the result will also be funded, providing a venue for further scrutiny.
Until those checks are reported, the appropriate description is that OpenAI has announced an AI-produced proof and released formal materials for review—not that the field has confirmed the refutation. Any practical algorithmic consequences would require additional work, and no immediate change to the performance of SAT solvers is established by the announcement.
Key Questions
What does the OpenAI result claim?
The preprint claims a deterministic, subexponential-time algorithm with polynomial memory for exactly counting satisfying assignments in k-SAT when k is fixed. Applied to 3-SAT, the claim contradicts the Exponential Time Hypothesis if the proof is correct.
Has the result been independently confirmed?
No independent verdict is reported in the supplied material. OpenAI has released the preprint and Lean code, and an external group is set to examine the computer-science result.
Does this solve P vs. NP?
No. ETH is a separate, stronger hardness hypothesis. Refuting it would not settle whether every efficiently checkable problem can also be solved efficiently; P vs. NP remains open.
Does Lean verify the preprint’s strongest runtime claim?
According to the source report’s examination of the formalization, no. Lean checks a slightly weaker bound, while the sharpest runtime bound stated in the preprint is not covered by the formalized proof.
Will this make SAT problems faster to solve now?
No immediate practical speed-up has been established. The claimed algorithm remains superpolynomial, and OpenAI said practical improvements would require significant further research.
Source: ThorstenMeyerAI.com
