🔍 Read the full analysis: OpenAI’s AI Claim, Unpacked: The Exponential Time Hypothesis And Its Limits 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 long-standing assumption about how quickly certain logic problems can be solved. A Lean formalization has been machine-checked, but it covers a slightly weaker runtime bound than the preprint and has not yet received independent expert review. The result does not solve P vs. NP or make practical computing faster today.
OpenAI said on October 26 that an unreleased AI model had produced a proof refuting the Exponential Time Hypothesis (ETH), a foundational conjecture in theoretical computer science. The company released a preprint and a Lean formalization that has been machine-checked, but the work has not yet been independently reviewed, and the formalization verifies a slightly weaker runtime bound than the paper claims.
The result concerns k-SAT, a family of logic problems that asks whether Boolean variables can be assigned true or false so that every rule in a formula is satisfied. The preprint claims a deterministic algorithm can exactly count solutions for any fixed k in subexponential time and polynomial memory. For 3-SAT, that would contradict ETH, which says no algorithm can solve the problem in subexponential time.
OpenAI says its system found 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 publisher’s account says the Lean release contains 3,977 files and about 710,000 lines of code, while the formal statement itself is much shorter. Those details are reported by ThorstenMeyerAI.com, which says it examined the announcement, preprint, formal statement and code.
The distinction between the paper and the checked code matters. According to the source report, Lean verifies a bound that is weaker than the sharpest runtime stated in the preprint; that strongest version is not covered by the formalization. Machine checking can establish that a formal proof follows from its definitions and rules, but it does not by itself establish that the formal statement matches every claim in the paper or that the approach is understood and accepted by the field.
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 preprint’s result survives review, it would overturn ETH and, according to the paper as described in the source report, also refute the stronger Strong Exponential Time Hypothesis and three related hypotheses. ETH has been used to support hardness results across theoretical computer science, so its status affects the assumptions researchers rely on when proving that algorithms cannot run substantially faster.
The claimed algorithm is not a practical shortcut for everyday computing. The source report describes its runtime as roughly 2 to the power of n divided by the square root of log n. That is subexponential, but still grows much faster than polynomial time. OpenAI itself said practical improvements would require significant further research. The result, if confirmed, would first change theoretical understanding—not immediately speed up software or devices.
The process is also being watched as a test of how AI-generated mathematics can earn trust. The reported workflow combines an AI-produced argument, a public formalization and review by researchers, including people involved in developing ETH. Each step addresses a different question: whether the argument is sound, whether the formal proof checks, and whether experts agree that the result and its implications are correctly stated.
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From SAT Algorithms to ETH
In k-SAT, each rule contains no more than k variables or their negations. With n variables there are 2^n possible assignments, so trying every combination gives an exponential-time method. Researchers have developed faster algorithms by reducing the base of that exponential, but ETH asserts that the exponential dependence cannot be eliminated for 3-SAT.
Russell Impagliazzo, Ramamohan Paturi and Francis Zane formulated ETH in 2001. The source report cites earlier algorithmic improvements, including Schöning’s 1999 method and later PPSZ refinements, as examples of progress that still leaves an exponential runtime. OpenAI’s claim is therefore not that no one had found faster algorithms, but that a subexponential exact algorithm exists under the conditions set out in the preprint.
ETH is related to, but stronger than, the P vs. NP problem. P vs. NP asks whether problems whose answers can be checked efficiently can also be solved efficiently; it remains open. Refuting ETH would not prove P = NP, because a subexponential algorithm can still be much slower than a polynomial-time one. The source report also cautions that ETH’s acceptance is not universal: it cites computer scientist Scott Aaronson’s 2017 description of the field as having no consensus that ETH is true.
“there’s by no means a consensus in the field that ETH is true”
— Scott Aaronson, in a 2017 survey cited by ThorstenMeyerAI.com
Proof Scope and Expert Review
The central open question is whether independent experts will accept the preprint’s argument and its full implications. ThorstenMeyerAI.com reports that a group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the result; Impagliazzo and Paturi were among ETH’s original authors. The review has not yet concluded, and no independent endorsement is reported in the source material.
There is also a gap between the strongest runtime claim in the paper and the bound checked in Lean. The formalization supports a weaker bound, according to the report, so readers should not treat the paper’s sharpest figure as already verified by the released code. The source material does not establish whether the difference is a matter of formalization scope, a substantive proof issue, or both.
OpenAI’s broader research announcement included claims in mathematics and physics as well as computer science. The source report says those projects have different release and review arrangements. Their announcements are not independent confirmation of the ETH result, and the available material does not say when the ETH review group will publish its conclusions.
Reviewers to Examine the Proof
The next milestone is the independent examination of the preprint and formalization by the outside group. That review will need to address both the mathematical argument and the relationship between the paper’s strongest runtime claim and the weaker bound checked in Lean. No review timetable is given in the source material.
OpenAI has said it will fund a conference on the result, according to the report. Discussion there may help researchers test the proof, clarify its assumptions and assess what follows for results that depend on ETH. Until reviewers report back, the appropriate status is a substantial, machine-checked claim that remains under expert scrutiny—not a settled change to computer science.
Key Questions
What is the Exponential Time Hypothesis?
ETH is the conjecture that 3-SAT cannot be solved in subexponential time. It is used as an assumption in proofs about the difficulty of many computational problems.
Has OpenAI proved that P equals NP?
No. The reported result concerns ETH, which is stronger than the claim that P differs from NP. A subexponential algorithm can still take far longer than polynomial time, and P vs. NP remains open.
Has the proof been independently verified?
Not yet, according to the source material. A Lean formalization has been machine-checked, and an outside group is expected to examine the work. The report says Lean checks a slightly weaker runtime bound than the paper’s sharpest claim.
Will this make computers faster now?
No immediate practical speed-up is established. The claimed algorithm remains far slower than polynomial time, and OpenAI said practical improvements would require significant further research.
Source: ThorstenMeyerAI.com
