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TL;DR
OpenAI says an unreleased AI model produced a proof that a subexponential-time algorithm can exactly count solutions to fixed-width SAT problems, contradicting the Exponential Time Hypothesis. Lean checked a formalized version, but the sharpest runtime bound in the preprint is not covered, and independent expert review has not been completed. The result does not resolve P versus NP or imply immediate practical speed gains.
OpenAI said October 26 that an unreleased AI model produced a proof refuting the Exponential Time Hypothesis (ETH), a longstanding conjecture about the difficulty of solving certain logic problems. The company published a preprint and a Lean formalization that it says a computer has checked, but independent expert review has not yet been completed, and the formalized result does not cover the paper’s sharpest runtime bound.
The preprint claims a deterministic algorithm can exactly count the solutions of any k-SAT formula, for fixed k, in subexponential time while using polynomial memory. For 3-SAT, that would contradict ETH, which says no algorithm can solve the problem in time below an exponential bound of the form 2^(c·n) for some fixed positive c. The preprint also claims to refute the stronger Strong Exponential Time Hypothesis and three related hypotheses.
OpenAI says its system found the disproof on October 16, and a Lean formalization followed on October 17. The preprint, dated October 17 and revised October 25, and the formalization are public. The source account reports that the Lean release contains 3,977 files and about 710,000 lines; those totals are the account’s count. The formal statement itself is short, but the machine-checkable development is much larger.
There is a significant qualification: according to the source account’s examination of the Lean scope notes, Lean checks a slightly weaker runtime bound than the sharpest bound in the paper. That does not by itself establish that the paper’s full claim is wrong, but it means the public formalization does not machine-check every strongest statement as presented. OpenAI’s announcement also described two other research claims, in mathematics and physics; the ETH work is being released for outside examination.
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 by computer scientists to support claims that particular problems cannot be solved substantially faster than existing methods. If the result holds up, researchers may need to revisit arguments that rely on ETH, including hardness results in areas such as algorithms and complexity theory. The claim concerns a foundational boundary on computation, not a new product or a demonstrated improvement to everyday software.
The immediate importance is also methodological. OpenAI attributes the result to an AI system, while Lean provides a formal check of the parts encoded in its proof. Formal verification can help identify logical errors, but it does not mean that experts have agreed the mathematical definitions, translation, scope, or full paper are sound. The independent review is therefore central to determining how much confidence the result deserves.
There is no reported practical speed-up for users or businesses. The claimed runtime is roughly 2^(n divided by the square root of log n), which remains superpolynomial and can still grow very rapidly. OpenAI itself says practical improvements would require significant further research. Nothing in the announcement shows that existing SAT workloads become easy to solve.
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ETH, SAT and the Claim’s Scope
In k-SAT, a formula consists of rules that each involve at most k variables, and the task is to determine whether assignments of true or false values satisfy all the rules. A formula with n variables has 2^n possible assignments to check by brute force. Researchers have developed faster algorithms, but those improvements have historically reduced the exponential base rather than removed exponential growth.
ETH was formulated by Russell Impagliazzo, Ramamohan Paturi and Francis Zane in 2001. It posits that 3-SAT still requires exponential time, in a formal sense. The new paper’s claimed subexponential algorithm would challenge that assumption. The source account says a group led by Impagliazzo, Paturi and Ryan Williams will examine the result; Impagliazzo and Paturi were among ETH’s original authors.
This is not a proof that P = NP. P versus NP asks whether every problem whose proposed solution can be checked efficiently can also be solved efficiently. ETH makes a stronger claim about the time needed for a particular class of problems. Refuting ETH would weaken that stronger assumption but would not, on its own, settle P versus NP, which remains listed as unsolved by the Clay Mathematics Institute.
What Review Still Needs to Establish
Independent validation remains pending. The source account says the outside group led by Impagliazzo, Paturi and Williams will examine the proof, but it does not report a completed review or an expert verdict. A computer check confirms only that the encoded formal proof follows from the definitions and rules supplied to Lean; it does not automatically verify that the formalization captures every claim in the paper.
The scope difference also remains material: the account says the Lean development proves a slightly weaker runtime bound than the preprint’s sharpest result. It is not yet clear from the supplied material how reviewers will assess that gap, whether the full bound can be formalized, or whether the paper’s claimed refutation of the additional hypotheses survives scrutiny. The announcement supplies no basis for treating those questions as settled.
The broader context also calls for care. OpenAI’s earlier research repository had withdrawals and revisions after a sign error was found, according to the source account. That history is not evidence that this proof is incorrect, but it reinforces why publication, formal checking and independent mathematical review are separate steps.
Independent Review and Formal Scope
The next milestone is the outside group’s examination of the preprint and Lean code. OpenAI says it asked external experts, described as “shepherds,” how to release each research result. For the ETH work, the materials were made public at the group’s request, and OpenAI has agreed to fund a conference on the result, according to the source account.
Readers should look for a public assessment that addresses both the mathematical argument and the mismatch between the paper’s strongest runtime bound and the bound checked in Lean. Until reviewers report their findings, the result is best described as an AI-produced claim with a machine-checked formalization under review, not as a settled change to complexity theory. No date for a final review or conference is provided in the source material.
Key Questions
What does OpenAI say the AI proved?
OpenAI says an AI model produced a proof of a deterministic, subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT formulas. For 3-SAT, that would contradict ETH if the claim is valid.
Has the result been independently verified?
Not according to the supplied source material. Lean checked a formalized version, while an outside group is expected to review the work. The machine check and independent expert review are distinct steps.
Does this settle P versus NP?
No. ETH is a stronger hypothesis about the time required for particular problems. Refuting ETH would not by itself prove P = NP or resolve the P-versus-NP problem.
Will SAT problems become faster to solve in practice?
That has not been shown. The claimed runtime remains superpolynomial, and OpenAI said practical improvements would require significant further research.
What is the main caveat in the formal proof?
The source account says Lean checks a slightly weaker runtime bound than the sharpest bound in the preprint. The full claim and the significance of that difference remain matters for review.
Source: ThorstenMeyerAI.com
