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TL;DR
OpenAI says an unreleased AI model produced a proof of an algorithm that would refute the Exponential Time Hypothesis, a long-standing conjecture about the difficulty of solving SAT problems. The work has a public Lean formalization, but its formalized runtime bound is weaker than the preprint’s sharpest claim, and independent experts have not yet completed their review. The result does not resolve P versus NP or show that practical computing will improve soon.
OpenAI said October 26 that an unreleased internal AI model produced a proof of a subexponential-time algorithm for exactly counting solutions to fixed-width SAT problems, a result that would refute the Exponential Time Hypothesis (ETH). The preprint and a Lean formalization are public, but the claim has not yet been independently reviewed, and the formalization checks a weaker runtime bound than the paper’s sharpest version.
The result concerns k-SAT, a family of logic problems in which each rule involves at most a fixed number of variables. The preprint claims a deterministic algorithm that exactly counts satisfying assignments in subexponential time and polynomial memory. For 3-SAT, that contradicts ETH, which says that solving such instances requires time exponential in the number of variables. The authors also say the result refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses.
OpenAI reports that its model found the disproof on October 16 and that the Lean formalization followed on October 17. The preprint is dated October 17 and was revised October 25. The source article counted about 710,000 lines across 3,977 files in the released Lean code. Lean is software that checks whether a formal proof follows from its stated definitions and rules; it does not, by itself, establish that the formal statement matches every claim made in the paper.
That distinction matters here: the source article says the formal statement verifies a slightly weaker runtime bound than the preprint’s sharpest bound. It also reports that an independent group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the work. Impagliazzo and Paturi were among ETH’s original authors. Their 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
If the paper’s claim survives expert scrutiny, it would overturn a widely used assumption in theoretical computer science. ETH is used to show that many problems are unlikely to have efficient exact algorithms, by relating their difficulty to that of SAT. A counterexample could require researchers to revisit some of those conditional hardness results and the algorithms or lower bounds built on them.
The claimed improvement is subexponential, not polynomial: the source describes the runtime as roughly 2 raised to n divided by the square root of log n. That remains far slower than a polynomial-time algorithm as problem size grows. OpenAI also says practical improvements would require significant further research. The result therefore does not mean ordinary software, businesses or devices will become faster immediately.
It is also not a solution to P versus NP. That open question asks whether every problem whose answers can be checked efficiently can also be solved efficiently. Refuting ETH would weaken a stronger hardness assumption, but would not establish that P equals NP or that P differs from NP. The development is consequential both for the mathematical claim and for the process: an AI-produced proof is being offered alongside a machine-checkable formalization and an outside review.
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Why ETH Has Shaped Complexity Research
SAT asks whether truth values can be assigned to variables so that all stated logical rules hold. In 3-SAT, each rule can involve up to three variables. With n variables, a direct search has 2^n possible assignments. Researchers have found algorithms faster than checking every assignment, but those algorithms have still taken exponential time, with improvements reducing the base of the exponent.
ETH was formulated by Impagliazzo, Paturi and Francis Zane in 2001. It proposes that 3-SAT cannot be solved in time 2 raised to c times n for any fixed positive c. The OpenAI preprint’s claimed subexponential algorithm challenges that proposition. The source article also notes that researchers have not held a uniform view of ETH: OpenAI described it as widely accepted, while Scott Aaronson’s 2017 survey cautioned that there was no consensus on whether it is true.
The announcement is part of a wider OpenAI research release. According to the company’s October 26 post, it released work in mathematics and computer science, while a separate physics result is being stewarded by an outside group and its AI-produced materials are being withheld. OpenAI also said it had published more than 700 AI-generated manuscripts on October 6. The source article reports that three were withdrawn and fourteen revised the next day after a sign error was found, underscoring why formal checking and expert scrutiny matter.
What the Proof Review Must Resolve
The central open issue is whether the result is correct and whether the paper’s strongest runtime claim follows from its argument. The source article says the Lean proof covers a weaker bound, so readers should not treat the formalization as machine verification of every version of the claim. The independent group has not yet reported its conclusions.
It is also not yet clear how the result will affect specific hardness results that rely on ETH. That depends on the final statement, its proof, and how other researchers adapt their work. OpenAI’s account of the model’s role and the publication process comes from the company; the material provided does not describe an independent audit of how the AI generated the proof.
Independent Review and Follow-Up
The next major milestone is the outside examination led by Impagliazzo, Paturi and Williams. The group is expected to explain and verify the computer-science result, but the source material gives no publication date or timetable for its findings. The preprint and Lean code remain available for scrutiny in the meantime.
Readers should watch for a detailed review addressing both the mathematical argument and the relationship between the formalized theorem and the preprint’s strongest bound. Until that work is published, the responsible description is that OpenAI claims a refutation of ETH, with a related formal proof checked by Lean—not that the field has accepted ETH as refuted. OpenAI says practical applications, if any, would require further research.
Key Questions
What is the Exponential Time Hypothesis?
ETH is the hypothesis that 3-SAT cannot be solved in subexponential time as the number of variables grows. It is used to support conditional claims about the difficulty of other computational problems.
Does this result solve P versus NP?
No. A refutation of ETH would challenge a stronger assumption about computational hardness. It would not settle whether P equals NP, which remains open.
Has the proof been independently verified?
Not yet, according to the source material. OpenAI released a Lean formalization, and an outside group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is set to examine the work.
What does Lean verification establish?
Lean checks that a formal proof follows from the definitions and rules encoded in the system. The source article says the formalization checks a weaker runtime bound than the preprint’s sharpest claim, so it does not automatically verify every claim in the paper.
Will the claimed algorithm make computers faster soon?
There is no confirmed near-term practical improvement. The claimed algorithm is still superpolynomial, and OpenAI says translating the result into practical gains would require significant further research.
Source: ThorstenMeyerAI.com
