Is The Exponential Time Hypothesis False? What OpenAI’s AI Results Show

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OpenAI said October 26 that an unreleased internal AI model produced a proof of a subexponential-time algorithm for exactly counting solutions to k-SAT, a central problem in theoretical computer science. If the claim holds, it would refute the Exponential Time Hypothesis (ETH), but the proof has not yet been independently reviewed by experts.

The result concerns a deterministic algorithm that counts satisfying assignments for any fixed-width k-SAT formula using subexponential time and polynomial memory. For 3-SAT, that would contradict ETH, which holds that solving 3-SAT requires exponential time. The preprint also says the result refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses.

OpenAI says its model found the result on October 16, with a Lean formalization produced the next day. The preprint, dated October 17 and revised October 25, and the formalization are public. The source account reports that the formalization contains 3,977 files and about 710,000 lines of code. Lean checks a slightly weaker runtime bound than the sharpest one stated in the paper, so the formalized claim does not cover the preprint’s strongest bound.

OpenAI describes the manuscripts it released as “a form of raw research data”. For the computer science result, it says an outside group will explain and verify the work. A group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is due to examine the proof; Impagliazzo and Paturi co-authored the hypothesis that the result challenges. Machine checking establishes that a formal statement follows from its encoded assumptions, but it does not by itself settle whether the encoding captures the paper’s full claim or whether the result withstands expert scrutiny.

What an ETH Refutation Would Change

ETH is used to support hardness results across theoretical computer science: researchers can use it to argue that certain algorithms are unlikely to run substantially faster than known approaches. A valid refutation would change the assumptions behind those results and could prompt researchers to revisit work that relies on ETH. It would not, by itself, supply practical algorithms for those problems.

The runtime described in the source is roughly 2 raised to n divided by the square root of log n. That is subexponential, but still much slower than polynomial time. OpenAI said practical improvements would require significant further research; the announcement offers no evidence that ordinary computing tasks will become faster immediately.

The result also draws attention to a research process in which an AI system proposes a proof and a proof assistant checks a formalized version. That can make errors easier to detect, but the gap between the paper’s strongest bound and the weaker bound covered by Lean shows why human review of both the mathematics and its formalization remains necessary.

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From SAT Algorithms to ETH

In k-SAT, each rule is a clause involving at most k variables, and the task is to determine which assignments make every clause true. With n variables there are 2^n possible assignments, although algorithms have improved on checking every one. The preprint describes k-SAT for k at least 3 as a canonical NP-complete problem.

ETH, introduced by Impagliazzo, Paturi and Francis Zane in 2001, makes a stronger claim than P ≠ NP: it says 3-SAT cannot be solved in subexponential time. Rejecting ETH would not settle whether P equals NP. The Clay Mathematics Institute lists P vs. NP as unsolved, and the source material emphasizes that the announced algorithm remains slower than every polynomial-time algorithm.

OpenAI says it released three research claims on October 26: the ETH result, a mathematics result concerning Sogge’s local smoothing conjecture, and a physics result concerning quantum chromodynamics and a proposed completion of the Standard Model. The company says outside groups are handling the releases differently. The ETH preprint and code are public, while the physics materials are being held back at the stewarding group’s request.

The Proof Still Needs Expert Review

Independent verification is pending. The source says a group led by Impagliazzo, Paturi and Williams will examine the result, but it does not report that the review has concluded or that experts have endorsed the proof. The AI-generated work is therefore an announced claim supported by a formalization, not a result that has completed that review.

It is also unclear whether the formalized, weaker runtime bound is sufficient to support every conclusion in the paper, including the claims about stronger and related hypotheses. Lean checks the proof represented in its code; reviewers must still compare that representation with the paper’s arguments and confirm that the assumptions and conclusions match.

The source account also notes that OpenAI’s earlier research release had errors: its repository history showed three manuscripts withdrawn over a sign error and 14 revised. That history does not establish a problem with the ETH proof, but it underscores the need to check each claim rather than infer reliability from the volume of published code. The article’s author reports having scanned the Lean code; that is not a substitute for independent expert verification.

Review and Formalization Checks

The next major milestone is the outside group’s examination of the ETH proof and its formalization. Reviewers will need to assess the mathematical argument, clarify the difference between the preprint’s sharpest bound and Lean’s weaker checked bound, and determine which stated hypotheses follow from the verified result.

OpenAI says it has agreed to fund a conference on the result. Further details about the review timetable, any changes to the paper or code, and the reviewers’ conclusions were not provided in the source material. Until those steps are reported, the appropriate status is a high-profile, machine-checked claim awaiting independent review, not a settled change to computer science.

Key Questions

What does OpenAI say its AI proved?

OpenAI says its internal model found a deterministic subexponential-time algorithm for exactly counting solutions to k-SAT when k is fixed. Applied to 3-SAT, the claim would contradict ETH.

Has the proof been independently verified?

Not according to the available account. A Lean formalization checks a related, slightly weaker runtime bound, and an outside group is expected to review the work. Expert sign-off has not been reported.

Does this solve P vs. NP?

No. ETH is a stronger hypothesis about the time needed to solve 3-SAT. A refutation would not establish that P equals NP, and P vs. NP remains open.

Will the result make software or computing faster now?

No immediate practical speedup has been established. The stated running time is still slower than polynomial, and OpenAI said practical improvements would require significant further research.

Source: ThorstenMeyerAI.com