OpenAI Says Its AI Proved The Exponential Time Hypothesis False. Here Is What Is Actually Proven

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OpenAI said on October 26 that an unreleased AI model had found an algorithm that would refute the Exponential Time Hypothesis (ETH), a long-standing conjecture about how quickly certain logic problems can be solved. The company released a research preprint and a Lean formalization that checks a related, weaker runtime bound, but outside experts have not yet completed their review.

The preprint describes a deterministic algorithm that exactly counts solutions to any k-SAT formula, for fixed k, in subexponential time using polynomial memory. For 3-SAT, that result contradicts ETH, which says there is no algorithm that solves the problem in subexponential time. The preprint also claims to refute the stronger Strong Exponential Time Hypothesis and three related hypotheses.

OpenAI says its system found the disproof on October 16, and that a Lean formalization followed the next day. The preprint, dated October 17, was revised October 25. The company published the research material after consulting outside academics. The source account reports that the Lean release contains 3,977 files and roughly 710,000 lines of code; the formal statement itself is much shorter.

There is an important qualification: Lean checks a weaker runtime bound than the sharpest bound in the paper. The formalization therefore does not verify every strongest claim in the preprint. Formal checking can confirm that a specified proof follows from its definitions and assumptions; it does not, by itself, establish that the paper’s full result has passed independent expert scrutiny. That review is pending.

What the ETH Claim Would Change

If the result survives review, it would overturn a widely used assumption in theoretical computer science. ETH has served as a basis for proving that many algorithms cannot run below certain time limits. A counterexample could require researchers to revisit results that rely on the hypothesis, though it would not automatically invalidate each such result: the consequences would depend on how each proof uses ETH.

The result does not settle P versus NP. P versus NP asks whether every problem whose proposed solutions can be checked efficiently can also be solved efficiently. The reported algorithm remains superpolynomial, meaning it is still slower than any polynomial-time algorithm as input size grows. OpenAI has said practical improvements would require significant further research, and the source account’s arithmetic indicates the speed-up grows slowly with problem size. No immediate improvement to everyday computing follows from the announcement.

The news also tests a research process: an AI system generated a mathematical result, and a proof assistant checked a formal version of part of it. That may offer a way to inspect complex AI-produced arguments, but the gap between the paper’s strongest bound and the checked formal statement shows why machine verification and expert assessment are distinct steps.

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From SAT Algorithms to ETH

k-SAT asks whether a set of logical rules can all be satisfied by assigning each variable a true or false value. With n variables, checking every assignment takes 2ⁿ possibilities. Researchers have developed faster-than-brute-force algorithms, but ETH posits that 3-SAT still requires exponential time: roughly, at least 2 to the power of c·n for some fixed positive c.

The source material places ETH’s formulation in work by Russell Impagliazzo, Ramamohan Paturi and Francis Zane in 2001. It also notes that earlier algorithms reduced the exponential base without eliminating exponential growth. The reported AI result instead gives a subexponential bound, described in the source as roughly 2 to the power of n divided by the square root of log n.

OpenAI announced two other AI-produced research claims alongside the ETH work: a proof concerning local smoothing in harmonic analysis, and results involving quantum chromodynamics and a proposed completion of the Standard Model. Those projects have separate release and review arrangements. They do not independently confirm the computer-science claim.

What Review Has Yet to Establish

Independent review is not complete. OpenAI says a group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the computer-science result. Impagliazzo and Paturi were among the original authors of ETH. The source account says the group will explain and verify the work, but it provides no completed review or timetable for a verdict.

It is also not clear whether reviewers will validate the strongest runtime bound in the paper, which the current Lean formalization does not cover, or whether later formal work will bridge that gap. The exact practical value of the algorithm is likewise unsettled. OpenAI’s announcement is the source of the claim that its model produced the proof; the announced result should not be treated as established consensus before outside scrutiny.

Independent Checks and Formal Proof

The next key step is the outside group’s examination of the preprint and its proof. Readers can also compare the paper with the public Lean code and its formal statement, keeping in mind that the checked bound is weaker than the paper’s sharpest one. A clear account of any gaps, corrections or additional formalization would help establish precisely which conclusions follow.

OpenAI says it has agreed to fund a conference on the result. No conference date or final review schedule is given in the source material. Until the reviewers report, the claim remains a significant but unconfirmed challenge to ETH, not a settled change to computer-science theory.

Key Questions

What does OpenAI say its AI proved?

OpenAI says an internal AI model found a deterministic, subexponential-time algorithm for exactly counting solutions to fixed-width k-SAT. For 3-SAT, that would contradict the Exponential Time Hypothesis.

Has the proof been independently verified?

Not yet, according to the source material. A Lean formalization has checked a weaker runtime bound, while an outside group is expected to review the research claim.

Does this solve P versus NP?

No. The reported algorithm is still slower than polynomial time, and P versus NP remains an open problem. Refuting ETH would challenge a stronger assumption about computational hardness, not settle P versus NP.

Will the result make computers solve hard problems faster now?

There is no confirmed immediate practical effect. OpenAI says practical improvements would require significant further research, and the reported speed-up grows slowly with input size.

What is the main caveat about the Lean check?

The formalization checks a weaker runtime bound than the sharpest bound claimed in the preprint. The distinction means the strongest version of the paper’s claim still needs scrutiny beyond the current formal check.

Source: ThorstenMeyerAI.com