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TL;DR
OpenAI says an unreleased internal AI model produced a proof of a subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT, which would refute ETH. A Lean formalization checks a weaker runtime bound, and independent expert review has not yet taken place. The result does not solve P vs. NP or mean practical software will become faster.
OpenAI said October 26 that an unreleased internal AI model had found a proof of an algorithm that, if correct, would refute the Exponential Time Hypothesis (ETH), a longstanding assumption about the difficulty of solving certain logic problems. The company released a preprint and a Lean formalization, but independent expert review is still pending, and the machine-checked result establishes a weaker runtime bound than the sharpest one in the paper.
The preprint claims a deterministic algorithm that exactly counts satisfying assignments for any fixed-width k-SAT formula in subexponential time, using polynomial memory. For 3-SAT, the claimed result contradicts ETH, which says that solving such instances requires exponential time 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, and that the Lean formalization followed the next day. The preprint is dated October 17 and was revised October 25. The source account reports that the public formalization contains 3,977 files and about 710,000 lines; those counts are the author’s tally, not a figure independently verified here. Lean is a proof assistant that checks whether a formal argument follows from its stated definitions and rules. In this case, the formalization covers a slightly weaker runtime bound than the paper’s sharpest claim.
OpenAI has asked an outside group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams to examine the computer science result. Impagliazzo and Paturi were among the original authors of ETH. OpenAI says the group will explain and verify the work, and that the preprint and code were released at its request. That process has not yet produced an independent public verdict.
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 experts validate the result, it would overturn a widely used assumption in theoretical computer science. ETH has supported lower-bound arguments showing that many problems are unlikely to have fast algorithms, provided ETH holds. A counterexample would require researchers to revisit results that depend on it, checking which conclusions need the assumption and which survive without it.
The result is not a solution to P vs. NP. ETH makes a stronger claim about the difficulty of 3-SAT than the basic assertion that P differs from NP; refuting ETH does not establish that every efficiently checkable problem can be efficiently solved. Nor does a subexponential algorithm make the problem quick in practice. The source describes the claimed time as roughly 2 to the power of n divided by the square root of log n, which remains far slower than polynomial time. OpenAI itself says practical improvements would require significant further research.
The immediate significance is also methodological. The announcement presents a chain from an AI-generated argument to a formal proof check and then human expert scrutiny. Lean checking can catch gaps relative to the formalized statement, but it does not by itself confirm that the formal statement matches every claim in the paper or that the result is useful in applications. The gap between the paper’s strongest bound and the formalized weaker one makes that distinction especially relevant.
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The Hypothesis Behind the Claim
k-SAT asks whether Boolean variables can be assigned true or false so that every clause in a formula is satisfied. In 3-SAT, each clause contains at most three variables or their negations. A direct search over n variables considers as many as 2^n assignments, though existing algorithms reduce the work by improving the exponential base.
ETH was introduced in 2001 by Impagliazzo, Paturi and Francis Zane. In broad terms, it says 3-SAT cannot be solved in time 2^(c·n) for any fixed positive constant c. Researchers have used ETH and related assumptions to establish conditional hardness results across computer science. The source account notes that views on ETH are not unanimous: it cites Scott Aaronson’s 2017 description of the hypothesis as an ambitious strengthening of P ≠ NP, while cautioning that the field has not had a consensus that ETH is true.
This announcement is part of a larger OpenAI release of AI-generated research. The company also announced results in mathematics and physics on October 26. The source account says an earlier release of more than 700 manuscripts was followed by corrections: repository history recorded three withdrawals for a sign error and revisions to 14 others. That history is a reason to distinguish a formal check of a particular proof from review of a research program or its other claims.
What Review Still Needs to Establish
The central open question is whether independent researchers can verify the paper’s argument and its claimed runtime. The outside review group has been named, but the source material gives no completed review or timetable for a conclusion. Until that work is public, OpenAI’s announcement remains a claim supported by a released preprint and a formalization, not an independently endorsed result.
It is also unclear how the paper’s strongest bound relates to the Lean-checked version in full detail. The source account says the formalization proves a weaker bound, so the machine check should not be described as verifying every part of the strongest claim. The account also does not establish whether the algorithm can be implemented efficiently or applied to practical workloads.
The source material is an account of OpenAI’s announcement and the author’s inspection of its files; it is not itself the independent expert report. The extent to which ETH-dependent results would need revision will depend on the final proof and on which conclusions rely specifically on ETH.
Independent Review of the Proof
The next major milestone is the outside group’s examination of the preprint and formalization. Its members are expected to explain and verify the result, but no publication date or review schedule is given in the source material. Readers should look for a public account of the group’s findings, including whether it confirms the paper’s strongest runtime or only the weaker bound checked in Lean.
Further work will also be needed to clarify the algorithm’s implications for existing ETH-based results and to test whether any practical applications follow. OpenAI has said it will fund a conference on the result; the source account does not provide a date. Until review and follow-up research are complete, claims about immediate computing or commercial gains would go beyond what is established.
Key Questions
Has OpenAI proved that P equals NP?
No. The claimed result would refute ETH, a stronger hardness hypothesis about SAT. It does not resolve the open P vs. NP question.
What has Lean checked?
OpenAI released a Lean formalization that checks a proof of a slightly weaker runtime bound than the sharpest bound stated in the preprint. The source account does not say that Lean has checked the full strongest claim.
Has the result been independently verified?
Not yet, according to the available account. An outside group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is expected to examine it. No completed verdict is reported.
Will this make computers faster soon?
There is no confirmed near-term practical improvement. The claimed algorithm remains far slower than polynomial time, and OpenAI says practical gains would require significant further research.
Why does the ETH claim matter?
Researchers use ETH to support conditional hardness results: conclusions that certain problems are unlikely to have fast algorithms if ETH is true. If the refutation is validated, those results may need to be reassessed, though not all would necessarily fall.
Source: ThorstenMeyerAI.com
