🔍 Read the full analysis: OpenAI’s AI Claim About The Exponential Time Hypothesis, Explained on ThorstenMeyerAI.com
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TL;DR
OpenAI says an unreleased AI model produced a proof that refutes the Exponential Time Hypothesis, a foundational conjecture about the difficulty of solving 3-SAT. Lean checked a formalized version of the proof, but experts have not yet completed independent review, and the formalization does not cover the paper’s sharpest runtime bound. The result does not resolve P vs. NP or offer an immediate practical speed-up.
The preprint claims a deterministic algorithm that exactly counts the solutions to any k-SAT formula, for fixed k, using subexponential time and polynomial memory. For 3-SAT, that would contradict ETH. The paper also says the result refutes the stronger Strong Exponential Time Hypothesis (SETH) and three related hypotheses. These are claims made in the preprint; the source material does not report that outside experts have verified them.
OpenAI says its model found the disproof on October 16, with the Lean formalization following on October 17. The preprint is dated October 17 and was revised October 25. The formal statement is brief, while the released Lean project contains 3,977 files and about 710,000 lines of code, according to the source author’s count. Lean is software that checks whether a formal mathematical proof follows from its stated definitions and rules.
There is a qualification: the Lean formalization checks a slightly weaker runtime bound than the one highlighted in the paper. The sharpest bound is not covered by that formalization. OpenAI has made the preprint and code public at the request of an outside group that will examine the computer science result. No independent sign-off is reported so far.
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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Why the ETH Claim Matters
ETH is used to support claims that many computational problems cannot be solved much faster than by searching an exponentially large set of possibilities. If the preprint’s result survives scrutiny, researchers would need to revisit some hardness results that rely on ETH or related assumptions. The scope and consequences would depend on the proof’s exact assumptions and on which results use the hypotheses.
The claimed algorithm is subexponential, not polynomial-time. The source describes its runtime as roughly 2 to the power of n divided by the square root of log n. That remains far slower than polynomial growth, and the source author says the speed-up over exponential search grows very slowly with problem size. OpenAI also says practical improvements will require significant further research. The result does not resolve P vs. NP, which remains an open problem.
The immediate importance is also methodological: an AI-produced proof has been translated into a form that a proof assistant can check, while human specialists review the mathematical claim. That may offer a model for evaluating future AI-generated research, but formal checking alone does not settle whether the formal statement matches the paper’s strongest claim or whether the result has been independently understood and assessed.
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From SAT to ETH
In k-SAT, a formula consists of rules that each involve at most k variables, and the task is to determine whether there is an assignment of true or false values that satisfies every rule. With n variables, a direct search can examine as many as 2^n assignments. Researchers have developed faster algorithms, but ETH asserts that 3-SAT still requires exponential time: 2 to the power of c times n, for some fixed c greater than zero.
Russell Impagliazzo, Ramamohan Paturi and Francis Zane formulated ETH in 2001. The source says ETH has informed hundreds of papers, while also noting that the degree of support for the hypothesis has been debated. A 2017 survey by Scott Aaronson, as quoted in the source, said there was “by no means a consensus in the field that ETH is true.”
ETH is stronger than the familiar P vs. NP question. P vs. NP asks whether every problem whose proposed answers can be checked efficiently can also be solved efficiently. Refuting ETH would challenge a stronger assumption about the limits of computation; it would not show that every efficiently checkable problem can be solved efficiently.
Review and Runtime Gaps
Independent review is pending. The source says a group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the computer science result. Impagliazzo and Paturi were among ETH’s original authors. Their involvement establishes who is expected to review the work, but does not indicate that they have endorsed it.
It remains unclear whether the reviewers will validate the full argument, whether the paper’s strongest runtime bound can be formalized, and how the result would affect specific hardness results if confirmed. The available source also does not provide a completed expert assessment of the proof’s assumptions or implications. Until that work is done, the distinction between the paper’s claim and the Lean-checked statement matters.
OpenAI’s announcement included two other research claims, in mathematics and physics, but their handling differs: the mathematics materials were released, while a group stewarding the physics work chose to withhold its AI-produced materials. Those claims are separate from the ETH result and should not be treated as corroboration of it.
Independent Experts Examine the Proof
The next reported milestone is the outside group’s examination of the ETH preprint and its formalization. OpenAI says it has agreed to fund a conference on the result. The source does not give a date for the review or conference, so the timing of any public assessment remains unknown.
Readers should look for a detailed account of whether the proof establishes the paper’s advertised result, how the gap between the preprint’s sharpest bound and the Lean statement is handled, and what consequences follow for work that assumes ETH or SETH. Until those questions are addressed, the result is best described as a significant AI-generated, machine-checked claim awaiting human review, not a settled change to practical computing.
Key Questions
What is the Exponential Time Hypothesis?
ETH says that solving 3-SAT requires exponential time as the number of variables grows, for some fixed positive constant in the exponent. It is a conjecture used to support lower-bound claims in theoretical computer science.
What does OpenAI say its model proved?
OpenAI says an unreleased model produced a deterministic, subexponential-time algorithm for exactly counting satisfying assignments of fixed-k SAT formulas. For 3-SAT, the preprint says this refutes ETH. The claim has not yet received the independent review described in the source.
Did Lean verify the entire claim?
Lean checked a formalized version of the proof, but the source says its statement gives a slightly weaker runtime bound than the paper’s sharpest result. That strongest bound is not covered by the formalization.
Does this settle P vs. NP?
No. Refuting ETH would challenge a stronger hypothesis about computational difficulty. P vs. NP remains open, and the claimed algorithm is still slower than polynomial time.
Will this make computers solve SAT problems faster now?
No immediate practical speed-up is established. The source says the claimed algorithm’s advantage grows slowly with problem size and quotes OpenAI saying practical improvements require significant further research. The result also awaits independent expert review.
Source: ThorstenMeyerAI.com
