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TL;DR
OpenAI says an unreleased AI model produced a proof that a deterministic algorithm can exactly count solutions to fixed-width SAT problems in subexponential time, which would refute the Exponential Time Hypothesis. A Lean formalization checks a slightly weaker bound, and independent experts have not yet reviewed the result. The claim does not resolve P versus NP or mean practical problems will become faster soon.
OpenAI said October 26 that an unreleased internal AI model produced a proof challenging the Exponential Time Hypothesis, a long-standing conjecture about the difficulty of solving certain logic problems. The company released a preprint and Lean code for a machine-checked formalization, but independent expert review is still pending, and the formalized result proves a slightly weaker runtime bound than the sharpest claim in the paper.
The preprint claims a deterministic algorithm can exactly count solutions to k-SAT in subexponential time for each fixed clause width k. For 3-SAT, that result would contradict ETH, which says the problem cannot be solved in subexponential time. The authors also say the work refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses. Those are claims in a preprint, not yet conclusions endorsed by independent reviewers.
OpenAI says its model found the result on October 16, and that a Lean formalization followed on October 17. The company published the preprint and formalization after consulting outside experts. The source account reports that the formalization comprises 3,977 files and about 710,000 lines; those figures are its own count. Lean verifies that a formal statement follows from its encoded definitions and proof steps. Here, however, the formal statement covers a weaker runtime bound than the preprint’s sharpest one.
The source account says a review 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. OpenAI has also announced results in mathematics and physics, but the three projects have different release and review arrangements; the physics materials, for example, are being held back at the request of the group overseeing that 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
ETH is used to support lower bounds: arguments that particular computational tasks cannot be solved below certain time costs. If the preprint’s central claim survives scrutiny, researchers would need to revisit results that rely on ETH and related assumptions. That could affect theoretical computer science, including the analysis of algorithms and the stated limits of computational problems. It would not automatically invalidate every paper that invokes ETH; the consequences would depend on each result’s assumptions and proof.
The immediate significance is also methodological. OpenAI’s claim puts an AI-generated proof and machine checking into a high-profile mathematical review process. A formal proof can help identify gaps in a derivation, but it does not by itself settle whether the formal statement matches the intended result or whether the paper’s interpretation is correct. Human experts still need to assess the definitions, the relationship between the code and the preprint, and the scope of what has actually been proved.
There is no reported near-term effect on ordinary software or commercial computing. The claimed algorithm remains slower than polynomial time, and OpenAI says practical improvements would require significant further research. The result is potentially important for theory, not evidence that everyday SAT-solving tasks will suddenly become faster.
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The Longstanding SAT Hardness Claim
SAT asks whether truth values can be assigned to variables so that every logical rule is satisfied. In k-SAT, each rule, or clause, contains at most k variables or their negations. With n variables, checking every possible assignment takes 2ⁿ possibilities. Researchers have developed faster-than-brute-force methods, but the cited source describes them as still exponential for 3-SAT.
The Exponential Time Hypothesis, introduced by Impagliazzo, Paturi and Francis Zane in 2001, says 3-SAT cannot be solved in subexponential time. It is stronger than the famous P-versus-NP question, which asks whether every problem with answers that can be checked efficiently can also be solved efficiently. Refuting ETH would not prove P = NP: a subexponential algorithm can still take more than polynomial time.
The source reports that OpenAI’s claimed algorithm runs roughly in time 2 to the power of n divided by the square root of log n. That is subexponential but remains far from a polynomial-time bound. It also notes that the status of ETH as a belief is not unanimous: theoretical computer scientist Scott Aaronson wrote in a 2017 survey that there was no field-wide consensus that ETH is true.
Independent Review and Proof Scope
The central uncertainty is whether independent researchers will validate the preprint’s reasoning and the relationship between its claims and the Lean formalization. The source says the Lean statement establishes a weaker bound than the paper’s sharpest runtime claim, so the formal check should not be described as covering every headline claim in its strongest form. The review group has been named, but no final assessment is reported.
It is also unclear how much the claimed algorithm would change the practical performance of SAT solvers or affect specific results that rely on ETH. The source says practical gains would require substantial further research and gives no demonstrated real-world speed-up. OpenAI’s account of the model’s work and timeline is attributable to the company; the material provided does not include an independent reconstruction of how the AI reached the proof.
Reviewers to Test the Claim
The next milestone is the external group’s examination of the preprint, formal statement and Lean code. Reviewers will need to determine whether the mathematical argument holds, whether the formalized theorem captures the relevant claims, and what follows for ETH and related hypotheses. The source account says OpenAI agreed to fund a conference on the result, which may provide a forum for scrutiny, but it does not give a date.
Until the reviewers report, the accurate description is that OpenAI has announced an AI-produced proof claim and released materials for inspection. Whether the work changes accepted complexity-theory results, and whether any practical algorithmic benefit follows, remains unsettled.
Key Questions
What does OpenAI say its AI proved?
OpenAI says an internal model produced a proof of a deterministic subexponential-time algorithm for exactly counting solutions to k-SAT when k is fixed. For 3-SAT, that would refute the Exponential Time Hypothesis.
Has the result been independently verified?
Not yet, according to the source material. OpenAI released a Lean formalization, and an external group is expected to examine the result. The source says the formalized bound is weaker than the sharpest bound claimed in the preprint.
Does this solve P versus NP?
No. Refuting ETH would challenge a stronger hypothesis about the time needed for 3-SAT, but it would not establish that P equals NP. The claimed algorithm remains slower than polynomial time.
Will this make SAT problems faster in practice?
No practical improvement is established in the material provided. OpenAI says significant further research would be needed, and the claimed theoretical runtime does not show that existing real-world workloads will run faster.
Source: ThorstenMeyerAI.com
