🔍 Read the full analysis: OpenAI’s AI And A Famous Complexity Hypothesis: What The Results Establish on ThorstenMeyerAI.com
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TL;DR
OpenAI says an internal AI model produced a proof refuting the Exponential Time Hypothesis, a longstanding conjecture about the difficulty of solving certain logic problems. The proof has been checked in Lean, but independent expert review is pending; the formalized result also establishes a weaker runtime bound than the preprint’s sharpest claim. The result does not solve P versus NP or make practical computing tasks faster.
OpenAI said on October 26 that an unreleased internal AI model produced a proof refuting the Exponential Time Hypothesis (ETH), a conjecture about how quickly computers can solve certain logic problems. The company published a preprint and a Lean formalization, but the claim has not yet been independently reviewed, and the formalized statement is weaker than the preprint’s sharpest runtime bound.
The preprint claims a deterministic algorithm can exactly count solutions to any k-SAT formula, for fixed k, in subexponential time and polynomial memory. For 3-SAT, that would contradict ETH, which holds that solving such problems requires exponential time. The authors also say the result refutes the stronger Strong Exponential Time Hypothesis and three related hypotheses.
OpenAI says its system found the disproof on October 16, and that the Lean formalization followed the next day. The preprint is dated October 17 and was revised October 25. The source article reports that the public Lean repository contains 3,977 files and about 710,000 lines, based on its own count. Lean is software that checks whether a proof follows from its formal definitions and rules; that check is not the same as independent expert review of the underlying research.
A key qualification is that the Lean formalization checks a slightly weaker runtime bound than the one highlighted in the preprint. The formalization does not cover the paper’s sharpest bound. The source article describes the claimed runtime as roughly 2 to the power of n divided by the square root of log n, which remains far slower than polynomial time.
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 use it to argue that particular problems cannot have substantially faster algorithms, provided the hypothesis holds. If the proof survives scrutiny, some of those arguments would need to be revisited or reformulated. The result would also matter as a test of whether AI-generated mathematical work can contribute to research when paired with a formal proof checker and outside scrutiny.
But the claim does not mean that practical computing tasks will suddenly become faster. The proposed algorithm is still subexponential, not polynomial, and its very slow-growing advantage over exponential-time approaches may be of little immediate use on real-world inputs. OpenAI itself said practical improvements would require significant further research, according to the source article.
Nor does the result settle P versus NP. That question asks whether every problem whose solutions can be checked efficiently can also be solved efficiently. ETH is a stronger, more specific hypothesis about exponential-time lower bounds. Refuting ETH would challenge that additional assumption, while leaving P versus NP unresolved.
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How ETH Became a Hardness Benchmark
In k-SAT, each rule contains at most k variables, and the task is to determine whether true-or-false assignments can satisfy all the rules. With n variables, trying every assignment takes 2^n possibilities. Researchers have developed algorithms faster than brute force, but the hypothesis at issue says the exponential dependence cannot be eliminated for 3-SAT.
Russell Impagliazzo, Ramamohan Paturi and Francis Zane introduced ETH in 2001. The source article notes that researchers have improved exact SAT algorithms over time, including methods that reduce the base of the exponential, without removing the exponential form. It also cautions against portraying ETH as an uncontested belief: computer scientist Scott Aaronson wrote in a 2017 survey that there was no consensus in the field that ETH is true.
The ETH announcement was one of three research claims OpenAI described on October 26. The company also announced work concerning harmonic analysis and a physics result involving quantum chromodynamics and a proposed completion of the Standard Model. The source article says the outside groups overseeing those projects chose different release arrangements. The ETH preprint and code were made public at the reviewing group’s request.
Review, Scope and Runtime Questions
The central open issue is whether independent specialists agree that the paper’s mathematical argument is sound and that its implications match the authors’ description. OpenAI says an outside group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams will examine the result. Impagliazzo and Paturi were among ETH’s original authors, but the review has not yet been completed.
It is also unclear how the gap between the preprint and the Lean formalization will be resolved. The formal checker verifies the statement encoded in its files; it does not, by itself, establish a stronger statement absent from that formalization. Readers will need to see whether the sharpest bound can be formalized and whether reviewers accept the paper’s full claims.
The source article also describes earlier problems in OpenAI’s research repository: on October 7, three manuscripts were withdrawn over a sign error and 14 others were revised. Those changes do not establish that the ETH proof is wrong, but they reinforce why independent review and careful comparison between the paper and its formal code matter.
Independent Review and Formalization
The next major step is the external group’s examination of the preprint and Lean files. The review should clarify whether the proof supports the stated ETH and SETH conclusions, whether the formalization captures the relevant argument, and what can be said about the stronger runtime bound that is not currently covered.
OpenAI has also agreed to fund a conference on the result, according to the source article. Discussion there and any subsequent revisions could give researchers a clearer account of the proof’s scope and its consequences for results that rely on ETH. Until that work is public, the accurate status is an announced, machine-checked proof claim awaiting independent review, not a settled change to computer science.
Key Questions
What does OpenAI say the AI proved?
OpenAI says an internal model produced a proof of an algorithm that exactly counts solutions to fixed-k SAT formulas in subexponential time and polynomial memory. For 3-SAT, the preprint says this refutes ETH. Independent review is pending.
Does this solve P versus NP?
No. P versus NP remains open. ETH is a stronger hypothesis about the time required for particular problems, so refuting ETH does not settle whether every efficiently checkable problem is efficiently solvable.
Has the proof been independently verified?
Not yet, according to the source article. The proof has a public Lean formalization, and an outside group led by Russell Impagliazzo, Ramamohan Paturi and Ryan Williams is expected to examine it.
What does the Lean check establish?
Lean checks whether the proof encoded in its formalization follows from the stated definitions and rules. The source article says the formalized result has a slightly weaker runtime bound than the preprint’s sharpest claim.
Will the result make computers faster right away?
There is no reported immediate practical speed-up. The claimed algorithm is still slower than polynomial time, and OpenAI said practical improvements would require significant further research.
Source: ThorstenMeyerAI.com
