🔍 Read the full analysis: What To Know About OpenAI’s AI And 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 k-SAT, a result that would refute the Exponential Time Hypothesis. A Lean formalization checks a weaker runtime bound than the preprint’s sharpest claim, and independent experts have not yet completed their review. The result does not solve P vs. NP and has no immediate practical speed-up.
OpenAI said on October 26 that an unreleased internal AI model produced a proof of an algorithm that would refute the Exponential Time Hypothesis (ETH), a long-standing conjecture about the difficulty of solving certain logic problems. The company released a preprint and a Lean formalization, but the result has not yet received independent expert sign-off, and the formalized statement covers a weaker runtime bound than the paper’s sharpest claim.
The preprint claims a deterministic algorithm that exactly counts the solutions of any k-SAT formula, for fixed k, in subexponential time and polynomial memory. In 3-SAT, where each rule involves no more than three variables, that would contradict ETH. The authors also say the result refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses. Those conclusions depend on the proof being correct and on how its statements are established.
OpenAI says its model found the result on October 16, and that the Lean formalization followed on October 17. The preprint is dated October 17 and was revised October 25. ThorstenMeyerAI.com reports counting 3,977 files and about 710,000 lines in the published Lean material. Lean is a proof-assistant system that checks whether a formal proof follows from its stated definitions and rules. The source says the formal statement is much shorter than the full codebase.
There is a gap between the paper and the machine-checked result: according to the source, Lean checks a slightly weaker runtime bound, not the sharpest one asserted in the preprint. OpenAI has asked an external group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams to examine the computer-science work. Impagliazzo and Paturi were among the original authors of ETH. The review is not complete.
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 in theoretical computer science to support claims that particular problems cannot be solved quickly. If this proof survives scrutiny, researchers would have to revisit results whose hardness arguments rely on ETH or stronger assumptions. That could change the theoretical limits used to compare algorithms, even if it does not immediately change software or services.
The claimed algorithm remains subexponential, not polynomial-time. The source describes its runtime as roughly 2 to the power of n divided by the square root of log n. That is still a steep cost as problem size grows. OpenAI itself said practical improvements would require significant further research, and the source’s own arithmetic suggests the improvement over exponential growth becomes pronounced only very slowly as inputs get larger.
The development also tests a process for evaluating AI-produced mathematics: a model generates a proof, formal software checks a specified claim, and specialists assess the result and its significance. A formal check can catch errors within the encoded statement, but it does not by itself establish that the paper’s strongest claim is the one checked or that the work has been independently understood and validated.
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Why ETH Has Guided Hardness Research
In k-SAT, a formula contains rules made up of up to k variables or their negations. The task is to determine whether assignments of true and false can satisfy all the rules, or, in the result described here, to count how many assignments do so. With n variables, a direct search can require checking 2^n possibilities.
Researchers have found algorithms faster than trying every assignment, but those improvements have remained exponential. The source cites Schöning’s 1999 algorithm, which takes about (4/3)^n steps, and later PPSZ refinements with a bound around 1.307^n. ETH, proposed by Impagliazzo, Paturi and Francis Zane in 2001, asserts that 3-SAT still requires exponential time, in the sense that no algorithm removes the exponential dependence altogether.
ETH is stronger than the famous P vs. NP question, which asks whether every problem whose answers can be checked efficiently can also be solved efficiently. A refutation of ETH would not resolve P vs. NP. The source also cautions against presenting ETH as universally accepted: OpenAI described it as widely believed, while computer scientist Scott Aaronson wrote in a 2017 survey that there was no consensus in the field that ETH is true.
The announcement was one of three research claims OpenAI released on October 26. The company also described a mathematics result concerning Sogge’s local smoothing conjecture and a physics result concerning quantum chromodynamics and a grand unified completion of the Standard Model. OpenAI says outside experts advised how each project should be released. Those other claims are separate from the ETH proof and have their own review arrangements.
““there’s by no means a consensus in the field that ETH is true””
— Scott Aaronson, in a 2017 survey cited by ThorstenMeyerAI.com
The Proof Still Awaits Review
The central open question is whether independent specialists accept the proof and whether the paper’s strongest runtime claim can be established. The published Lean formalization checks a weaker bound than the sharpest bound in the preprint, according to ThorstenMeyerAI.com. The source says the external group has been asked to examine the work, but provides no completed review or final verdict.
It is also not yet clear how much the result would alter practical algorithms, if validated. The stated running time is still far above polynomial for large inputs, and OpenAI says more research is needed before practical gains can be assessed. A formal proof check supports the particular encoded theorem; it does not automatically settle every interpretation or consequence of the paper.
There is relevant history in the project’s earlier releases. The source reports that OpenAI withdrew three manuscripts on October 7 after identifying a sign error and revised 14 others. That does not establish an error in this result, but it underscores why publication, formal checking and independent review are distinct stages.
Expert Review and Formal Checks
The next major development is the external group’s examination of the ETH preprint and its Lean materials. The group is led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams; the source says OpenAI has also agreed to fund a conference on the result. No review timetable or final conclusion is specified in the material provided.
Readers should watch for a public account of whether the experts accept the proof, identify gaps or request revisions, and whether the formalization is extended to cover the preprint’s sharpest runtime bound. Until those steps are reported, the accurate description is that OpenAI has announced an AI-generated proof claim and released supporting materials—not that ETH has been conclusively refuted.
Key Questions
What is the Exponential Time Hypothesis?
ETH says that 3-SAT cannot be solved in subexponential time: the problem requires an exponential amount of computation as the number of variables grows. It is a hypothesis used to support hardness results in theoretical computer science.
Has OpenAI solved P vs. NP?
No. The reported result concerns ETH, a stronger assumption about the hardness of 3-SAT. P vs. NP remains open, and the claimed algorithm is still slower than any polynomial-time algorithm.
Has the proof been verified?
OpenAI released a Lean formalization, and the source reports that it checks a slightly weaker runtime bound than the preprint’s sharpest claim. An external expert review is pending, so independent sign-off has not been reported.
Will this make computing faster now?
There is no reported immediate practical speed-up. The claimed algorithm remains computationally demanding, and OpenAI said practical improvements will require significant further research.
Source: ThorstenMeyerAI.com
