🔍 Read the full analysis: The Exponential Time Hypothesis Isn’t Settled By OpenAI’s AI Claim on ThorstenMeyerAI.com
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TL;DR
OpenAI says an unreleased internal model found a subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT, contradicting the Exponential Time Hypothesis if the result holds. A Lean formalization checks a related, slightly weaker bound, but independent expert review is pending; the claim does not resolve P versus NP or promise immediate practical speedups.
OpenAI said on October 26 that an unreleased internal AI model produced a proof of an algorithm that, if correct, would contradict the Exponential Time Hypothesis (ETH), a central conjecture in theoretical computer science. The company has released a preprint and a Lean formalization, but the result has not yet been independently reviewed, and the formalized statement does not cover the paper’s sharpest runtime bound.
The preprint describes a deterministic algorithm that exactly counts solutions to k-SAT in subexponential time and polynomial memory, for each fixed clause width k. For 3-SAT, the authors say the result contradicts ETH, which holds that solving 3-SAT requires exponential time in the number of variables. The preprint also claims consequences for the stronger Strong Exponential Time Hypothesis and three related hypotheses.
OpenAI says its system found the disproof on October 16, and that a Lean formalization followed the next day. The source report counts 3,977 files and about 710,000 lines of Lean code. Its account says the formalization checks a slightly weaker runtime bound than the preprint gives; the paper’s sharpest bound is not covered by that formal proof. A computer-checked proof can help rule out certain logical errors within the formalized statement, but it is not the same as independent experts confirming the argument, its scope or its significance.
OpenAI has made the preprint and code public at the request of an outside group that will examine the computer-science result. The source report says the group is led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams, with Impagliazzo and Paturi among the original authors of ETH. OpenAI also says it has agreed to fund a conference on the result. Those details, including the review arrangement, are reported from OpenAI’s account and the source material.
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 to support conditional lower bounds: results showing that certain computational tasks are unlikely to admit substantially faster algorithms, provided ETH is true. If the claim survives review, researchers would need to revisit arguments that depend on ETH and related assumptions. The reach of any revision would depend on the exact theorem and which downstream results rely on the hypothesis.
The stated algorithm is subexponential, not polynomial-time. The source material describes its runtime as roughly 2 to the power of n divided by the square root of log n. That growth remains far beyond what would make large instances easy in practice. OpenAI itself said practical improvements would require significant further research, so the announcement does not mean software or businesses will see an immediate speedup.
Nor does the result settle P versus NP. That separate open problem asks whether every problem whose solutions can be checked efficiently can also be solved efficiently. Refuting ETH would challenge a stronger assumption about the difficulty of 3-SAT; it would not establish that P equals NP or that P does not equal NP.
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From SAT Algorithms to ETH
In k-SAT, each rule is a clause containing at most k variables or their negations, and the task is to determine whether assignments can satisfy all clauses. With n variables, direct enumeration considers up to 2ⁿ assignments. Researchers have developed algorithms faster than brute force, but ETH posits that 3-SAT still requires time exponential in n, rather than a subexponential-time solution.
Russell Impagliazzo, Ramamohan Paturi and Francis Zane formulated ETH in 2001. It has been used as a basis for conditional hardness results across theoretical computer science. Its status has never been the same as a proven theorem: the source material notes that OpenAI described it as widely believed, while a 2017 survey by Scott Aaronson cautioned that there was no consensus that ETH is true.
The claim is part of a broader OpenAI research release announced October 26. According to the source, the company also presented claims in harmonic analysis and physics, with different outside groups advising on their release. The computer-science work is the item at issue here; the existence of those other claims does not independently verify it.
Review and Formal Proof Gaps
The main unresolved issue is whether independent researchers will validate the preprint’s argument and its claimed implications. The source material says the outside review is forthcoming; no expert sign-off is reported yet. Until the reviewers report, the result should be described as OpenAI’s claim, not as an established refutation of ETH.
There is also a specific difference between the paper and the machine-checked result: according to the source, Lean formalizes a slightly weaker runtime bound, not the sharpest one in the preprint. The available information does not explain in enough detail how that gap affects the paper’s headline claim or what further formalization may be planned. The source report also says its line and file counts are its own count, rather than a figure independently confirmed by OpenAI.
More broadly, a formal checker verifies that a proof follows the encoded definitions and rules; the quality of the formal statement and whether it matches the intended mathematical claim still matter. The available material does not establish that reviewers have completed those checks, nor does it provide a timetable for their conclusions.
Independent Review and Follow-Up
The next milestone is the outside group’s examination of the preprint and formalization. Its assessment should clarify whether the claimed algorithm follows from the proof, how the weaker Lean bound relates to the paper’s sharper bound, and which consequences for ETH and related hypotheses are justified. OpenAI has said the group will explain and verify the work; the source material gives no date for a final report.
Researchers may also examine the released code and statement directly, and future work could extend the formalization to cover the sharper theorem. Until review and any revisions are public, the accurate status is a significant AI-generated claim with a machine-checked component—not a settled change to computer science’s assumptions or a practical algorithm ready for deployment.
Key Questions
What is OpenAI claiming about ETH?
OpenAI says an internal AI model produced a proof of a deterministic, subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT. If the result is correct, its application to 3-SAT contradicts the Exponential Time Hypothesis.
Has ETH been refuted?
Not as an independently established result. OpenAI has released a preprint and Lean formalization, but the source material says outside expert review has not yet concluded. The formalization also checks a slightly weaker bound than the paper’s sharpest claim.
Does this prove P equals NP?
No. ETH is a stronger hypothesis about the time needed to solve problems such as 3-SAT. A refutation would not resolve the separate P-versus-NP problem, which remains open.
Will this make computing faster right away?
There is no reported immediate practical improvement. The proposed algorithm remains subexponential rather than polynomial-time, and OpenAI said practical applications would require significant further research.
What happens next?
An outside group is expected to review the proof and its formalization. The source material reports no timetable for the group’s findings, so the result’s status and any changes to the formal proof remain pending.
Source: ThorstenMeyerAI.com
