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TL;DR
OpenAI says an unreleased AI model produced a proof of a subexponential-time algorithm for exactly counting solutions to fixed-width SAT formulas, which would refute the Exponential Time Hypothesis. Lean checked a formalized version with a weaker runtime bound than the preprint claims, and independent review has not yet concluded. The result does not solve P vs. NP or imply an immediate practical speed-up.
OpenAI said on October 26 that an unreleased internal AI model had produced a proof of a subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT problems, a result that would refute the Exponential Time Hypothesis (ETH). The company released a preprint and a Lean formalization, but the work has not yet completed independent expert review, and the formalized theorem is weaker than the sharpest runtime claim in the paper.
The preprint claims 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, the claimed result contradicts ETH, which says that solving 3-SAT requires exponential time in the number of variables. The paper also says its result refutes the stronger Strong Exponential Time Hypothesis (SETH) and three related hypotheses. These are claims made in the preprint, not findings that have already received independent expert approval.
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 report describes the public Lean release as containing 3,977 files and about 710,000 lines of code; those counts are the report author’s tally. Lean is a proof assistant that checks whether a formal proof follows from its stated definitions and rules. The formal statement itself is much shorter than the full codebase.
A key qualification is that Lean checks a slightly weaker runtime bound than the strongest one stated in the paper. The preprint’s sharpest bound is not covered by the formalization, according to the source report’s review of the scope notes. OpenAI has asked an external group to examine the computer-science result; that group is led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams. Impagliazzo and Paturi were among the original authors of ETH.
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 preprint’s result survives review, it would overturn a widely used assumption about the limits of exact algorithms. Researchers use ETH and related hypotheses to show that many computational problems cannot be solved efficiently, even when they are not known to be NP-complete themselves. A valid counterexample could require those conditional hardness arguments to be revisited and could alter how theoretical computer scientists classify the difficulty of problems.
That potential change is not the same as an immediate improvement to software or commercial systems. The algorithm described remains superpolynomial: its running time grows faster than every polynomial in the input size, even though it grows more slowly than an exponential bound. OpenAI itself says practical improvements would require significant further research. The source report also says its own arithmetic suggests the speed-up grows very slowly with problem size; that is an interpretation of the proposed bound, not a measured performance result.
The development also puts attention on a research workflow: AI-generated proof, machine checking and human scrutiny. A formal checker can catch errors in a proof that has been encoded correctly, but the community must still assess whether the formal statement matches the paper’s claims and whether the result’s assumptions and implications have been understood correctly.
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From SAT Algorithms to ETH
In k-SAT, each rule involves at most k variables and asks whether at least one specified variable or its negation is true. The task is to find an assignment satisfying all rules, or, in the new paper’s formulation, to count the satisfying assignments exactly. With n variables, a direct search can examine 2^n assignments. Researchers have developed faster algorithms, but those improvements have traditionally reduced the base of the exponential rather than removed exponential growth.
ETH was formulated by Russell Impagliazzo, Ramamohan Paturi and Francis Zane in 2001. It says, roughly, that 3-SAT cannot be solved in subexponential time. The hypothesis is stronger than the claim that P does not equal NP. Refuting ETH would not establish P = NP: the algorithm in the preprint is still slower than polynomial time, and the Clay Mathematics Institute continues to list P versus NP as an unsolved problem.
There is also no unanimous view that ETH is true. OpenAI described it as widely believed and cited its use in hundreds of papers. The source report notes that Scott Aaronson’s 2017 survey said there was no field-wide consensus. That distinction matters: a result can challenge a powerful and widely used research assumption without overturning a settled theorem.
What Reviewers Still Need to Check
The central uncertainty is whether independent experts will validate the proof and its implications. The review group has been named, but the source material does not report a completed assessment or a timetable for one. Until that review is finished, the ETH refutation should be described as a claim in a preprint rather than an established result.
There is also a gap between the paper and the formal verification: the strongest stated runtime bound is not covered by Lean. The source material does not provide enough detail to establish how much that difference affects the paper’s broader claims, or what further formalization may be needed. Lean’s check confirms the encoded proof of the encoded statement; it does not, by itself, confirm every claim in the manuscript.
Finally, the result concerns a theoretical algorithm, not demonstrated performance on real-world workloads. No practical speed-up, product change or effect on ordinary users was reported. The relationship between the claimed result and each of the additional hypotheses listed in the preprint also awaits scrutiny by researchers outside OpenAI.
External Review and Formal Checks
The next major step is the examination by the external group led by Impagliazzo, Paturi and Williams. Its review will determine whether the argument is sound, whether the formalization covers the claims readers are being asked to accept, and what the result establishes about ETH and related hypotheses. The source report says OpenAI agreed to fund a conference on the result, but it gives no date.
Readers can inspect the public preprint and Lean code while that process proceeds. The most informative updates will be a detailed review, any revisions to the theorem or its formal proof, and an explanation of the difference between the paper’s sharpest bound and the checked statement. Until then, the result is a substantial, machine-checked research claim—not a completed consensus or a practical computing breakthrough.
Key Questions
Has OpenAI’s AI definitively disproved ETH?
Not yet as an independently established result. OpenAI announced the claim and published a preprint with a Lean formalization. An external expert group is expected to examine the work, and that review has not been reported as complete.
Does the result solve P vs. NP?
No. The claimed algorithm is subexponential but still slower than polynomial time. A refutation of ETH would challenge a stronger hypothesis than P ≠ NP, but it does not settle whether P equals NP.
What did Lean verify?
Lean checked a formal proof of the statement encoded in the public formalization. According to the source report, that statement uses a slightly weaker runtime bound than the preprint’s sharpest claim, which is not covered by the formalization.
Will this make everyday computing faster now?
No immediate practical improvement has been reported. The result is theoretical, and OpenAI said practical improvements would require significant further research.
Source: ThorstenMeyerAI.com
