🔍 Read the full analysis: The Exponential Time Hypothesis, Explained Through OpenAI’s AI Claim on ThorstenMeyerAI.com
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TL;DR
OpenAI says an unreleased model produced a proof of an algorithm that would refute the Exponential Time Hypothesis, a longstanding conjecture about the difficulty of solving certain logic problems. A Lean formalization checks a slightly weaker runtime bound than the paper claims, and independent expert review has not yet been completed. The result does not resolve P versus NP or make practical computing faster now.
OpenAI said on October 26 that an unreleased AI model produced a proof of an algorithm that would refute the Exponential Time Hypothesis (ETH), a conjecture underpinning many results about the difficulty of computational problems. The company released a preprint and a Lean proof-checking formalization, but the claim has not yet been independently reviewed, and the formalization covers a weaker runtime bound than the paper’s sharpest claim.
The preprint describes a deterministic algorithm that exactly counts solutions to a k-SAT formula in subexponential time when k is fixed. For 3-SAT, that would contradict ETH, which holds that no algorithm can solve the problem in subexponential time. The preprint also says the result refutes the Strong Exponential Time Hypothesis and three related hypotheses.
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. Lean is software that checks whether a formal proof follows from specified definitions and rules. The source account reports that the public formalization contains 3,977 files and about 710,000 lines; those counts are its own tally.
The formalized result is not identical to the strongest statement in the paper: according to the source account, Lean checks a slightly weaker runtime bound, while the preprint’s sharpest bound is not covered. OpenAI has asked an outside group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams to examine the computer science work. Impagliazzo and Paturi were among ETH’s original authors. That review is a next step, not a completed endorsement.
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 in theoretical computer science to support claims that particular problems cannot be solved substantially faster than known methods. If the proof survives expert scrutiny, researchers may need to revisit results whose hardness arguments rely on ETH or related assumptions. The scope of any such revision would depend on the proof’s exact statement and on which results use the affected hypotheses.
The finding would not mean that everyday computing suddenly becomes faster. The algorithm described in the source runs in roughly 2 to the power of n divided by the square root of log n, still an exponential-time bound. OpenAI itself said practical improvements would require significant further research. The source also says that, by its arithmetic, the speed-up grows very slowly with problem size.
Nor does this settle P versus NP. That question asks whether every problem whose proposed solutions can be checked efficiently can also be solved efficiently. ETH is a stronger assumption about the time needed for specific problems; refuting it does not establish that P equals NP, or resolve the P-versus-NP problem in either direction. P versus NP remains listed as unsolved by the Clay Mathematics Institute.
The result also puts attention on how AI-generated mathematics is evaluated. A machine-checkable proof can test whether a formal argument follows its encoded rules, but it does not by itself establish that the formal statement matches every claim in a research paper or that the work’s significance has been correctly interpreted. Formal checking and independent expert review address different parts of that task.
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The Conjecture Behind the Claim
In k-SAT, a formula is made up of rules, each involving at most k variables, and the task is to determine whether some assignment of true and false values satisfies every rule. With n variables, trying every assignment takes 2ⁿ possibilities. Researchers have developed faster algorithms, but these still take exponential time.
ETH, formulated by Russell Impagliazzo, Ramamohan Paturi and Francis Zane in 2001, proposes that 3-SAT cannot be solved in subexponential time: in simplified terms, its running time must remain at least 2 to the power of a positive constant times n. That is why a valid subexponential algorithm would challenge the hypothesis. The claim concerns the mathematical running time as problem size grows, not a reported benchmark on practical workloads.
OpenAI’s October 26 announcement included three research claims, not only the ETH result. The company described a mathematics proof concerning local smoothing and a physics result involving quantum chromodynamics and a grand unified completion of the Standard Model. The ETH preprint and Lean code were made public after consultation with outside experts, whom OpenAI calls “shepherds.” The source account says the physics materials were held back at the request of the group overseeing that work.
The release follows OpenAI’s October 6 publication of more than 700 AI-generated manuscripts. The repository’s history, as described in the source material, records three manuscripts withdrawn over a sign error on October 7 and revisions to 14 others. That earlier episode makes the distinction between a model producing a result, a formal system checking a proof, and researchers evaluating the work particularly relevant.
The Proof’s Remaining Checks
The central open issue is whether independent researchers accept the proof and its implications. The review group led by Impagliazzo, Paturi and Williams has been tasked with examining the result, but the source material provides no final assessment or timetable for one.
There is also a gap between the preprint’s sharpest runtime claim and the statement checked in Lean. The formalization provides a machine-checked result for a weaker bound, according to the source account; readers should not treat that as formal verification of every detail of the paper’s strongest claim. The precise relationship between the two statements, and whether the stronger result can also be checked, remains a matter for further scrutiny.
OpenAI’s account of how the model produced the result and how the review process is organized is the company’s own description. The source material does not provide the reviewers’ conclusions, a completed independent replication, or evidence of practical performance gains. Until those questions are addressed, the result is best described as a significant AI-produced claim supported by a formalized proof component, not an independently settled change to computer science.
Independent Review and Formalization
The next milestone is the outside group’s examination of the preprint and Lean code. That review can clarify whether the reasoning is sound, how the formalized bound compares with the paper’s strongest bound, and what consequences follow for work that assumes ETH. OpenAI also said it agreed to fund a conference on the result, which could provide a venue for technical discussion.
Until those steps produce public findings, the central claims remain under examination. Researchers and readers will be looking for a clear account of what has been formally checked, what the paper proves beyond that formalization, and whether the claimed algorithm changes any known practical methods. No immediate change to ordinary computing applications has been established.
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 in theoretical computer science to support hardness results for other problems.
What does OpenAI say its model proved?
OpenAI says its model produced an algorithm that exactly counts solutions to fixed-k SAT formulas in subexponential time. For 3-SAT, that would contradict ETH. The paper’s claim has not yet received independent review.
Has Lean checked the entire strongest claim?
Not according to the source account. The public Lean formalization checks a slightly weaker runtime bound; the sharpest bound in the preprint is not covered by that formalization.
Does this solve P versus NP?
No. P versus NP remains open. Refuting ETH would challenge a stronger assumption about the difficulty of particular problems, but would not establish whether P equals NP.
Will computers get faster because of this result?
No immediate practical speed-up has been shown. The claimed algorithm remains exponential-time, and OpenAI said practical improvements would require significant further research.
Source: ThorstenMeyerAI.com
