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TL;DR
OpenAI says an unreleased AI model found an algorithm that, if the proof holds, refutes the Exponential Time Hypothesis (ETH), a longstanding conjecture about the difficulty of solving certain logic problems. A Lean formalization has machine-checked a version of the proof, but independent expert review is still ahead, and the result does not resolve P vs. NP or make practical computing faster today.
OpenAI said on October 26 that an unreleased AI model found a deterministic algorithm that exactly counts solutions to fixed-k SAT problems in subexponential time, a result that would refute the Exponential Time Hypothesis if the proof is correct. The company published a preprint and a Lean machine-checked formalization, but independent review by computer scientists has not yet taken place.
The result concerns k-SAT, a family of problems that asks whether truth values can be assigned to variables so that every logical clause is satisfied. The preprint describes an algorithm that exactly counts satisfying assignments for formulas with a fixed clause size, using time roughly 2 to the power of n divided by the square root of log n, where n is the number of variables. For 3-SAT, that subexponential runtime conflicts with ETH, which holds that 3-SAT cannot be solved in subexponential time.
OpenAI says its system discovered the disproof on October 16 and that the Lean formalization followed the next day. The preprint is dated October 17 and was revised October 25. The source report says the public formalization contains 3,977 files and about 710,000 lines. Lean checks that a formal proof follows from its stated definitions and rules; that check does not by itself establish that the paper’s informal explanation is clear, that every modeling choice is appropriate, or that the result has been independently assessed by field experts.
There is a gap between the preprint and the formalized result: according to the source report, Lean verifies a slightly weaker runtime bound, while the paper’s sharpest bound is not covered by the formalization. OpenAI says an external group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the computer-science work. Impagliazzo and Paturi were among the researchers who introduced ETH. That review has not yet produced a public verdict.
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 independent review confirms the proof, it would overturn a widely used assumption in theoretical computer science about how quickly hard SAT problems can be solved. Researchers use ETH and related assumptions to establish that many algorithms cannot run below certain time bounds. A valid refutation would require those results to be revisited where they rely on ETH, though it would not automatically invalidate findings based on other assumptions or proofs.
The result is also a test of a research process involving AI-generated mathematics and formal verification. A machine-checkable proof can catch certain errors and make a result more auditable, but it is not the same as independent human scrutiny of the theorem’s significance, the translation into formal definitions, or the limits of what has been established. The distinction matters as AI systems produce more technical research claims.
There is no immediate practical speed-up promised. The stated algorithm remains superpolynomial and grows very quickly with problem size. The source report says OpenAI itself described practical improvements as requiring significant further research. For businesses and users, the near-term news is about the possible revision of a theoretical assumption, not a new tool that makes everyday computing faster.
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From SAT Algorithms to ETH
SAT asks whether a set of logical clauses can all be made true at once. In k-SAT, each clause contains at most k variables or their negations. Trying every assignment takes 2^n possibilities for n variables, while decades of research have produced algorithms that improve on brute force without removing exponential growth. ETH, proposed in 2001 by Impagliazzo, Paturi and Francis Zane, formalizes the expectation that 3-SAT still requires exponential time.
ETH is stronger than the famous P vs. NP question. P vs. NP asks whether every problem whose proposed answer can be checked efficiently can also be solved efficiently; it remains open. Refuting ETH would weaken a stronger assumption about the rate at which particular problems can be solved, but it would not prove P = NP. The source material also notes that researchers have not agreed universally that ETH is true, so describing it as an established fact would overstate its status.
The ETH claim appeared alongside two other research announcements from OpenAI: a mathematics result concerning local smoothing and related conjectures, and a physics claim concerning quantum chromodynamics and a grand unified completion of the Standard Model. OpenAI said different outside groups were asked to guide the release of the manuscripts. The ETH preprint and Lean code were made public; the physics materials were held back at the request of the group overseeing them, according to OpenAI.
Formal Checks and Open Questions
The main unresolved issue is whether independent experts agree that the preprint establishes the claimed theorem. The Lean check is meaningful evidence that the formalized statement follows within the system, but the source report says it covers a weaker runtime bound than the paper’s strongest claim. The scope and relationship between those two results need close examination.
It is also not yet clear when the reviewing group will publish its findings, what changes it may request, or whether the paper will pass peer review. OpenAI’s account of how the work was produced and released is the company’s own description. The source report notes that an earlier repository release had three manuscripts withdrawn over a sign error and 14 revised, illustrating why public research claims still need checking; those earlier corrections do not by themselves determine whether this proof is valid.
The result’s consequences would also depend on the exact theorem that survives review and on which existing hardness arguments rely on ETH rather than other assumptions. Even if the central claim is verified, it would not establish that SAT can be solved in polynomial time or settle P vs. NP.
Review of the Proof and Bound
The next public milestone is the external group’s examination of the preprint and its formalization. Readers should look for a clear account of whether the paper’s strongest bound is proved, how it relates to the weaker Lean-checked statement, and whether the reviewers identify errors or request revisions. OpenAI has also said it agreed to fund a conference on the result.
Until that scrutiny is complete, the accurate status is an announced result with a machine-checked formal component, not an independently confirmed refutation of ETH. Further updates may clarify the proof’s scope, the formalization, and whether the claim appears in a peer-reviewed publication.
Key Questions
Has OpenAI proved that P equals NP?
No. The announced result concerns ETH, a stronger hypothesis about the time needed to solve SAT problems. P vs. NP remains open, according to the source material.
What does the Lean check establish?
Lean checks that a formalized theorem follows from its definitions and rules. The source report says the formalization covers a slightly weaker runtime bound than the preprint’s sharpest claim, and expert review is still pending.
Will this make computers solve SAT problems faster now?
There is no immediate practical improvement established. The reported algorithm is still superpolynomial, and OpenAI said practical gains would require significant further research.
Who is reviewing the computer-science result?
OpenAI says an external group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine it. No public review verdict is included in the source material.
Source: ThorstenMeyerAI.com
