🔍 Read the full analysis: What OpenAI’s AI Results Really Show About The Exponential Time Hypothesis 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 SAT formulas, which would refute the Exponential Time Hypothesis. A Lean formalization checks a weaker runtime bound than the preprint’s sharpest claim, and independent expert review has not yet been completed. The result does not resolve P versus NP or imply practical speedups now.
OpenAI said October 26 that an unreleased internal AI model found an algorithm that, if the proof holds, refutes the Exponential Time Hypothesis (ETH), a long-standing conjecture about the difficulty of solving certain logic problems. The company published a research preprint and a Lean formalization, but independent expert review is pending, and the formalized result does not cover the preprint’s sharpest runtime bound.
The preprint claims a deterministic algorithm that exactly counts satisfying assignments for any fixed-width k-SAT formula in subexponential time and polynomial memory. For 3-SAT, that would contradict ETH, which says the problem cannot be solved in subexponential time. The authors also say the result refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses. Those are claims in the preprint, not conclusions independently endorsed by reviewers.
OpenAI says its system found the disproof on October 16, with the Lean formalization following a day later. The paper is dated October 17 and was revised October 25. The source report counts the public Lean release at 3,977 files and about 710,000 lines; the formal theorem itself is much shorter. Lean is proof-assistant software that checks whether a formal proof follows from its stated definitions and rules. Its check supports the formalized statement, but it does not by itself establish that every claim in the paper is represented in that statement.
That distinction matters here: according to the source report’s examination of the formalization and scope notes, Lean checks a slightly weaker runtime bound than the preprint’s strongest one. The full research claim therefore still needs scrutiny by mathematicians and theoretical computer scientists. OpenAI says an external group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the computer-science work; Impagliazzo and Paturi were among ETH’s original authors.
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 central claim survives review, it would overturn ETH and affect results that rely on the hypothesis to show that particular problems are hard to solve efficiently. OpenAI says ETH has been used in hundreds of papers to establish hardness results. Researchers would need to revisit which of those conclusions depend on ETH and whether they can be proved under weaker assumptions. The result would be a substantial change in complexity theory, not a blanket end to computational difficulty.
The claimed algorithm is subexponential, not polynomial-time. The source report describes its running time as roughly 2 to the power of n divided by the square root of log n. That still grows faster than any polynomial, and the report’s own arithmetic suggests the speedup over exponential-time methods grows very slowly with problem size. OpenAI also says practical improvements would require significant further research. There is no basis in the announcement for expecting ordinary software or commercial workloads to become faster immediately.
The finding also puts attention on a possible research process: an AI system proposes a proof, formal software checks a precisely stated version, and outside experts assess the mathematics and its relation to the paper. Formal verification is a valuable check, but it is not a substitute for establishing that the formal statement matches the headline claim, that assumptions are appropriate, and that the result has been independently understood.
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From SAT Algorithms to ETH
In k-SAT, each rule asks that at least one of up to k variables or their negations be true. The task is to determine whether assignments exist that satisfy all the rules; exactly counting such assignments is a related, stronger output task. With n Boolean variables, exhaustive search considers 2^n assignments. Researchers have developed algorithms faster than brute force, but still with exponential running times.
ETH, formulated by Impagliazzo, Paturi and Francis Zane in 2001, says 3-SAT cannot be solved in subexponential time. It is stronger than the assertion P ≠ NP: refuting ETH would not show that every problem whose solutions can be checked quickly can also be solved quickly. The P-versus-NP problem remains open, and the Clay Mathematics Institute lists it as unsolved.
OpenAI presented the computer-science work as one of three AI-produced research results announced October 26, alongside claims in harmonic analysis and physics. The company says outside researchers are helping determine how the manuscripts should be released and reviewed. The source report also notes that an earlier OpenAI release of more than 700 AI-generated manuscripts was followed by the withdrawal of three papers over a sign error and revisions to 14 others. That history makes the distinction between generated research, formal checks and independent review relevant to this announcement.
What Review Must Still Establish
The key open question is whether independent researchers verify the full theorem claimed in the preprint, including the strongest runtime bound, and whether the Lean formalization accurately captures the relevant arguments and assumptions. The source report says the public formalization proves a weaker bound. Until reviewers complete their examination, the strongest version should be treated as an announced claim, not an established result.
It is also not yet clear what revisions, if any, the review group will request, how long its examination will take, or whether the result will be published in a peer-reviewed venue. OpenAI’s account of how the model produced the proof and how the collaborators were selected is the company’s own description. No independent assessment of the model’s role or of the result’s practical consequences is provided in the source material.
The External Review Process
The next substantive milestone is the examination by the external group led by Impagliazzo, Paturi and Williams. The preprint and Lean code are public, so reviewers and other researchers can inspect both the mathematical argument and the formalized statement. A clear account of whether the checked theorem matches the paper’s strongest claim will be central to assessing the announcement.
Readers should look for the reviewers’ findings, any corrections to the preprint or formalization, and publication details. Until those appear, the responsible conclusion is limited: OpenAI reports an AI-generated proof that would refute ETH, and a Lean system has checked a related formal statement. Whether the broader claim holds remains under review; no practical computing change has been established.
Key Questions
What does OpenAI say its AI system proved?
OpenAI says an internal model found a deterministic, subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT formulas. Applied to 3-SAT, that would contradict ETH. Independent review has not yet confirmed the paper’s claim.
Does this solve P versus NP?
No. P versus NP remains open. ETH is a stronger conjecture about the time needed for 3-SAT; refuting it would not establish that all problems with quickly checkable answers can be solved quickly.
What did Lean check?
Lean checked a formalized proof statement against its rules. The source report says that statement establishes a weaker runtime bound than the sharpest bound claimed in the preprint, so the formal check does not automatically validate every part of the paper’s strongest claim.
Will the result make SAT solvers faster now?
That has not been shown. The claimed algorithm is still superpolynomial, and OpenAI says practical improvements would require substantial further research. No immediate change to commercial or everyday computing follows from the announcement.
Who is reviewing the result?
OpenAI says an external group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the computer-science work. The source material does not give a completion date or report findings from that review.
Source: ThorstenMeyerAI.com
