🔍 Read the full analysis: Can AI Disprove The Exponential Time Hypothesis? OpenAI’s Claim Examined on ThorstenMeyerAI.com
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TL;DR
OpenAI says an unreleased internal model found a deterministic, subexponential-time algorithm for exactly counting solutions to fixed-k SAT problems, which would contradict the Exponential Time Hypothesis for 3-SAT. A Lean formalization checks a slightly weaker bound than the preprint’s sharpest result, and independent experts have not yet completed their review. The claim does not resolve P versus NP or mean SAT problems will become practically easier soon.
OpenAI said on October 26 that an unreleased internal AI model found an algorithm that would refute the Exponential Time Hypothesis (ETH), a longstanding conjecture about the difficulty of solving certain logic problems. The company published a preprint and Lean proof code, but independent expert review is still pending, and the formalized result covers a slightly weaker runtime bound than the paper’s sharpest claim.
The preprint claims a deterministic algorithm that exactly counts satisfying assignments for any fixed-k SAT formula in subexponential time and polynomial memory. For 3-SAT, that would contradict ETH, which says no algorithm can solve such instances in subexponential time. The paper says the result also refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses.
OpenAI says its system found the result on October 16, with a Lean formalization following the next day. The preprint is dated October 17 and was revised October 25. The source report says the published formalization contains 3,977 files and about 710,000 lines; those are the report author’s counts. Lean checks a slightly weaker runtime bound than the preprint’s strongest one, so the formalized statement does not verify every detail of the sharpest claim.
OpenAI has made the preprint and code public at the request of an external group that will examine the computer-science result. That review is not complete. A proof checked by Lean has passed a formal consistency check of its encoded steps, but researchers must still assess whether the paper’s argument addresses the conjecture as stated and whether its assumptions and formalization match the intended result.
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 used to support lower-bound arguments across theoretical computer science: researchers rely on it to say that particular problems are unlikely to have substantially faster algorithms. If the preprint’s claim survives scrutiny, those arguments would need to be revisited where they depend on ETH or stronger hypotheses. The potential effect is therefore broad for theory, but it is not an immediate change to everyday computing.
The claimed algorithm remains slower than every polynomial-time algorithm. The source report says its running time is roughly 2 raised to the power of n divided by the square root of log n, and describes the improvement over exponential time as growing very slowly with problem size. OpenAI itself says practical improvements would require significant further research. The result does not establish that real-world SAT instances can now be solved faster in practice.
The development also tests a process for evaluating AI-generated mathematics: an AI system proposes a proof, formal software checks encoded steps, and outside researchers review the result. That process can help identify errors, but formal checking is not a substitute for independent scrutiny of the mathematical claim, the formal statement, and the relationship between them.
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ETH and the SAT Problem
In k-SAT, each rule requires at least one of a limited number of variables or their negations to be true. The question is whether there is an assignment of true and false values that satisfies all rules. With n variables, checking every possible assignment takes 2 to the power of n combinations. Researchers have developed faster-than-brute-force algorithms, but ETH says the exponential dependence cannot be eliminated for 3-SAT.
ETH was proposed in 2001 by Russell Impagliazzo, Ramamohan Paturi and Francis Zane. It is stronger than the familiar P-versus-NP question: P versus NP asks whether every problem whose proposed solutions can be checked efficiently can also be solved efficiently. Refuting ETH would not resolve P versus NP, which remains open. The source report also notes that expert opinion on ETH has not been unanimous; it cites Scott Aaronson’s 2017 observation that there was no field-wide consensus that the hypothesis was true.
OpenAI presented the computer-science result alongside two other claimed research results, in mathematics and physics. It says outside experts, described as “shepherds,” helped determine how each project would be released. The source report also notes that OpenAI withdrew three manuscripts and revised 14 others after an earlier release of AI-generated research, underscoring why review and correction matter in evaluating the current claims.
Questions Before the Proof Is Settled
The central open question is whether independent experts will validate the preprint’s argument and its claimed implications for ETH and the stronger hypotheses. The external group’s examination is ongoing, and the source material does not report a completed verdict. It is also unclear whether reviewers will find that the Lean formalization adequately corresponds to the paper’s sharpest runtime bound, since the encoded result is slightly weaker.
OpenAI’s account of how the work was produced, the timing of the discovery and its collaboration arrangements comes from the company. The source report says the outside collaborators are unpaid, sign no non-disclosure agreements, speak for themselves, and that OpenAI says it does not seek academic credit. Those details do not by themselves settle the mathematical claim. No practical performance gains or changes to existing software are established by the announcement.
Independent Review and Formal Checks
The next milestone is the external group’s review of the preprint and Lean materials. Its findings could clarify whether the proof supports the announced refutation, whether the formal statement captures the paper’s strongest result, and what corrections or extensions may be needed. The source report identifies a group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams as examining the work; Impagliazzo and Paturi were among ETH’s original authors.
Until that review is reported, the result should be described as OpenAI’s claim, supported by a published preprint and a formalized related statement, rather than as a settled change to computer science. Researchers may also seek to reproduce the argument, inspect the code and test whether the claimed algorithm can be improved or applied to practical problems. OpenAI has agreed to fund a conference on the result, according to the source report, but no date or review timetable is given.
Key Questions
What does OpenAI say its AI system proved?
OpenAI says its internal model found a deterministic algorithm for exactly counting solutions to fixed-k SAT formulas in subexponential time and polynomial memory. Applied to 3-SAT, that would contradict ETH, according to the preprint.
Has the result been independently verified?
Not yet, according to the source material. Lean code has been published, and an external group is examining the work, but its review has not been reported as complete.
Does this settle P versus NP?
No. ETH is a stronger hypothesis about the time needed for problems such as 3-SAT. A refutation of ETH would not resolve the separate P-versus-NP problem, which remains open.
Will SAT problems become faster to solve now?
No practical speed-up is established by the announcement. The claimed algorithm is still superpolynomial, and OpenAI says practical improvements would require significant further research.
Source: ThorstenMeyerAI.com
