# OpenAI says its internal models refuted the Exponential Time Hypothesis and proved the Kakeya conjecture in every dimension

The complexity theorists who first proposed the hypothesis are leading an unpaid review of a Lean-checked algorithm that counts k-CNF solutions in time below 2 to the power of any constant times n.

- Published: 2026-10-27T06:18:03.036Z
- Canonical: https://polylog.news/ai/2026-10-27/openai-says-its-internal-models-refuted-the-exponential-time
- Publisher: Polylog (AI desk)
- Section: tech
- Sources: [OpenAI News](https://openai.com/index/building-scientific-understanding-with-academic-communities), [Hacker News](https://github.com/openai/math/blob/main/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf), [Polylog editors](https://polylog.news)

On October 26, OpenAI [published three results](https://openai.com/index/building-scientific-understanding-with-academic-communities) that it says unreleased internal models produced. The results concern problems in theoretical computer science, harmonic analysis and mathematical physics that specialists have worked on for decades. The company also explained how outside researchers can check the work. All three are claims, and none has been independently confirmed yet.

The computer science result is the sharpest. A [preprint dated October 25](https://github.com/openai/math/blob/main/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf) describes a deterministic algorithm that exactly counts the satisfying assignments of k-CNF formulas with fixed clause width, running in subexponential time using polynomial space, [according to coverage of the paper](https://fourweekmba.com/ai-openai-says-its-ai-refuted-the-exponential-time-hypothesis/). The Exponential Time Hypothesis (ETH) holds that 3-SAT requires time of 2 to the power of some constant times n. Any subexponential bound for this problem would refute it. OpenAI says it found the natural-language argument on October 16 and [formalized it in Lean the next day](https://ai-tldr.dev/releases/openai-eth-false-kakeya-results/).

The mathematics claim is a proof, in every dimension, of the local smoothing conjecture that Christopher Sogge proposed in 1991. That result would also settle the Kakeya and Stein restriction conjectures in all dimensions, [The Next Web reported](https://thenextweb.com/news/openai-exponential-time-hypothesis-false-kakeya-yang-mills). Local smoothing had previously been proved only in two dimensions, a result from 2019, and Hong Wang and Joshua Zahl [proved the three-dimensional Kakeya case in 2025](https://aiweekly.co/alerts/openai-hands-eth-kakeya-yang-mills-proofs-to-reviewers). A [summary of the announcement](https://t.me/ai_machinelearning_big_data/11215) from a Russian-language research channel describes local smoothing as a bound on how strongly waves can concentrate. The physics claim is a rigorous existence proof for four-dimensional quantum field theories, including quantum chromodynamics, [according to Unite.AI](https://www.unite.ai/openai-says-its-models-produced-three-proofs-in-math-cs-and-physics/).

Russell Impagliazzo and Ramamohan Paturi of UC San Diego and Ryan Williams of MIT are organizing the review of the ETH claim. Impagliazzo and Paturi co-authored the original hypothesis. Williams called the work ["a claimed refutation of ETH"](https://ai-tldr.dev/releases/openai-eth-false-kakeya-results/) and said the review group has signed no nondisclosure agreements and will receive no compensation. A Lean check confirms that a proof establishes its encoded statement, but it does not confirm that the formal definitions match the intended conjecture, and [reviewers have flagged that gap](https://ai-tldr.dev/releases/openai-eth-false-kakeya-results/).

OpenAI's recent record calls for caution. On October 6 it released 722 manuscripts in 372 result families. It withdrew three of them the next day, after a sign error invalidated one result and, through dependencies, two others that relied on it. The repository states that 42% of top-line results are Lean-formalized, but [independent recounts put the share at 24% to 36%](https://tech-insider.org/openai-math-papers-lean-verification-42-percent-2026/). [Scientific American reported](https://www.scientificamerican.com/article/openai-unleashes-hundreds-more-math-results-upon-a-field-already-in-shock/) widespread skepticism among mathematicians. Terence Tao has raised concerns about results claimed by people who may not understand the proofs well enough to defend them, and Tristan Buckmaster has asked whether some results complete unpublished human work.

Even if it holds, the ETH result has no near-term practical effect. The bound is superpolynomial, so the P versus NP question is untouched, and the crossover point below 2 to the n lies far beyond any formula size anyone will ever actually solve. SAT solvers and deployed cryptography are unaffected. The effect would fall on fine-grained complexity theory, where many conditional lower bounds assume ETH. OpenAI made the announcement the same day Anthropic filed to go public, and a lab competing for the same investor capital gains from putting out a capability claim on that date.

## What this means

The mechanism is that a machine-checkable proof moves the bottleneck from producing mathematics to judging it. A Lean formalization narrows any dispute to one question, whether the formal statement encodes the real conjecture, and a small number of senior specialists can answer that quickly. If the claims hold, OpenAI gains in recruiting, in credibility for its research work with enterprise customers and in standing with investors, at the same moment Anthropic's public offering is setting valuation benchmarks for frontier labs. Theorists whose lower bounds assume ETH would have to restate their results. Commentary has split between calling this proof that AI has solved mathematics and dismissing it because of OpenAI's October withdrawals, and both positions underweight how narrow and specific the open question actually is. The precedent is Vinay Deolalikar's 2010 claimed proof that P does not equal NP, which outside experts refuted within weeks through open review. The difference now is that a formal artifact turns a diffuse mathematical argument into a single check of whether one statement matches another. The desk's read is that the ETH question gets resolved within weeks, while the Kakeya and quantum field theory claims take months to years to settle. If the review group confirms that the Lean statement encodes the standard Exponential Time Hypothesis, expect the field to accept a theorem with large theoretical consequences and no practical ones. If reviewers find a definitional mismatch, such as a restricted formula class or a non-standard parameterization, the claim would shrink to a narrower result, and a second high-profile retraction within one month would damage OpenAI's scientific credibility.

## What to watch

- A written statement from Impagliazzo, Paturi and Williams on whether the Lean theorem statement matches the standard formulation of the Exponential Time Hypothesis. That is the single check that decides the claim.
- Whether mathematical physicists confirm that the quantum chromodynamics existence claim also covers a mass gap. If it does not, the result falls short of the Clay Millennium problem and should be described that way.
- Whether OpenAI names or releases the internal models behind the proofs. A product release would turn a research claim into a commercial capability customers can test themselves.