🔍 Read the full analysis: OpenAI Says Its AI Found A Way Around A Complexity Assumption—What’s Proven? on ThorstenMeyerAI.com
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TL;DR
OpenAI says an unreleased AI model found a deterministic, subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT, which would refute the Exponential Time Hypothesis for 3-SAT. A Lean formalization checks a slightly weaker bound than the preprint’s sharpest claim, and independent expert review has not yet been completed. This does not solve P vs. NP or establish an immediately practical speedup.
OpenAI said on October 26 that an unreleased internal AI model produced a proof of a subexponential algorithm for exactly counting solutions to fixed-width k-SAT, a result that would refute the Exponential Time Hypothesis (ETH) for 3-SAT. The company released a preprint and Lean proof-checking code, but the claim has not yet received independent expert review, and the formalized result covers a slightly weaker runtime bound than the paper’s sharpest claim.
The preprint claims a deterministic algorithm using subexponential time and polynomial memory to count satisfying assignments for any k-SAT formula when k is fixed. In k-SAT, each rule contains at most k variables or their negations; the task is to count the assignments that make all rules true. For 3-SAT, the claimed running time is roughly 2 raised to n divided by the square root of log n, where n represents the number of variables. That is still slower than polynomial time, but it would be subexponential and contradict ETH as commonly formulated.
OpenAI says its model found the disproof on October 16, with a Lean formalization following the next day. The company published the preprint, dated October 17 and revised October 25, alongside the formalization. The source report counted 3,977 files and about 710,000 lines in the Lean code. Lean is a proof assistant: it can check that a formal proof follows from its stated definitions and rules. That check is meaningful, but it does not by itself establish that the formal statement matches every strongest claim in the paper.
According to the source report’s inspection of the Lean scope notes and formal statement, the formalized proof checks a slightly weaker runtime bound than the preprint’s sharpest bound. The paper also says the result refutes the Strong Exponential Time Hypothesis and three related hypotheses. Those broader claims, and how they relate precisely to the formalized result, need scrutiny from researchers outside OpenAI. A review group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is expected to examine the computer-science work.
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 result withstands review, it would alter a widely used assumption in theoretical computer science: that 3-SAT cannot be solved in subexponential time. ETH has supported many conditional hardness results, where researchers show that certain algorithms or computational tasks cannot be substantially improved unless ETH fails. A valid refutation would require those results to be reconsidered, though it would not automatically invalidate every paper that uses ETH; each conclusion would need to be checked against its assumptions.
The result is also a test of a proposed research process: AI generation followed by formal verification and human review. Lean checking can catch errors within a formal proof, while expert review can assess the mathematics, the translation from the paper to the formal statement, and the importance of the result. Those are related but distinct checks. The gap between the paper’s strongest bound and the formalized bound makes that distinction especially relevant here.
There is no announced near-term practical payoff. The claimed algorithm remains superpolynomial, and OpenAI says practical improvements would require significant further research. The result does not show that ordinary hard problems will suddenly become easy, nor does it settle P versus NP.
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ETH Is Stronger Than P vs. NP
ETH grew out of research on the difficulty of satisfiability problems. In 2001, Impagliazzo, Paturi and Francis Zane proposed the hypothesis that 3-SAT requires exponential time in the number of variables, for some fixed positive exponent. Researchers have developed algorithms faster than trying every possible assignment, but those improvements have still taken exponential time, with a smaller base in the exponent.
P vs. NP remains open. It asks whether every problem whose proposed answer can be checked efficiently can also be solved efficiently. ETH makes a more specific and stronger prediction about the time needed to solve 3-SAT; disproving ETH would not establish that P equals NP. The Clay Mathematics Institute lists P versus NP as unsolved, and the source material notes that experts have not held uniform confidence in ETH itself.
This announcement follows a larger OpenAI research release on October 6, according to the source report. That repository’s history records withdrawn and revised manuscripts after a sign error was found. The episode adds a practical reason to distinguish an AI system’s proposed result, a formal proof check, and independent mathematical review rather than treating publication as the end of verification.
The Formal Proof’s Remaining Gaps
Independent review is pending. The review group has been identified, but the source material does not report a completed assessment or a public endorsement by its members. Until that work is done, the preprint’s claims should be treated as claims rather than an established change to complexity theory.
It is also not clear from the material provided how the paper’s sharpest runtime bound is connected to the weaker bound checked in Lean, or whether further formalization will cover the difference. Reviewers will need to examine both the mathematical argument and the relationship between the paper and its formal statement. The source material does not provide their findings, a review timetable, or a final judgment on the claims about SETH and the related hypotheses.
Reviewers Must Test the Claim
The next milestone is the external group’s examination of the preprint and formalization. That review will determine whether the mathematical argument holds, whether the Lean code represents the relevant claims, and how much of the paper’s stated result is supported by the checked proof. OpenAI has said it will fund a conference on the result, but the source material gives no date for that event.
Until researchers publish their assessment, the careful description is that OpenAI reports an AI-generated ETH refutation, with a machine-checked formalization for a somewhat weaker bound and human review still to come. Practical consequences, if any, would depend on further research and on whether the result survives scrutiny.
Key Questions
Has OpenAI proved that P equals NP?
No. The announced result concerns ETH, a hypothesis about the time needed to solve 3-SAT. P versus NP remains an open problem, and the claimed algorithm is still slower than polynomial time.
What does the AI result claim?
The preprint claims a deterministic, subexponential-time algorithm for exactly counting solutions to k-SAT when k is fixed. For 3-SAT, that would contradict ETH if the claim is correct.
Has the proof been independently verified?
Not yet, according to the source material. A Lean formalization has been published, and an external research group is expected to examine the work. Formal checking and independent expert review are separate steps.
Does Lean check the paper’s strongest runtime claim?
The source report says the Lean formalization checks a slightly weaker bound than the preprint’s sharpest runtime claim. The relationship between those statements remains a key point for reviewers.
Will this make computers solve hard problems faster now?
No immediate practical improvement has been established. The claimed algorithm remains superpolynomial, and OpenAI says practical gains would require significant further research.
Source: ThorstenMeyerAI.com
