🔍 Read the full analysis: Was The Exponential Time Hypothesis Disproved? Understanding OpenAI’s AI Claim on ThorstenMeyerAI.com
Get the latest gadgets delivered free with Prime
- Fast, free delivery on millions of items
- Prime Video, Amazon Music and more included
- Member-only deals all year
TL;DR
OpenAI says an unreleased AI model produced a proof of a 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 related, slightly weaker result, but independent expert review is pending; the claim does not resolve P versus NP or show that practical problems will become faster.
OpenAI said on October 26 that an unreleased internal AI model produced a proof of a subexponential-time 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 formalization, but the work has not yet received independent expert sign-off, and the formalized statement is weaker than the preprint’s sharpest runtime claim.
The 67-page 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 holds that solving such instances requires time exponential in the number of variables. The paper also says the result refutes Strong ETH and three related hypotheses; these are claims in the preprint, not conclusions independently established through review.
OpenAI says its model found the disproof on October 16, and that the Lean formalization followed the next day. The preprint is dated October 17 and was revised October 25. The company published the paper and code at the request of an outside group that will examine the work. According to the source material’s count, the Lean release contains 3,977 files and about 710,000 lines; the formal statement itself is much shorter.
Formal checking provides a specific kind of assurance: Lean can verify that a proof follows from its stated definitions and rules, subject to the formalization and system. It does not by itself establish that the result has been independently reviewed, that the paper’s strongest bound has been encoded, or that the algorithm is useful on real-world instances. The source reports that Lean checks a slightly weaker runtime bound than the paper’s sharpest one.
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.
Powered by Thorsten Meyer AI
What an ETH Refutation Would Change
ETH is a widely used working assumption in theoretical computer science. Researchers rely on it to argue that certain problems cannot be solved substantially faster than exponential-time methods. If the claimed result holds as stated, those conditional hardness arguments would need to be revisited, and the result could change how researchers analyze the limits of algorithms.
The immediate significance is also methodological. OpenAI presents the work as an example of an AI system producing a difficult mathematical result that can be checked in a proof assistant. That combination may offer researchers a way to test AI-generated claims more rigorously. But the distinction between a checked formal statement and the full claim in a paper matters: here, the formalization does not cover the preprint’s sharpest runtime bound, while outside review remains outstanding.
There is no reported near-term practical speed-up. The algorithm described is still superpolynomial: its stated running time is roughly 2 to the power of n divided by the square root of log n. OpenAI says practical improvements will require significant further research. The result, if verified, would change a theoretical assumption, not make routine computing tasks suddenly faster.
Products Worth Considering
Logitech MX Mechanical Wireless Illuminated Keyboard, Tactile Quiet, Graphite
DJI Osmo Mobile 8 3-Axis Smartphone Gimbal Stabilizer
Logitech MX Keys S Wireless Keyboard, Graphite
As an affiliate, we earn on qualifying purchases.
From SAT Algorithms to ETH
In k-SAT, a formula consists of rules, each involving at most k variables, and the task is to determine whether assignments of true or false can satisfy all the rules. With n variables, trying every assignment takes 2^n possibilities. For k of at least three, SAT is a canonical NP-complete problem, meaning a broad class of difficult problems can be translated into it.
Researchers have found algorithms faster than exhaustive search, but those methods have still taken exponential time. ETH, proposed by Russell Impagliazzo, Ramamohan Paturi and Francis Zane in 2001, asserts that 3-SAT cannot be solved in subexponential time. The OpenAI preprint’s claim would overturn that stronger hypothesis, not settle the separate question of whether P equals NP. That question remains open.
The result is part of a wider OpenAI research release. The company announced three AI-produced claims: the ETH result, a mathematics result concerning local smoothing and related conjectures, and a physics result involving quantum chromodynamics and a proposed completion of the Standard Model. OpenAI says different outside groups are stewarding the work and determining how it should be released. The source also notes that an earlier batch of AI-generated manuscripts was revised after a sign error was found, underscoring why verification matters.
The Proof Still Needs Expert Review
The main unresolved issue is whether independent experts agree that the proof establishes the claimed result. OpenAI says an outside group will examine it, led by researchers including Russell Impagliazzo and Ramamohan Paturi, two of ETH’s original authors. The review has not yet been completed, and the source material does not report an independent verdict.
There is also a gap between the paper’s strongest stated runtime and what the Lean code formally checks. The formalization verifies a slightly weaker bound, according to the source. How that gap affects the central conclusion depends on the exact theorem and on whether the argument linking the formalized result to the paper’s sharper claim holds up under review.
OpenAI’s descriptions of the model’s work and timeline are the company’s account. The source says it inspected the preprint and code, but does not provide a completed external assessment. It is therefore not yet accurate to describe ETH as definitively disproved. Nor does the result, even if confirmed, establish P = NP or imply that practical instances can now be solved efficiently.
Independent Review and Formalization
The next milestone is the outside group’s examination of the proof, preprint and Lean code. That review should clarify whether the argument is sound, whether the sharper runtime claim follows, and what changes—if any—are needed to the formalization. OpenAI says it will fund a conference on the result, which could provide a venue for researchers to scrutinize the work and discuss its implications.
Until those steps are complete, the claim remains a significant but unverified research announcement. Any assessment should distinguish what the machine-checked files establish from what the paper claims beyond that formal scope, and distinguish both from OpenAI’s broader account of how the AI system produced the work.
Key Questions
Has the Exponential Time Hypothesis been disproved?
Not yet as an independently confirmed result. OpenAI says its AI system produced a proof that would refute ETH, but expert review is pending.
Does this result prove P equals NP?
No. ETH is a stronger hypothesis about the time needed to solve 3-SAT. Refuting it would not settle the separate P-versus-NP problem.
What has Lean checked?
OpenAI released a Lean formalization that checks a related result, but the source says it covers a slightly weaker runtime bound than the preprint’s sharpest claim.
Will this make computers solve SAT 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.
Who is reviewing the claim?
An outside group is set to examine the work, with Russell Impagliazzo and Ramamohan Paturi among its leaders, according to the source material. Its conclusions have not yet been reported.
Source: ThorstenMeyerAI.com
