🔍 Read the full analysis: AI’s Role In OpenAI’s Exponential Time Hypothesis Claim, Explained 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 k-SAT, a result that would refute the Exponential Time Hypothesis. A Lean formalization checks a slightly weaker bound than the paper’s sharpest claim, and independent experts have not yet reviewed the result. The claim does not resolve P versus NP or imply immediate practical speedups.
OpenAI said on October 26 that an unreleased internal AI model found an algorithm that, if the published proof holds, refutes the Exponential Time Hypothesis (ETH), a longstanding conjecture about the difficulty of solving Boolean satisfiability problems. The company released a preprint and a Lean formalization, but independent expert review is pending, and the formalized result proves a slightly weaker runtime bound than the paper’s sharpest claim.
The preprint claims a deterministic algorithm that exactly counts solutions to k-SAT in subexponential time and polynomial memory, for each fixed clause width k. For 3-SAT, that would contradict ETH, which holds that solving such instances requires exponential time as the number of variables grows. The paper also says its result refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses.
OpenAI says the model found the disproof on October 16, with the Lean formalization following the next day. The preprint is dated October 17 and was revised October 25. The source report counted 3,977 files and about 710,000 lines in the formalization; that is the report’s own count, not a figure independently confirmed here. Lean is software that checks whether a proof follows from its stated formal assumptions.
The distinction between the paper and the formalization matters: according to the source report, Lean checks a slightly weaker runtime bound, while the preprint’s sharpest bound is not covered by the formal proof. OpenAI has made the preprint and code public at the request of an external group that will explain and examine the computer-science result. The group is led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams; Impagliazzo and Paturi helped formulate ETH.
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 researchers to argue that particular problems cannot be solved efficiently, and OpenAI says it has informed hardness results across hundreds of papers. If the proof survives scrutiny, some conclusions that rely on ETH would need to be revisited. That would be a major theoretical result, but it would not automatically provide practical algorithms for those problems.
The claimed runtime is roughly 2 to the power of n divided by the square root of log n, according to the source material. This is subexponential, but still far slower than polynomial time. OpenAI itself says practical improvements would require significant further research. The result is therefore about the limits of what theory rules out, not a promise that companies or users will see faster computing soon.
The process is also being watched as a test of how AI-generated mathematics can be evaluated. A formal proof checker can catch errors within a formalized argument, but it does not by itself establish that the paper’s informal claims match the formal statement or that the formalization covers every headline result. Expert scrutiny remains part of verification, not a step replaced by the software.
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How the SAT Hypothesis Took Shape
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 values can satisfy them. With n variables there are 2^n possible assignments, although algorithms can do better than checking each one. For k of at least three, satisfiability is a canonical NP-complete problem.
Researchers have developed algorithms that reduce the base of the exponential. The Exponential Time Hypothesis, introduced by Impagliazzo, Paturi and Francis Zane in 2001, makes a stronger claim: 3-SAT cannot be solved in subexponential time. ETH is related to, but stronger than, the unresolved P-versus-NP question. Refuting ETH would not show that P equals NP; the claimed algorithm remains superpolynomial.
OpenAI presented the computer-science work alongside two other research claims: a result in harmonic analysis and a physics result concerning quantum chromodynamics and a proposed completion of the Standard Model. The source report says the physics materials are being held back while a group led by Nobel laureate David Gross stewards that work. The ETH preprint and code, by contrast, were released for examination.
The release follows a larger OpenAI research publication on October 6, which included more than 700 AI-generated manuscripts, according to the source. The repository history reportedly shows that three manuscripts were withdrawn after a sign error and fourteen revised the next day. That history is relevant to the current claims because it illustrates why formal checks and outside review are being treated as separate safeguards.
What Reviewers Still Need to Check
The principal open question is whether independent researchers can verify the argument and its implications. The expert group has been tasked with examining the work, but the source material does not report a completed review or an expert endorsement. Until that process is finished, the ETH refutation remains a published claim awaiting independent scrutiny.
There is also a gap between the strongest statement in the preprint and what the Lean files formally certify. The source report says the formalization proves a weaker bound, but it does not provide enough detail here to assess the precise difference or whether additional formal work is planned. The count of files and lines is likewise the report’s own scan, not a measure of correctness.
More broadly, the source material does not establish whether the claimed algorithm can be implemented with useful performance, how it would affect specific applied workloads, or when researchers might reach a conclusion. ETH’s standing in the field is itself not unanimous: the cited survey cautions against describing it as a settled belief held by every expert.
Independent Examination of the Proof
The next milestone is the external group’s examination of the preprint and its formalization. OpenAI says the group will explain and verify the result; the source report also says the company has agreed to fund a conference on the work. No review timetable or conference date is specified in the material provided.
Researchers will need to assess both the mathematical argument and the relationship between the paper’s strongest runtime claim and the weaker bound checked in Lean. Further formalization, corrections, or revisions could follow. For now, the appropriate status is an AI-attributed result released for review, not a settled change to computer-science theory or an immediate change to practical computing.
Key Questions
What is the Exponential Time Hypothesis?
ETH says that 3-SAT cannot be solved in subexponential time as the number of variables grows. It is a conjecture used to support lower-bound results in theoretical computer science.
Does OpenAI’s result prove P equals NP?
No. The claimed algorithm is still superpolynomial, and refuting ETH does not resolve the separate P-versus-NP problem.
Has the proof been independently verified?
Not yet, according to the source material. Lean checked a formalized result, but an external expert group is still expected to examine the work. The formalization also checks a weaker bound than the paper’s sharpest claim.
Will this make computers solve SAT problems faster now?
No immediate practical speedup has been established. The claimed runtime remains superpolynomial, and OpenAI says practical improvements would require significant further research.
What is Lean’s role in the claim?
Lean is a proof-checking system used to verify that a formal argument follows from its specified assumptions. Its check supports the formalized statement; it does not replace independent review of the paper or show that the strongest published bound has been formalized.
Source: ThorstenMeyerAI.com
