🔍 Read the full analysis: OpenAI’s AI Claim About A Complexity Hypothesis, Carefully Explained on ThorstenMeyerAI.com
Get the latest gadgets delivered free with Prime
- Fast, free delivery on millions of items
- Prime Video, Amazon Music and more included
- Member-only deals all year
TL;DR
OpenAI says an unreleased internal AI model produced a proof of an algorithm that would refute the Exponential Time Hypothesis, a longstanding conjecture about the difficulty of solving SAT problems. A Lean formalization checks a weaker runtime bound than the preprint’s sharpest claim, and independent expert review is still pending. The result does not resolve P versus NP or make practical computing faster now.
OpenAI said on October 26 that an unreleased internal AI model produced a proof of a subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT problems, a result that would refute the Exponential Time Hypothesis (ETH). The claim is laid out in a public preprint and accompanied by a Lean formalization, but the work has not yet received independent expert sign-off, and the formalization checks a weaker runtime bound than the paper’s sharpest one.
The preprint claims a deterministic algorithm can exactly count solutions to any k-SAT formula, for fixed k, in time roughly 2^(n / √log n), using polynomial memory. Here, n represents the number of variables. Applied to 3-SAT, the claimed subexponential running time conflicts with ETH, which says that 3-SAT cannot be solved in subexponential time. The paper also says the result refutes the stronger Strong ETH (SETH) and three related hypotheses.
OpenAI says its system 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 account reports that the public Lean release contains 3,977 files and about 710,000 lines, while the formal statement itself is brief. Lean can check whether a formal proof follows from its stated assumptions and rules; that check does not by itself establish that the paper’s informal interpretation, scope or strongest bound matches the formalized result.
That distinction matters here: according to the source account’s review of the material, the Lean proof covers a slightly weaker runtime bound than the sharpest bound in the preprint. A group led by computer scientists Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is expected to examine the computer-science result. Impagliazzo and Paturi were among ETH’s original authors. OpenAI says it released the preprint and code at the group’s request.
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.
Powered by Thorsten Meyer AI
What an ETH Refutation Would Change
ETH is used in theoretical computer science to support claims that particular problems cannot be solved substantially faster than known methods. If the new result holds as described, researchers would have to revisit hardness results that depend on ETH or related assumptions. The significance is therefore mainly theoretical and methodological at this stage, rather than an immediate change to everyday computing.
The claimed algorithm remains far slower than polynomial time. A subexponential expression such as 2^(n / √log n) still grows rapidly as the input grows, and OpenAI itself says practical improvements would require significant further research. The result also does not establish P = NP: the question of whether every problem whose solutions are efficiently checkable can also be efficiently solved remains open. ETH is a stronger conjecture than P ≠ NP, so rejecting ETH would not settle that broader question.
The process may also matter beyond this specific claim. An AI-produced proof paired with a machine-checkable formalization offers a possible route for scrutinizing complex mathematical work. But the gap between the preprint’s strongest bound and the checked formal statement, along with the need for external review, shows why formal verification and expert assessment are not interchangeable.
Products Worth Considering
DJI Osmo Mobile 8 Standard Combo Phone Gimbal Stabilizer
Amazon Kindle Scribe Colorsoft 64GB (1st Generation)
Apple iPad Air (2025) 13‑inch Wi‑Fi 256 GB – Space Gray (M3 chip)
As an affiliate, we earn on qualifying purchases.
How ETH Became a Hardness Test
In k-SAT, a formula is made up of logical clauses, each involving at most k variables or their negations. The question is whether there is an assignment of true and false values that satisfies every clause; counting solutions asks how many such assignments exist. With n variables, a direct search can examine as many as 2^n assignments, although algorithms have improved on that brute-force approach.
The Exponential Time Hypothesis was proposed in 2001 by Impagliazzo, Paturi and Francis Zane. In broad terms, it says that solving 3-SAT still requires exponential time, rather than a subexponential algorithm. The source article notes that researchers have developed algorithms that reduce the exponential base, but ETH predicts that they cannot eliminate the exponential character of the problem.
OpenAI announced the ETH work alongside two other research claims: a proof concerning Sogge’s local smoothing conjecture in harmonic analysis, and a physics result concerning quantum chromodynamics and a proposed completion of the Standard Model. Those projects have separate review and release arrangements. The ETH preprint is part of a wider OpenAI research release; its claims should be evaluated on their own evidence, rather than treated as confirmed because they appeared alongside other announcements.
The Proof Still Needs Expert Review
The central open issue is whether independent specialists agree that the proof is correct and that its conclusions follow at the scope claimed in the preprint. The planned review group has not yet reported a verdict in the supplied material. Until that examination is complete, the ETH refutation remains an announced research claim, not an independently validated conclusion.
It is also unclear from the available account why the Lean formalization covers a weaker runtime bound, whether the difference can be closed, and how the formal statement relates in full detail to the paper’s strongest result. A machine check confirms the formalized proof under its definitions and assumptions; it does not automatically validate claims outside that formalization. The source account notes that earlier items in OpenAI’s research repository were withdrawn or revised after errors were identified, reinforcing the need to track versions and corrections without deciding the present proof’s status in advance.
OpenAI’s account says the system generated the proof and describes its release and review arrangements, but the supplied material does not provide a completed independent assessment or a public expert consensus. The status of the claim could change as reviewers examine the manuscript and code.
Reviewers Will Test the Claim
The next milestone is the examination by the external group led by Impagliazzo, Paturi and Williams. Their review is expected to address the proof’s logic, the relationship between the preprint and Lean formalization, and the implications for ETH and the other hypotheses named in the paper. No review timetable or final conclusion is given in the available material.
Readers should watch for a public assessment, revisions to the preprint or formalization, and any clarification of the runtime bound covered by Lean. Until then, the accurate description is that OpenAI has announced an AI-generated proof that claims to refute ETH, with a machine-checked formal component and independent review still pending.
Key Questions
What does OpenAI say its AI proved?
OpenAI says an internal model produced a proof of a deterministic, subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT formulas. For 3-SAT, that would contradict the Exponential Time Hypothesis.
Has the proof been independently confirmed?
Not in the supplied material. A Lean formalization is public, but an external group’s review is still pending. The source account also says Lean checks a weaker runtime bound than the sharpest claim in the preprint.
Does this mean P equals NP?
No. A refutation of ETH would not resolve P versus NP. The claimed algorithm is still superpolynomial, and P versus NP remains an open problem.
Will computers solve SAT problems faster now?
No immediate practical speed-up is established. The claimed runtime is still very large for growing inputs, and OpenAI says practical improvements would require significant further research.
What is Lean checking?
Lean checks that a proof written in its formal system follows from specified definitions and rules. It can help detect errors in the formalized argument, but it does not replace expert review of how that formal result relates to the paper’s claims.
Source: ThorstenMeyerAI.com
