🔍 Read the full analysis: OpenAI Says Its AI Found A Counterexample. What Does That Prove? on ThorstenMeyerAI.com
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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 k-SAT, which would refute the Exponential Time Hypothesis. Lean checked a formalization of a weaker runtime bound, but the result has not yet been independently reviewed. It does not resolve P versus NP or promise practical speedups.
OpenAI said on October 26 that an unreleased AI model found 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 formalized result checks a weaker runtime bound than the paper’s sharpest claim, and independent expert review has not been completed.
The preprint claims a deterministic algorithm that exactly counts solutions to any k-SAT formula, for fixed k, in subexponential time and polynomial memory. For 3-SAT, that would contradict ETH, which holds that no algorithm can solve 3-SAT in subexponential time. The paper also says the result refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses. Those are claims made in the preprint, not conclusions independently endorsed by reviewers so far.
OpenAI says its model found the disproof on October 16, and that a Lean formalization followed the next day. The preprint is dated October 17 and was revised October 25. Lean is a proof assistant: it checks whether a formal proof follows from specified mathematical definitions and rules. The source report counted about 710,000 lines across 3,977 files in the released formalization. However, its scope notes indicate that the checked statement has a weaker runtime bound than the strongest bound asserted in the paper.
A group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is set to examine the computer-science result, OpenAI said. Impagliazzo and Paturi were among the three researchers who introduced ETH with Francis Zane in 2001. OpenAI also announced work in mathematics and physics, but the ETH claim is the one for which the preprint and Lean code were released at the outside group’s request.
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 as a conditional basis for proving computational hardness. Researchers use it to argue that certain problems are unlikely to have fast algorithms: if ETH holds, a proposed efficient solution to one of those problems would imply an unexpectedly fast way to solve SAT. A valid refutation could force revisions to results that rely on the hypothesis and prompt researchers to reassess which complexity barriers are genuine.
The practical consequences are less immediate. The reported algorithm runs roughly in time 2 raised to n divided by the square root of log n, according to the source material. That is subexponential, but it remains much slower than polynomial time and may not provide useful speedups on real-world problems. OpenAI itself said practical improvements would require significant further research. The result, if confirmed, would change a theoretical assumption; it would not make ordinary software or industrial optimization suddenly faster.
The episode also tests a possible model for AI-assisted research: a system proposes a result, a proof assistant checks a formalized version, and mathematicians scrutinize both the formal statement and the connection to the paper. Machine checking strengthens confidence in the proof it covers, but it does not by itself show that the formalized theorem matches every claim in the manuscript or that the result is relevantly interpreted.
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The Hypothesis Behind the Claim
In k-SAT, a formula consists of logical rules, each involving at most k variables. The question is whether the variables can be assigned true or false values so that all rules are satisfied. With n variables, trying every assignment takes 2 to the power of n possibilities. Researchers have developed algorithms faster than exhaustive search, but the time still grows exponentially with n.
ETH, proposed in 2001 by Impagliazzo, Paturi and Zane, says that 3-SAT cannot be solved in subexponential time. The preprint’s claimed algorithm concerns exact counting of solutions, rather than only deciding whether at least one solution exists; the paper argues that this yields the contradiction to ETH. The source report also describes the algorithm as subexponential but not polynomial.
ETH is not the same as P versus NP. The latter asks whether every problem whose answer can be checked efficiently can also be solved efficiently, and remains open. ETH is a stronger hypothesis about the rate at which particular hard problems can be solved. Refuting ETH would not prove P equals NP, nor would it settle the question in the other direction.
What Reviewers Still Need to Check
The central open question is whether the claimed proof is correct and whether the published formalization establishes the theorem the paper needs. Lean checks only the formal statement encoded in the proof; the source report says that statement uses a weaker runtime bound than the manuscript’s sharpest result. Reviewers will need to assess the mathematical argument, the formalization’s scope and the link between the two.
The source material does not report a completed independent review or a final assessment from the invited experts. Nor does it establish what consequences the result would have for particular algorithms or hardness results if accepted. The reported runtime is subexponential, but how much it could help on practical instances is unknown. The claim that the work refutes three additional hypotheses also awaits scrutiny alongside the primary ETH claim.
Independent Review and Formal Scope
The next milestone is the examination by the outside group led by Impagliazzo, Paturi and Williams. That review is expected to address the proof and its implications; no completion date is given in the source material. The public preprint and Lean files allow other researchers to inspect the work in parallel.
Readers should watch for a reviewer account that spells out which theorem has been verified, whether the paper’s strongest runtime bound follows, and whether the additional hypothesis claims hold. Until those questions are resolved, the careful description is that OpenAI reports an AI-generated proof claim with a machine-checked, weaker formal result—not that the field has accepted ETH as refuted.
Key Questions
What does OpenAI say its AI proved?
OpenAI says an internal model found a deterministic subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT. The preprint argues that, for 3-SAT, this contradicts ETH. Independent review is still pending.
Did Lean verify the paper’s strongest claim?
Not according to the source report. Lean checked a formalization with a weaker runtime bound than the sharpest bound in the preprint. Reviewers must examine how the formal statement relates to the paper’s full claim.
Does this solve P versus NP?
No. P versus NP remains open. ETH is a stronger hypothesis about the time needed to solve certain problems; refuting it would not establish whether P equals NP.
Will SAT problems become faster to solve now?
There is no established immediate practical speedup. The reported algorithm is still slower than polynomial time, and OpenAI said practical improvements would require significant further research.
Who is reviewing the result?
OpenAI said an external group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams would examine the computer-science result. The source material does not give a review deadline or final verdict.
Source: ThorstenMeyerAI.com
