🔍 Read the full analysis: What Does OpenAI’s AI Work Actually Prove About Complexity Theory? 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 k-SAT, which would refute the Exponential Time Hypothesis. A Lean formalization checks a related, slightly weaker bound, but independent experts have not yet verified the result. It does not show that P equals NP, and practical effects are unknown.
OpenAI said on October 26 that an unreleased internal AI model produced a proof of a subexponential-time algorithm for exactly counting solutions to k-SAT, a result that would refute the Exponential Time Hypothesis (ETH) if the proof holds. The company released a research preprint and a Lean formalization, but independent experts have not yet signed off; the formalized statement also checks a slightly weaker runtime bound than the paper’s sharpest claim.
The preprint describes 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, this contradicts ETH, which holds that no algorithm can solve the problem in subexponential time. The paper says the result also refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses. These are claims made in the preprint, not conclusions independently established by outside review.
OpenAI’s account says its model found the disproof on October 16 and that a Lean formalization followed on October 17. The preprint is dated October 17 and was revised October 25. The source material reports that the public formalization contains 3,977 files and about 710,000 lines, while the formal statement itself is short. Lean is software that checks whether a proof follows from formally specified assumptions and rules. Its check supports the formalized theorem; it does not by itself establish that the paper’s stronger, unformalized runtime bound follows.
OpenAI says an outside group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the computer-science result. Impagliazzo and Paturi were among ETH’s original authors. The company has also announced two other AI-produced results: a mathematics proof concerning local smoothing and related conjectures, and a physics claim concerning quantum chromodynamics and a grand unified theory. Those are separate claims with separate review and release arrangements.
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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A Check on a Core Hardness Assumption
ETH is used in theoretical computer science to show that many problems are unlikely to have efficient algorithms: researchers often prove that making one of those problems much faster would also undermine ETH. If this result survives review, those conditional hardness arguments would need to be reassessed, and some algorithmic limits might require new foundations. The immediate effect would be on theory and proof, not a sudden change in software or computing services.
The proposed algorithm is subexponential but still far slower than polynomial time. The source describes its runtime as roughly 2 to the power of n divided by the square root of log n, where n is the number of variables. That growth remains immense for large instances. OpenAI says practical improvements would require significant further research; the result does not show that ordinary SAT problems will become easy to solve.
The process is also notable because it combines AI-generated mathematics with machine checking and review by researchers in the field. Formal verification can catch errors within the statement and proof encoded in a system, but independent experts still need to assess the assumptions, the relationship between the formal statement and the paper, and the result’s implications. The gap between the paper’s strongest bound and the checked bound makes that distinction especially relevant here.
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ETH Is Not the P-versus-NP Question
In k-SAT, each rule says that at least one of up to k variables or their negations must be true. For n variables, trying every possible assignment takes 2^n possibilities. Researchers have found algorithms faster than exhaustive search, but their running times remain exponential. ETH, proposed by Impagliazzo, Paturi and Francis Zane in 2001, states that 3-SAT cannot be solved in subexponential time.
P versus NP remains open. That problem asks whether every problem whose proposed solution can be checked efficiently can also be solved efficiently. Refuting ETH would overturn a stronger assumption about the rate at which certain problems can be solved; it would not prove that P equals NP. OpenAI’s announcement follows a larger release of AI-generated manuscripts on October 6. According to the source material, three manuscripts were withdrawn the next day because of a sign error and fourteen others were revised, illustrating why verification and review matter.
OpenAI has characterized ETH as widely used in research. But the source material also cites a 2017 survey by Scott Aaronson saying there was no field-wide consensus that ETH is true. The hypothesis has been influential, but its status should not be confused with a proven theorem.
The Strongest Bound Awaits Review
The central uncertainty is whether independent experts will accept the proof and whether its strongest claimed runtime bound is correct. According to the source material, Lean checks a slightly weaker bound than the sharpest one in the preprint. Formal verification is evidence about the statement encoded in Lean; it should not be described as formal confirmation of a stronger statement that is not covered by that formalization.
The review has not been completed, and no outside endorsement is reported in the material provided. The exact consequences for existing ETH-based results would depend on what experts validate and how the theorem’s assumptions and scope compare with those results. OpenAI’s account of the model’s role and the research timeline also comes from the company; the available material does not provide an independent reconstruction of how the system found the proof.
Researchers Begin Examining the Proof
The next milestone is the outside examination led by Impagliazzo, Paturi and Williams. They will need to evaluate the paper’s argument, the formalization, and the difference between the checked bound and the preprint’s sharpest bound. OpenAI says the preprint and Lean code were made public at the group’s request and that it agreed to fund a conference on the result.
Until that review produces findings, the careful description is that OpenAI has announced a proposed refutation of ETH and released material for scrutiny—not that the hypothesis has been conclusively overturned. Readers should look for a public account of the reviewers’ conclusions and any corrections or revisions to the preprint and formal proof.
Key Questions
Does OpenAI’s result prove that P equals NP?
No. The claimed result concerns ETH, a stronger assumption about the time needed to solve 3-SAT. The P-versus-NP problem remains open.
What does the proposed algorithm do?
The preprint claims a deterministic, subexponential-time algorithm that exactly counts satisfying assignments for k-SAT when k is fixed. For 3-SAT, that would contradict ETH if the proof is valid.
Has the proof been independently verified?
Not yet, according to the source material. A Lean formalization has been released, and an outside group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is expected to examine the result.
What does Lean checking establish here?
Lean checks that a specified formal proof follows from its encoded definitions and rules. The source material says the formalization checks a slightly weaker runtime bound than the sharpest bound in the preprint, so it does not formally cover that strongest claim.
Will this make SAT problems faster to solve in practice?
No immediate practical improvement is established. The claimed runtime is still far slower than polynomial time, and OpenAI says practical gains would require significant further research.
Source: ThorstenMeyerAI.com
