🔍 Read the full analysis: OpenAI’s AI And The Exponential Time Hypothesis: Reading The Fine Print 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. Lean has checked a formalized version, but that check covers a weaker runtime bound than the paper’s sharpest claim, and independent expert review is still underway. The result does not solve P vs. NP and is not expected to make practical computing faster now.
OpenAI said on October 26 that an unreleased internal AI model produced a proof of a subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT, a result that would refute the Exponential Time Hypothesis (ETH). The company released a preprint and a Lean formalization, but independent experts have not yet completed their review, and the formalized statement is weaker than the paper’s sharpest runtime claim.
The preprint claims a deterministic algorithm that exactly counts satisfying assignments for any k-SAT formula when k is fixed, using subexponential time and polynomial memory. For 3-SAT, that would contradict ETH, which holds that solving such problems requires exponential time in the number of variables. The authors also say the result refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses.
OpenAI says the 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 source material reports a public formalization comprising 3,977 files and about 710,000 lines of code; its short formal statement is also available to inspect. Lean checks the formalized proof, but the checked runtime bound is weaker than the sharpest bound in the paper. That distinction means the machine-checked result does not, by itself, validate every strongest claim in the preprint.
OpenAI presented the work alongside two other research announcements: a claimed proof concerning Sogge’s local smoothing conjecture in harmonic analysis, and a physics result involving quantum chromodynamics and a proposed completion of the Standard Model. The computer-science preprint and Lean code were made public at the outside group’s request. OpenAI says the other projects have separate review arrangements, including that the physics materials are being held back.
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 has been used to support conditional hardness results across theoretical computer science: researchers show that certain algorithms cannot be much faster if ETH is true. A valid refutation would require those results to be reconsidered or stated under other assumptions. The reach of that change would depend on the proof’s precise theorem and on how it can be applied to individual results.
The claimed algorithm is not a practical shortcut for ordinary computing. The source describes its running time as roughly 2 to the power of n divided by the square root of log n, still far slower than polynomial time. OpenAI says practical improvements would require significant further research. The immediate importance is methodological: the claim tests whether AI-produced mathematics can be made legible and checkable through public code and specialist scrutiny.
The result is also not a solution to P vs. NP. ETH is a stronger assumption about the difficulty of particular problems; disproving it does not establish that every efficiently checkable problem can be efficiently solved. P vs. NP remains open.
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From SAT Hardness to Formal Checks
In k-SAT, each rule places a constraint on at most k variables, and the question is whether values can be assigned to satisfy all rules. A brute-force approach tests 2^n assignments for n variables. Researchers have developed faster-than-brute-force algorithms, but ETH asserts that the exponential dependence cannot be eliminated for 3-SAT.
Russell Impagliazzo, Ramamohan Paturi and Francis Zane formulated ETH in 2001. OpenAI’s account says an outside group led by Russell Impagliazzo and Ramamohan Paturi, two of the hypothesis’s original authors, will examine the new result, alongside Ryan Williams. The source says the group is expected to explain and verify the work; its conclusions have not yet been reported.
The release follows OpenAI’s publication of more than 700 AI-generated manuscripts on October 6, according to the source. It says repository records later showed three papers withdrawn over a sign error and 14 revised. Those changes illustrate why a formal proof check and outside review matter: code can test whether a specified theorem follows from formalized steps, while experts must still assess whether the formal statement matches the paper’s claims and whether the result is correctly interpreted.
The Proof’s Remaining Review Gaps
The central question is whether independent reviewers accept the paper’s argument and whether the formalized theorem captures the result as described. The source says Lean checks a weaker runtime bound than the paper’s strongest claim, leaving that sharper bound outside the formalization. A proof assistant can verify a formal statement against its rules; that does not automatically establish that the statement is the same as every claim made in a research paper.
The expert review has not been completed, and no final assessment from the group is included in the source material. It is also unclear what practical algorithms, if any, could follow from the theoretical result, or how quickly researchers might derive consequences for existing conditional hardness results. OpenAI’s account of the model’s work and release process is the company’s own description, not independent confirmation of the result.
Independent Review and Formalization
The next milestone is the outside group’s examination of the ETH paper and its formal proof. Reviewers will need to assess the mathematical argument, the relationship between the preprint and Lean statement, and the gap between the formally checked bound and the paper’s sharpest claim. The code and preprint are public, giving other researchers material to examine as well.
OpenAI says it has agreed to fund a conference on the result. No date or schedule for the reviewers’ conclusions is provided in the source. Until those assessments appear, the result should be described as an AI-generated claim with a machine-checked formalization, not as a finding that has received broad independent sign-off.
Key Questions
Did OpenAI’s AI solve P vs. NP?
No. The announced result concerns the Exponential Time Hypothesis, a stronger claim about the difficulty of SAT problems. P vs. NP remains open.
What does the claimed ETH refutation say?
The preprint claims a deterministic, subexponential-time algorithm for exactly counting satisfying assignments to fixed-width k-SAT formulas. If valid for 3-SAT, the result would contradict ETH.
Has the proof been independently verified?
Not yet, according to the source. A Lean formalization has checked a version of the result, and an outside group is expected to review the work. The formalized runtime bound is weaker than the paper’s sharpest claim.
Will this make computers faster now?
No immediate practical improvement is established. The claimed running time remains far slower than polynomial time, and OpenAI says practical gains would require significant further research.
Why does Lean matter here?
Lean is a proof assistant that can check whether a formal proof establishes a specified mathematical statement. Its check adds a reproducible layer of verification, but experts still need to assess whether the formal statement matches the paper’s full claims.
Source: ThorstenMeyerAI.com
