🔍 Read the full analysis: What The Exponential Time Hypothesis Claim Reveals About AI Research on ThorstenMeyerAI.com
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TL;DR
OpenAI says an unreleased AI model found a subexponential-time algorithm for exactly counting solutions to fixed-width SAT formulas, contradicting the Exponential Time Hypothesis. Lean formalized and checked a weaker runtime bound than the preprint’s sharpest claim; independent review is pending, and the result does not resolve P vs. NP or promise near-term practical speed-ups.
OpenAI said on October 26 that an unreleased internal AI model had produced an algorithmic result that, if the proof holds, refutes the Exponential Time Hypothesis (ETH), a longstanding conjecture about the difficulty of solving Boolean formulas. The proof has been checked with the Lean proof assistant, but independent expert review has not been completed, and the formalized result covers a weaker runtime bound than the sharpest claim in the paper.
The preprint describes a deterministic algorithm that exactly counts satisfying assignments for k-SAT formulas, with k fixed, in subexponential time and polynomial memory. For 3-SAT, the claimed runtime is roughly 2 to the power of n divided by the square root of log n, where n is the number of variables. That is subexponential, but still much slower than polynomial time. The paper says the result also refutes the stronger SETH and three related hypotheses.
OpenAI says its system found the ETH result on October 16, and that the Lean formalization followed on October 17. The public preprint is dated October 17 and was revised October 25. The source article reports that the released Lean repository contains 3,977 files and about 710,000 lines; those are the source author’s counts. The formal statement itself is much shorter, but the formalized runtime bound is weaker than the preprint’s sharpest version.
OpenAI’s October 26 post announced three research claims: the ETH result, a proof concerning Sogge’s local smoothing conjecture in harmonic analysis, and a physics result concerning quantum chromodynamics and a grand unified completion of the Standard Model. The company said outside academic groups would help steward the releases. For the computer-science result, a group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is to examine the work. OpenAI says the collaborators are unpaid, sign no non-disclosure agreements and speak for themselves.
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 New Test for AI-Made Proofs
The result matters both for complexity theory and for how AI-generated research is evaluated. ETH has been used to support conditional lower bounds: results showing that certain algorithms are unlikely to run faster, assuming the hypothesis is true. If the proof survives scrutiny, researchers would need to revisit arguments that rely on ETH, and assess which conclusions still hold under weaker assumptions.
The immediate significance is not that common software or industrial optimization will suddenly become faster. The claimed algorithm remains superpolynomial in scale, and OpenAI itself says practical improvements would require significant further research. The result instead tests whether an AI system can generate a difficult theoretical argument and whether a formal proof assistant can make parts of that argument more reliable to inspect.
Lean verification is meaningful because it checks that a formalized proof follows from its stated definitions and rules. It does not establish that the formalization captures every claim in the preprint, nor does it replace scrutiny of the modeling choices and the relationship between the paper and code. The gap between the paper’s strongest bound and Lean’s weaker one is therefore material, not a technical footnote.
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The Hypothesis Behind the Claim
In k-SAT, each rule is a logical clause involving at most k variables, and the task is to determine whether an assignment of true or false values satisfies every clause. With n variables, trying every assignment requires 2^n possibilities. Researchers have developed faster-than-brute-force algorithms, but those methods still take exponential time.
ETH, proposed in work by Impagliazzo, Paturi and Francis Zane in 2001, says that 3-SAT cannot be solved in subexponential time: its running time must be at least 2 to the power of c times n for some fixed positive c. The announced result challenges that claim by presenting a subexponential algorithm for the related exact-counting task, which would also provide a way to decide whether a formula has any satisfying assignments.
This is not a proof that P equals NP. P versus NP asks whether every problem whose solutions can be checked efficiently can also be solved efficiently. A subexponential algorithm can still take far more than polynomial time, so the claimed result would overturn ETH while leaving the P-versus-NP question open.
What Review Must Still Establish
The central unresolved issue is whether independent experts can verify the argument and confirm that the formalized theorem matches the preprint’s claims. The review group has been named, but its conclusions are not yet available. The source article says experts have not signed off on the proof.
There is also a specific scope limitation: the Lean development checks a weaker runtime bound than the paper’s sharpest one. The public materials therefore do not yet establish, through the cited formalization, the strongest version of the announced result. The preprint’s further claims about SETH and related hypotheses also require careful assessment of their reasoning and assumptions.
More broadly, the source article notes that OpenAI’s earlier repository had corrections after three manuscripts were withdrawn over a sign error and fourteen others were revised. That history does not show that this proof is wrong, but it underscores why publication, formal checking and independent review are separate stages. OpenAI’s account of the model’s work and release process is the company’s own account, not an independent finding.
Independent Review and Formal Checks
The next milestone is the examination by the academic group led by Impagliazzo, Paturi and Williams. Its review can clarify whether the mathematical argument is sound, how the paper’s claims relate to the Lean code, and what additional formalization may be needed to cover the strongest runtime bound.
OpenAI has said it will fund a conference on the result. Any account of the claim’s status should distinguish the preprint, the current machine-checked formal statement and the reviewers’ eventual conclusions. Until that process reports back, the ETH refutation remains an announced research claim with a machine-checked component, not a result independently endorsed by the field.
Key Questions
What does the Exponential Time Hypothesis say?
ETH says that 3-SAT cannot be solved in subexponential time; broadly, it predicts that solving the problem requires an exponential-time algorithm in the number of variables.
Does this result solve P versus NP?
No. The claimed algorithm is subexponential, not polynomial. Refuting ETH would challenge a stronger complexity assumption but would leave the P-versus-NP problem open.
Has the proof been independently verified?
Not yet, according to the source material. OpenAI released a Lean formalization, and an academic group is expected to examine the work. Independent review is pending.
What did Lean check?
Lean checked a formalized version of the proof, but the source article says the formalization establishes a weaker runtime bound than the sharpest bound stated in the preprint.
Will this make software faster now?
No immediate practical speed-up has been established. The claimed algorithm remains far slower than polynomial time, and OpenAI said practical improvements would require significant further research.
Source: ThorstenMeyerAI.com
