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TL;DR
OpenAI says an unreleased AI model produced a proof that a deterministic algorithm can exactly count solutions to fixed-width SAT problems in subexponential time, contradicting the Exponential Time Hypothesis for 3-SAT. Lean checked a formalization, but independent expert review is pending, and its scope does not cover the paper’s sharpest runtime bound. The result does not resolve P vs. NP or promise near-term practical speedups.
OpenAI said on October 26 that an unreleased AI model had produced a proof refuting the Exponential Time Hypothesis (ETH), a long-standing conjecture about the difficulty of solving certain logic problems. The proof has a published Lean formalization, but independent experts have not yet signed off on the result, and the formalization covers a weaker runtime bound than the preprint’s sharpest claim.
The preprint claims a deterministic algorithm can exactly count satisfying assignments for any fixed-width k-SAT formula in subexponential time while using polynomial memory. For 3-SAT, that would contradict ETH, which holds that solving such instances requires exponential time. The paper says the result also refutes the Strong Exponential Time Hypothesis and three related hypotheses.
OpenAI says its system found the disproof on October 16, with a Lean formalization following on October 17. The company has made the preprint and code public at the request of an external group that will examine the work. The source account reports a formalization comprising 3,977 files and about 710,000 lines; that count is the account’s own, not an independently audited total.
The distinction between the paper and the formalized result matters. Lean checks whether a proof follows from its formal definitions and assumptions; it does not establish that the definitions capture the intended mathematical claim unless those are also reviewed. According to the source account, the Lean statement checks a slightly weaker bound than the preprint’s strongest runtime claim. The formalization is machine-checked, but expert review of the mathematical argument and its scope remains outstanding.
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 result withstands review, it would overturn a widely used assumption about the limits of algorithms for SAT and related problems. ETH has underpinned conditional hardness results across theoretical computer science: researchers often show that improving an algorithm for another problem would also yield a faster-than-expected solution to SAT. A valid refutation could require some of those conclusions to be revisited.
That would not mean hard computational problems suddenly become easy in practice. The reported algorithm runs in roughly 2 to the power of n divided by the square root of log n, according to the source material. This is subexponential but still far slower than polynomial time, and the source says OpenAI itself cautioned that practical improvements would require significant further research. No immediate software or commercial speedup has been established.
The announcement also tests a model for scientific trust: an AI system generates a result, formal software checks a proof, and outside researchers review the work. Each step addresses a different concern. A proof assistant can catch formal errors in what it checks, while human experts still need to assess the claim’s interpretation, the paper’s reasoning and the gap between the formal statement and the strongest published result.
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From SAT Hardness to Machine-Checked Proofs
In k-SAT, each rule is a logical clause involving at most k variables, and the task is to determine which assignments satisfy all clauses. With n variables, a direct search can test 2^n assignments. Researchers have developed faster methods, but those have remained exponential. ETH, proposed by Russell Impagliazzo, Ramamohan Paturi and Francis Zane in 2001, formalizes the expectation that 3-SAT cannot be solved in subexponential time.
ETH is stronger than the famous P vs. NP question. P vs. NP asks whether every problem whose proposed answers can be checked efficiently can also be solved efficiently; it remains open. Refuting ETH would remove a stronger hardness assumption, but would not prove P = NP or settle P vs. NP. OpenAI announced the computer-science claim alongside separate claims in mathematics and physics, each with a different external review and release plan.
The source account also describes a recent history that helps explain scrutiny of the formalization. OpenAI released more than 700 AI-generated manuscripts on October 6; repository records cited in that account show three were withdrawn after a sign error and 14 revised the next day. The account says about 42% of the repository’s top-line results were formalized in Lean. Those figures do not determine whether this ETH proof is correct, but they show why the checked scope and independent examination matter.
The Gap Between Claim and Check
The main open question is whether independent reviewers will accept the preprint’s argument and whether the formal statement adequately supports its strongest conclusions. The source material says the Lean proof checks a weaker runtime bound; it does not establish that the paper’s sharpest bound has been formally verified. The review group’s assessment and any changes to the paper are not yet available.
It is also unclear how broadly the claimed result will affect existing conditional hardness results. That depends on the proof’s final accepted form and on how researchers translate it into consequences for particular problems. The announcement does not show that practical SAT workloads, commercial optimization systems or other applications will become faster.
OpenAI’s account describes the model and development timeline, but the supplied material does not independently establish how the AI produced the proof or what role human researchers played in shaping it. Those details, alongside the reviewers’ findings, remain relevant to judging both the mathematical claim and the broader case for AI-assisted research.
External Review and Formal Scope
An external group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is expected to examine the computer-science result, according to the source material. Impagliazzo and Paturi were among ETH’s original authors. Their review could clarify whether the proof is sound, whether the formalization matches the paper, and what conclusions follow if the claim holds.
Readers should watch for a public review, revisions to the preprint or Lean code, and a direct explanation of the difference between the checked bound and the paper’s strongest bound. Until those appear, the accurate description is that OpenAI has announced a potentially major result with a machine-checked formalization—not that the research community has confirmed ETH is false.
Key Questions
What is the Exponential Time Hypothesis?
ETH is the conjecture that 3-SAT cannot be solved in subexponential time as the number of variables grows. It is used to support conditional hardness results in theoretical computer science.
Has OpenAI proved P = NP?
No. The announced claim concerns ETH, a stronger assumption about computational hardness. The P vs. NP problem remains open, and the reported algorithm is still slower than any polynomial-time algorithm.
Does Lean verification mean experts have confirmed the result?
No. Lean checks that a formal proof follows from its formal definitions and assumptions. Independent experts still need to assess the mathematics and whether the checked formal statement matches the paper’s strongest claim.
Will this make SAT-solving software faster now?
There is no confirmed immediate practical improvement. The reported runtime is subexponential but remains far from polynomial, and the source material says practical gains would require further research.
What happens next?
An external group led by Impagliazzo, Paturi and Williams is expected to review the work. Its findings, along with any revisions to the preprint or formalization, will determine how the claim is received.
Source: ThorstenMeyerAI.com
