🔍 Read the full analysis: OpenAI’s AI Hypothesis Claim: Parsing The Exponential Time Evidence on ThorstenMeyerAI.com
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TL;DR
OpenAI says an unreleased AI model produced a proof that refutes the Exponential Time Hypothesis, a major conjecture about the difficulty of solving Boolean formulas. A Lean formalization checks a weaker runtime bound than the preprint’s sharpest result, and independent expert review is still pending. The claim does not resolve P versus NP or show that practical computing will get faster soon.
OpenAI said on October 26 that an unreleased internal AI model produced a proof refuting the Exponential Time Hypothesis (ETH), a long-standing conjecture about how quickly computers can solve certain logic problems. The company released a preprint and a Lean formalization, but independent expert review is still pending, and the formalized result establishes a weaker runtime bound than the sharpest bound stated in the paper.
The preprint claims a deterministic algorithm can exactly count the solutions of any k-SAT formula, for fixed k, in subexponential time and polynomial memory. For 3-SAT, that would contradict ETH. The authors also say the result refutes the stronger Strong Exponential Time Hypothesis (SETH) and three related hypotheses. These are claims in a research preprint, not conclusions already accepted by the field.
OpenAI says the AI found the disproof on October 16, and that the Lean formalization followed the next day. The source material counts the public formalization at about 3,977 files and 710,000 lines; the formal statement itself is much shorter. Lean is software that checks whether a proof follows from specified definitions and rules. The check provides a substantial verification layer, but it does not by itself establish that experts agree the mathematical result is correctly framed or that the formal statement covers every claim in the paper.
The key qualification is the gap between the preprint and the formalization: according to the source material, Lean checks a slightly weaker bound. The paper’s sharpest runtime bound is not included in the formalized result. A review group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is expected to 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
ETH is used to support lower-bound arguments: researchers rely on it to show that particular computational tasks are unlikely to have fast algorithms. If the preprint’s result survives scrutiny, those arguments would need to be revisited where they depend on ETH or related assumptions. That would affect theoretical computer science, including analyses of algorithms and the difficulty of computational problems.
The claimed algorithm is not a practical speedup for everyday computing. The source describes its runtime as roughly 2 to the power of n divided by the square root of log n: subexponential, but still much slower than any polynomial-time algorithm. OpenAI says practical improvements would require significant further research. The result also does not solve P versus NP; that question remains open, and ETH is a stronger hypothesis than the claim that P does not equal NP.
The immediate significance is also methodological. The case combines an AI-produced proof, a machine-checked formalization and review by researchers in the field. That process may offer a model for evaluating AI-generated research, while the difference between the paper’s strongest claim and the formalized bound shows why the scope of a check matters.
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From SAT Algorithms to ETH
In k-SAT, each rule is a logical statement involving at most k variables, and the task is to determine whether an assignment of true or false values can satisfy all the rules. With n variables, a direct search considers 2^n possible assignments. Researchers have developed algorithms that improve on brute force, but ETH asserts that 3-SAT still requires exponential time: roughly, a lower bound of 2^(c·n) for some fixed positive c.
Russell Impagliazzo, Ramamohan Paturi and Francis Zane formulated ETH in 2001. It is distinct from P versus NP, which asks whether every problem whose solutions can be checked efficiently can also be solved efficiently. A refutation of ETH would challenge a stronger assumption about computational hardness; it would not settle the P-versus-NP question.
OpenAI presented the computer-science result alongside two other research claims, in mathematics and physics. The source says the mathematical work concerns local smoothing and related conjectures, while the physics work concerns existence results in quantum field theory. OpenAI said outside academic groups would guide how each manuscript was released. The ETH preprint and code were made public; the physics materials were held back. This announcement followed an earlier release of more than 700 AI-generated manuscripts on October 6, some of which OpenAI later revised or withdrew after errors were identified.
“there’s by no means a consensus in the field that ETH is true”
— Scott Aaronson, in a 2017 survey
The Proof Still Faces Expert Review
The most important open issue is whether independent researchers will validate the preprint’s reasoning and its implications. The review group’s work is not complete, and the source material says experts have not yet signed off. A formal proof checker can verify that a written proof follows a formal statement, but readers still need to know whether that statement matches the paper’s intended theorem and whether the result’s assumptions and interpretation hold up.
There is also a specific scope limit: the public Lean work checks a weaker runtime bound than the strongest one reported in the preprint. The available material does not establish that the sharpest bound has been formally checked. Nor does it provide the review group’s findings, a timeline for its report, or evidence that researchers have reached consensus on the result.
The source also cautions that ETH has not been accepted uniformly across the field. OpenAI describes it as a widely used hypothesis, while cited researchers have expressed reservations about how broadly it is believed. That disagreement does not decide whether this proof is correct, but it matters when describing what a refutation would overturn.
Independent Review and Formal Scope
The review group led by Impagliazzo, Paturi and Williams is the next major checkpoint. Its task is to examine the public preprint and clarify whether the argument establishes the claimed refutation, including how the paper’s strongest runtime statement relates to the weaker bound formalized in Lean. The source material gives no date for a conclusion.
Researchers can also inspect the released code and formal statement, while further work may address the gap between the paper and the machine-checked result. OpenAI says it has agreed to fund a conference on the result. Until expert review and any further formalization are reported, the careful description is that OpenAI claims an AI-generated ETH refutation, with a machine-checked result of narrower scope and broader claims still under review.
Key Questions
What is the Exponential Time Hypothesis?
ETH is the hypothesis that solving 3-SAT requires exponential time as the number of variables grows. It is a widely used assumption in theoretical computer science, though the source material notes that researchers do not all agree on its status.
Did OpenAI solve P versus NP?
No. The announced result would challenge ETH, a stronger hypothesis about computational hardness. P versus NP remains open.
Has the proof been verified?
A Lean formalization has been released, and the source says it checks a weaker runtime bound than the sharpest one in the preprint. Independent expert review has not yet concluded, so the full claim should not be treated as settled.
Will this make computers solve hard problems faster now?
There is no announced immediate practical speedup. The claimed algorithm remains slower than any polynomial-time algorithm, and OpenAI said practical improvements would require significant further research.
Source: ThorstenMeyerAI.com
