🔍 Read the full analysis: How To Read OpenAI’s Claim About AI And The Exponential Time Hypothesis on ThorstenMeyerAI.com
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TL;DR
OpenAI says an unreleased internal model produced a proof of a subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT, which would refute ETH. A Lean formalization checks a weaker runtime bound than the sharpest bound in the preprint, and independent experts have not yet completed their review. The result does not solve P vs. NP and has no immediate practical consequence.
OpenAI said on October 26 that an unreleased internal AI model had found a proof refuting the Exponential Time Hypothesis (ETH), a longstanding assumption about how hard certain logic problems are to solve. The company released a preprint and a Lean formalization, but the result has not yet been independently reviewed, and the formalized statement checks a weaker runtime bound than the paper’s sharpest claim.
The preprint claims a deterministic algorithm that exactly counts solutions to any fixed-width k-SAT formula in subexponential time and polynomial memory. For 3-SAT, such an algorithm would contradict ETH, which says that solving 3-SAT 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’s announcement says its system found the disproof on October 16, followed by a Lean formalization on October 17. The preprint is dated October 17 and was revised October 25. The source account reports that the public formalization contains 3,977 files and about 710,000 lines; that count is the account’s own tally, not an independently verified measure.
The Lean code matters because it lets a proof assistant check that a formal proof follows from its stated definitions and rules. But formal checking is not the same as independent expert assessment of the result’s mathematical significance, assumptions or relationship to the paper. The source account says the formal statement covers a weaker runtime bound than the preprint’s strongest bound. A review group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is expected to examine the 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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A Major Change to Hardness Assumptions
If the paper’s result survives expert scrutiny, it would change how researchers use ETH to argue that computational problems are hard. The hypothesis has supported lower-bound arguments across theoretical computer science, and OpenAI says it has been used in hundreds of papers. A refutation would require researchers to revisit results that depend on ETH, though it would not automatically invalidate every such result: each argument and its assumptions would need to be checked.
The claimed algorithm is subexponential, not polynomial-time. The source describes its running time as roughly 2 to the power of n divided by the square root of log n. That still grows faster than any polynomial, and the source says the improvement becomes useful only very slowly as problem size increases. OpenAI itself said practical improvements would require significant further research. The announcement is therefore important first as a theoretical result and as a test of AI-assisted proof verification, not as evidence that hard computing tasks will soon become easy.
It also does not settle P vs. NP. That question asks whether every problem whose answer can be checked efficiently can also be solved efficiently. ETH is a stronger assumption about the difficulty of particular problems; refuting it removes that assumption but does not establish that P equals NP or that NP-complete problems have polynomial-time algorithms.
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The Hypothesis Behind the Claim
In k-SAT, each rule requires at least one of a bounded number of variables or their negations to be true. With n variables, there are 2^n possible assignments to check by brute force. Algorithms have improved on that approach, but ETH says the exponential dependence cannot be eliminated for 3-SAT: its running time must remain at least 2^(c·n) for some fixed positive c.
Russell Impagliazzo, Ramamohan Paturi and Francis Zane formulated ETH in 2001. The source account notes that researchers have since developed faster exponential-time algorithms for SAT, including work that reduces the base of the exponent; those advances do not contradict ETH, which rules out eliminating exponential growth altogether. The new preprint claims to cross that boundary with a subexponential algorithm.
OpenAI presented the ETH work alongside two other research announcements: a claimed proof concerning local smoothing in harmonic analysis and a physics result involving quantum chromodynamics and a grand unified completion of the Standard Model. Those projects have separate release and review arrangements. The ETH preprint and Lean code were made public at the request of its outside academic group, according to the source account.
Review and Runtime Gaps Remain
The proof has not yet received independent expert sign-off. Lean checking can catch errors in a formalized argument, but it does not by itself resolve whether the formalization fully captures the paper’s central claims or whether experts agree with the interpretation of the result. The source account says the formalized bound is weaker than the preprint’s sharpest runtime claim, leaving a gap between what the code checks and what the paper asserts.
The review group’s examination is still ahead, and the available account does not provide its conclusions or a timetable for them. It is also not yet clear how much of the preprint’s broader claims—including the asserted refutations of SETH and related hypotheses—will be covered by formal verification or expert review. OpenAI’s account of the model’s work and timeline is the company’s statement, not an independently established account of how the proof was produced.
The source also reports that in an earlier OpenAI release of AI-generated manuscripts, three papers were withdrawn after a sign error and 14 were revised. That history does not show that the ETH proof is wrong, but it reinforces why publication, formal checking and independent review are separate stages.
Experts Begin Examining the Proof
The next milestone is the promised review by the group led by Impagliazzo, Paturi and Williams. Readers will need to see whether the reviewers validate the argument, identify corrections, or question the paper’s claims, and whether the strongest runtime bound can be formally checked. Until those steps are complete, the announcement should be described as OpenAI’s claim supported by a public preprint and a machine-checked formalization with a narrower scope.
Researchers may also assess what follows for results that rely on ETH. Any consequences will depend on the exact proof and on the assumptions used in individual papers. OpenAI says it has agreed to fund a conference on the result, which could provide a forum for discussion; no practical application or near-term performance gain has been established.
Key Questions
What is OpenAI claiming its model proved?
OpenAI says an internal model found a subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT formulas. For 3-SAT, that would contradict the Exponential Time Hypothesis.
Does the claim solve P vs. NP?
No. P vs. NP remains open. Refuting ETH would challenge a stronger assumption about the difficulty of SAT problems, but it would not show that NP-complete problems can be solved in polynomial time.
Has the proof been verified?
OpenAI released a Lean formalization that checks a version of the proof, but the source account says its bound is weaker than the paper’s sharpest runtime claim. Independent expert review is still pending.
Will this make SAT problems easier to solve in practice?
No immediate practical improvement has been shown. The claimed algorithm remains superpolynomial, and OpenAI said practical gains would require significant further research.
What happens next?
A group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is expected to examine the result. Its assessment and any further formal verification will help determine how the claim should change researchers’ use of ETH.
Source: ThorstenMeyerAI.com
