🔍 Read the full analysis: OpenAI Says AI Proved A Complexity Assumption Wrong—What’s The Evidence? on ThorstenMeyerAI.com
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TL;DR
OpenAI says an unreleased internal AI model found an algorithm that refutes the Exponential Time Hypothesis, a long-standing conjecture about the difficulty of solving SAT problems. A Lean formalization checks a weaker result than the sharpest bound in the preprint, and independent expert review is still pending. The claim does not resolve P versus NP or imply an immediate practical speed-up.
OpenAI said on October 26 that an unreleased internal AI model found a deterministic, subexponential-time algorithm for exactly counting solutions to k-SAT formulas, a result that would refute the Exponential Time Hypothesis (ETH). The company published a preprint and Lean proof-checking code, but the result has not yet received independent expert review, and the formalized statement is weaker than the preprint’s sharpest runtime claim.
The claim concerns exactly counting satisfying assignments for k-SAT, a family of logic problems in which each rule contains at most k variables or their negations. OpenAI’s preprint says its algorithm runs in roughly 2^(n/√log n) time, where n is the number of variables. For 3-SAT, that is subexponential and contradicts ETH, which holds that solving 3-SAT requires exponential time of the form 2^(c·n) for some fixed positive c.
OpenAI says the AI system found the disproof on October 16, and that the Lean formalization followed a day later. The preprint is dated October 17 and was revised October 25. Lean is software that checks mathematical proofs against formal rules. The release includes a formal statement and a codebase that the source account counted at about 3,977 files and 710,000 lines. But that account reports the Lean proof checks a slightly weaker runtime bound; the preprint’s sharpest bound is not covered by the formalization.
The preprint also claims to refute the stronger Strong Exponential Time Hypothesis (SETH) and three related hypotheses. OpenAI has asked an outside group to explain and verify the computer-science result. The group is led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams; Impagliazzo and Paturi were among ETH’s original authors. OpenAI says the collaborators are unpaid, sign no nondisclosure agreements and speak for themselves, and that the company does not seek academic credit.
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 many algorithms cannot run faster than a particular exponential rate. Researchers use it to establish conditional lower bounds: if ETH holds, certain problems cannot have algorithms below stated time limits. If the new result survives review, those arguments would need to be reconsidered where they rely on ETH or the stronger SETH.
The claim is not that SAT becomes easy in practice. A running time near 2^(n/√log n) remains far above polynomial time, and the source account says OpenAI itself cautions that practical improvements would require substantial further research. The result also does not establish P = NP. It challenges a stronger assumption about the rate at which some problems must take time, not the open question of whether every efficiently checkable problem can be efficiently solved.
For readers beyond complexity theory, the immediate significance is also methodological. The episode tests whether an AI-produced mathematical argument can be assessed through public artifacts and formal proof checking, followed by scrutiny from specialists. A machine check can help expose gaps in a proof, but it does not by itself establish that the formal statement matches every headline claim or that the result’s assumptions and interpretation are sound.
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The Assumption Behind the Claim
In k-SAT, a formula is made up of logical clauses, such as “A or not-B or C,” and the task is to determine whether assignments of true or false can satisfy all clauses. With n variables, brute force tests up to 2^n assignments. Researchers have developed faster methods, but those improvements have historically reduced the exponential base rather than eliminated exponential growth.
ETH was formulated in work by Impagliazzo, Paturi and Francis Zane in 2001. It asserts that 3-SAT cannot be solved in subexponential time. SETH makes a stronger claim about the difficulty of k-SAT as k grows. The source material notes that algorithms have improved over time: Schöning’s 1999 method used about (4/3)^n steps, while later PPSZ refinements have reported lower exponential bases. These advances did not, by themselves, refute ETH.
ETH is stronger than the basic P-versus-NP question. P versus NP asks whether problems whose proposed answers can be checked efficiently can also be solved efficiently; the Clay Mathematics Institute still lists it as unsolved. A refutation of ETH would not answer that question. The source account also cautions against treating ETH as an agreed certainty: computer scientist Scott Aaronson wrote in a 2017 survey that there was “by no means a consensus in the field that ETH is true.”
What the Formal Proof Covers
The main outstanding question is whether specialists will validate the proof and its implications. No independent sign-off is reported yet. The Lean code checks a weaker bound than the preprint’s strongest claim, so readers should distinguish the formally verified statement from the full runtime result asserted in the paper. The available source material does not provide the exact weaker bound or a completed review.
It is also unclear how quickly the external group will complete its examination, whether reviewers will request changes, or whether the result will be accepted in its current form. OpenAI’s account says the group will explain and verify the work, but does not give a review timetable. The separate mathematics and physics claims in the same announcement have different release and review arrangements and should not be treated as independently established by the computer-science result.
External Review and Formal Checks
The next substantive milestone is the external group’s examination of the ETH preprint and its formalization. Reviewers will need to assess the argument, confirm that the Lean statement corresponds to the mathematical claim being made, and clarify the gap between the formalized bound and the paper’s sharpest bound. The published materials allow other researchers to inspect the work while that process proceeds.
OpenAI says it has agreed to fund a conference on the result, but the source material gives no date. Until expert review is complete, the appropriate description is that OpenAI has announced a proposed refutation supported by a machine-checked formalization of a weaker statement—not that the field has conclusively abandoned ETH. Any practical algorithmic consequences would require separate work.
Key Questions
What is the Exponential Time Hypothesis?
ETH is a conjecture that 3-SAT cannot be solved in subexponential time: its running time must remain exponential in the number of variables, up to a positive constant in the exponent. It is a hypothesis used in theoretical computer science, not a proved theorem.
What evidence has OpenAI released?
OpenAI released a preprint and Lean formalization. Lean checks a formal mathematical statement, but the source account says the checked statement has a weaker runtime bound than the preprint’s sharpest claim. Independent expert review has not yet been reported as complete.
Does this mean P equals NP?
No. The claim targets ETH, a stronger assumption about the time needed for certain problems. P versus NP remains open, and the proposed algorithm is still superpolynomial in the stated form.
Will this make SAT problems faster to solve now?
There is no reported immediate practical speed-up. The proposed running time remains extremely large as problem size grows, and the source material says practical improvements would require further research. The result’s practical reach is also subject to review.
Who is reviewing the claim?
OpenAI says an outside group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine and explain the computer-science result. The company has not given a completion date in the source material.
Source: ThorstenMeyerAI.com
