OpenAI says its AI refuted ETH. Computer scientists are still checking the proof.

OpenAI announced on October 26, 2026 that one of its internal AI systems produced a claimed refutation of the Exponential Time Hypothesis, publishing a manuscript and a Lean formalization describing a deterministic procedure that exactly counts satisfying assignments of fixed-width conjunctive normal formulas in subexponential time. OpenAI asked complexity theorists Russell Impagliazzo, Ramamohan Paturi and Ryan Williams to organize an open examination, and Williams told TechCrunch that parts of the argument looked routine while "two or three" sections appeared unfamiliar enough to require wider community scrutiny. The formalized Lean theorem carries a weaker running-time bound than the sharper bound stated in the accompanying manuscript, and the claim remains unaccepted by the field.

OpenAI says its AI refuted ETH. Computer scientists are still checking the proof. - OpenAI says its model found a subexponential algorithm for counting satisfying assignments of fixed-width Boolean formulas, which would contradict ETH. <sup \ 1\ https://openai.com/index/building-scientific-understanding-with-academic-communities/ </sup - ETH is related to P versus NP, but the two conjectures ask different questions. A subexponential algorithm can still take much longer than polynomial time. <sup \ 2\ https://www.claymath.org/millennium/p-vs-np/ </sup <sup \ 3\ https://people.csail.mit.edu/virgi/stoc18diameter.pdf </sup - OpenAI says the result was formalized in Lean, but the formalized theorem has a weaker running-time bound than the sharper bound stated in the accompanying manuscript. <sup \ 1\ https://openai.com/index/building-scientific-understanding-with-academic-communities/ </sup <sup \ 4\ https://github.com/openai/math/blob/main/lean/docs/378.md </sup - If the claim survives, research that relies on ETH would need to be reassessed. Immediate effects on cryptography and practical computing would depend on further work. <sup \ 1\ https://openai.com/index/building-scientific-understanding-with-academic-communities/ </sup <sup \ 5\ https://techcrunch.com/2026/10/27/frontier-ai-labs-say-their-models-refuted-the-exponential-time-hypothesis-what-does-that-mean/ </sup OpenAI says one of its internal AI systems has produced a result that, if independently validated, would overturn the Exponential Time Hypothesis, a 27-year-old assumption about the difficulty of solving certain Boolean logic problems. The company released a manuscript and a Lean formalization on October 26, 2026, and asked complexity theorists Russell Impagliazzo, Ramamohan Paturi and Ryan Williams to organize an open examination. \ 1\ https://openai.com/index/building-scientific-understanding-with-academic-communities/ That wording matters. OpenAI has announced a claimed refutation, not a result that the field has accepted. Williams told TechCrunch that parts of the argument looked routine while “two or three” sections appeared unfamiliar enough to require wider community scrutiny. \ 5\ https://techcrunch.com/2026/10/27/frontier-ai-labs-say-their-models-refuted-the-exponential-time-hypothesis-what-does-that-mean/ What ETH says The Exponential Time Hypothesis, or ETH, concerns satisfiability problems. In a Boolean satisfiability problem, a formula combines variables and constraints, and the task is to determine whether any assignment makes the formula true. In 3-SAT, each constraint contains at most three literals. A proposed assignment can be checked quickly, while finding one appears to require searching through a rapidly growing set of possibilities. \ 1\ https://openai.com/index/building-scientific-understanding-with-academic-communities/ \ 2\ https://www.claymath.org/millennium/p-vs-np/ ETH says that, for some fixed number of literals per clause, solving these problems requires time with an exponent that grows linearly with the number of variables, up to polynomial factors. The precise formulations are technical, but the central idea is that the exponent cannot be reduced to a term that grows strictly slower than the number of variables. \ 3\ https://people.csail.mit.edu/virgi/stoc18diameter.pdf OpenAI’s published claim goes in the opposite direction. Its public Lean scope describes a deterministic procedure that exactly counts satisfying assignments of fixed-width conjunctive normal formulas in subexponential time. The formalized running time is a polynomial factor multiplied by 2 raised to a term proportional to n times the square of log log n divided by the square root of log n. That exponent grows more slowly than every fixed positive multiple of n, which is the feature that would contradict ETH. \ 4\ https://github.com/openai/math/blob/main/lean/docs/378.md Why this is not a solution to P versus NP P versus NP asks whether every problem whose answer can be checked efficiently can also be solved efficiently. The Clay Mathematics Institute describes the question through examples such as finding a valid route or selection when checking a proposed answer is easy. It remains one of the major open problems in theoretical computer science. \ 2\ https://www.claymath.org/millennium/p-vs-np/ ETH is a narrower and stronger claim about the running time of particular NP-complete problems. A proof that 3-SAT can be solved in subexponential time would weaken a major hardness assumption, but subexponential time is still generally slower than polynomial time. For example, 2 raised to the square root of n grows faster than every polynomial in n. An ETH refutation therefore would not by itself show that P equals NP. \ 2\ https://www.claymath.org/millennium/p-vs-np/ \ 3\ https://people.csail.mit.edu/virgi/stoc18diameter.pdf \ 4\ https://github.com/openai/math/blob/main/lean/docs/378.md The result could still affect the broader P-versus-NP research program. OpenAI says ETH has been used in hundreds of papers to derive conditional hardness results in fields including graph theory, computational biology, databases and computational geometry. If the algorithm is correct, researchers would need to determine which arguments relied specifically on ETH and whether they can be rebuilt under weaker assumptions. \ 1\ https://openai.com/index/building-scientific-understanding-with-academic-communities/ What Lean verification does — and does not — establish Lean is a proof assistant that checks whether a formal statement follows from specified definitions, axioms and previously established lemmas. OpenAI says its system generated a natural-language disproof on October 16, 2026, and that the result was formalized in Lean on October 17. \ 1\ https://openai.com/index/building-scientific-understanding-with-academic-communities/ That is a meaningful check on the formal derivation, but it is not the same as independent scientific acceptance. Reviewers must still determine whether the formal theorem captures the intended computational model, whether the input encoding is appropriate, whether the running-time analysis has the claimed interpretation and whether the natural-language explanation accurately describes the formal result. \ 1\ https://openai.com/index/building-scientific-understanding-with-academic-communities/ \ 4\ https://github.com/openai/math/blob/main/lean/docs/378.md The public scope note is especially important because it distinguishes the formalized result from the sharper bound in the October 25 manuscript. The Lean statement gives a bound involving the square of log log n, while the manuscript claims a bound with a single log log n factor. Both are subexponential, but they are not the same theorem. \ 4\ https://github.com/openai/math/blob/main/lean/docs/378.md The outside group has not certified the claim. OpenAI says the researchers are unpaid, independent and free to publish their conclusions. The group has asked for raw data to be made public and for a conference to examine the implications. Its involvement is therefore part of the verification process, not evidence that the proof has already passed it. \ 1\ https://openai.com/index/building-scientific-understanding-with-academic-communities/ \ 5\ https://techcrunch.com/2026/10/27/frontier-ai-labs-say-their-models-refuted-the-exponential-time-hypothesis-what-does-that-mean/ What would change if the claim survives The first consequence would be theoretical. Papers that use ETH to argue that a problem cannot be solved below a particular time threshold would need to be revisited. Some results might remain valid under other assumptions; others could require new bounds or new proofs. The scale of that revision cannot be known until researchers understand exactly which theorem has been established and how broadly its algorithm can be applied. \ 1\ https://openai.com/index/building-scientific-understanding-with-academic-communities/ The practical consequences would arrive more slowly. A subexponential algorithm can be dramatically faster in asymptotic terms while remaining unusable for real inputs because of large constants, complex preprocessing or heavy memory requirements. OpenAI itself says that finding better algorithms and converting the theoretical advance into practical improvements will require substantial further research. \ 1\ https://openai.com/index/building-scientific-understanding-with-academic-communities/ Cybersecurity is an obvious area for review because complexity assumptions appear in analyses of some cryptographic constructions. Williams told TechCrunch that OpenAI had checked whether the result immediately broke cryptographic systems and said the lab was reasonably confident that it did not. That is an initial assessment, not a universal security guarantee. \ 5\ https://techcrunch.com/2026/10/27/frontier-ai-labs-say-their-models-refuted-the-exponential-time-hypothesis-what-does-that-mean/ As of October 28, 2026, the careful description is conditional: OpenAI has released an AI-generated claim, a manuscript and a Lean formalization; independent complexity theorists are examining whether they establish a genuine refutation of ETH. TechCrunch also reported that Anthropic released a separate ETH refutation after OpenAI’s announcement, but two company claims do not amount to two independently accepted theorems. \ 5\ https://techcrunch.com/2026/10/27/frontier-ai-labs-say-their-models-refuted-the-exponential-time-hypothesis-what-does-that-mean/ \ 6\ https://stanfordtechreview.com/articles/openai-anthropic-exponential-time-hypothesis-false Companies mentioned Further sources The stories that matter, in one email. Free — unsubscribe anytime.