🔍 Read the full analysis: How OpenAI’s AI Results Relate To The Exponential Time Hypothesis on ThorstenMeyerAI.com
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TL;DR
OpenAI says an unreleased AI model produced a proof of a subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT, which would refute the Exponential Time Hypothesis. A Lean formalization checks a slightly weaker result, and independent experts have yet to review the work; the claim does not solve P vs. NP or promise immediate practical speedups.
OpenAI said October 26 that an unreleased internal AI model had found a subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT, a result that would refute the Exponential Time Hypothesis (ETH). The proof has a public Lean formalization, but the formalized statement is weaker than the preprint’s sharpest runtime bound, and independent expert review has not yet concluded.
The claim concerns counting satisfying assignments for k-SAT formulas, where each rule contains at most a fixed number k of variables. The preprint says a deterministic algorithm can do this in subexponential time with polynomial memory. Applied to 3-SAT, the result contradicts ETH, which says that solving 3-SAT requires exponential time in the number of variables. The preprint also claims to refute the Strong Exponential Time Hypothesis and three related hypotheses.
OpenAI says the model found the disproof on October 16 and that the Lean formalization followed on October 17. The preprint, dated October 17 and revised October 25, and the formalization were released publicly at the request of an outside group examining the work. The source account reports that the Lean repository contains 3,977 files and about 710,000 lines; those are its own counts, not an independent audit.
Formal verification checks whether a proof follows from its encoded definitions and rules. It does not, by itself, establish that the paper’s strongest claim has been formalized: according to the source account, Lean checks a slightly weaker runtime bound, while the sharpest bound in the preprint is outside the formalization’s scope. The account says a group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the result; Impagliazzo and Paturi were among ETH’s original authors.
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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A Challenge to ETH-Based Hardness
ETH is used in theoretical computer science to support conditional lower bounds: results showing that certain algorithms are unlikely to run quickly, assuming ETH holds. If the new proof withstands scrutiny, researchers would need to revisit conclusions that depend on that assumption. The scale of that work is not yet established, and the announcement does not show that particular algorithms or products will become faster.
The stated runtime is roughly 2 to the power of n divided by the square root of log n. That is subexponential, but still grows far faster than any polynomial-time bound. OpenAI itself says practical improvements would require significant further research. The immediate importance is theoretical: the claimed result would alter a widely used assumption about computational hardness, not deliver a near-term tool for businesses or consumers.
The process also matters. OpenAI presents the result as AI-generated research checked with a proof assistant and then sent to outside specialists. That combination could offer a model for evaluating machine-produced mathematics, but the gap between the preprint’s strongest claim and the formalized statement shows why both technical review and careful comparison of claims remain necessary.
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What ETH Says About 3-SAT
In k-SAT, a formula consists of logical clauses, each involving at most k variables, and asks whether some assignment of true or false values satisfies every clause. There are 2^n possible assignments for n variables, so a direct search takes exponential time. Researchers have developed faster-than-brute-force algorithms, but those remain exponential for 3-SAT.
ETH, proposed in work by Impagliazzo, Paturi and Francis Zane in 2001, says 3-SAT cannot be solved in subexponential time. It is stronger than the claim P ≠ NP. Refuting ETH would therefore not settle P vs. NP, which remains an open problem listed by the Clay Mathematics Institute. The source material also cautions that belief in ETH is not unanimous: it cites Scott Aaronson’s 2017 survey as saying there was no field-wide consensus that ETH is true.
OpenAI announced two other research claims alongside the computer-science result: a proof concerning local smoothing in harmonic analysis and a claim about quantum chromodynamics and a grand unified completion of the Standard Model. The source account says those projects have separate outside groups overseeing them; their status does not independently verify the ETH result.
The Formal Proof’s Remaining Gap
The central unresolved question is whether independent specialists accept the argument. The source material says the expert group will examine the work, but reports no completed review or verdict. Until that process is finished, the ETH refutation remains a claim in a preprint, not a result confirmed by independent peer review.
There is also a difference between what the paper claims and what the Lean code checks: the formalization covers a slightly weaker runtime bound. The source account says it reviewed the code and scope notes, but readers should not treat that account as a substitute for the planned expert examination. The details of how reviewers will report disagreements, and the timing of a conclusion, are not specified.
OpenAI’s announcement also includes claims beyond the ETH result, but the material provided does not establish that those separate mathematical and physics claims have been independently verified. Nor does it establish any practical performance gains from the proposed algorithm.
Expert Review and Formal Checks
The next milestone is the examination by the outside group led by Impagliazzo, Paturi and Williams. Its assessment will need to address both the mathematical argument and the relationship between the preprint’s strongest runtime bound and the weaker bound checked in Lean. No completion date is given in the source material.
OpenAI has also agreed to fund a conference on the result, according to the source account. Further public explanations, corrections or formalization work may clarify the claim and its limits. Until reviewers report their findings, the accurate description is that OpenAI has announced an AI-produced proof challenging ETH, with a related formal proof checked by Lean—not that the hypothesis has been conclusively disproved.
Key Questions
What did OpenAI say its AI model proved?
OpenAI said an internal model produced a proof of a deterministic, subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT. For 3-SAT, that would contradict ETH.
Has the result been independently confirmed?
Not yet, according to the source material. A group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is expected to examine it.
Does this solve P vs. NP?
No. Refuting ETH would challenge a stronger assumption about the difficulty of 3-SAT, but it would not settle whether P equals NP. P vs. NP remains open.
What does Lean verification confirm?
Lean checks that a formalized proof follows from its encoded rules. The source account says the formalization covers a slightly weaker runtime bound than the sharpest one claimed in the preprint, so it does not check every part of the strongest claim.
Will computers or businesses see an immediate speedup?
No immediate practical improvement is established. The claimed algorithm remains superpolynomial, and OpenAI says practical gains would require significant further research.
Source: ThorstenMeyerAI.com
