Anthropic: Claude has also refuted one of computer science's big assumptions, hours after OpenAI

Anthropic says an internal Claude model has refuted the Exponential Time Hypothesis (ETH), one of the most widely used assumptions in theoretical computer science. The announcement came only hours after OpenAI made a similar claim. Two competing AI companies now say their own models independently found the same mathematical breakthrough.

Anthropic posted the paper and a machine-checked proof in its public GitHub repository for formal mathematics on Tuesday 27 October at 03:07 UTC (04:07 Norwegian time). The same day, the paper appeared on the preprint service Hexagon, labelled "Contributed by Anthropic". The paper is dated 26 October and lists Claude as its author.

What Anthropic claims

ETH states that the classic logic problem 3-SAT cannot be solved substantially faster than in exponential time. Anthropic describes a deterministic algorithm that solves 3-SAT on n variables in 2^o(n) time, faster than the hypothesis allows. According to the company, the algorithm also solves the harder variant #3-SAT, which counts how many solutions a formula has.

The paper says the result also refutes the stronger variant SETH, but not the so-called Orthogonal Vectors hypothesis, on which many other results rest. A literature search carried out by Claude found, according to Anthropic, more than 800 research papers that prove at least one result only under ETH or an assumption that implies it.

Found without human help

According to the paper's provenance statement, an internal Claude research model found the core of the algorithm on 16 October, using 14 million output tokens and with no human input. Anthropic researchers then worked with Claude to improve the presentation. The proof is formalised in the Lean 4 proof language, and according to Anthropic the formalisation was also written by AI models. The company says the check took about eight minutes on 24 virtual CPUs.

Anthropic itself writes that OpenAI "independently and concurrently" claims to have proven a similar result. OpenAI announced its claim on Monday 26 October. Which result actually came first cannot be checked from what the companies have published.

Major caveats

Anthropic is clear about the limits. The result is purely asymptotic and, in the company's words, "says nothing about solving formulas in practice". The paper states that the algorithm does not beat ordinary exhaustive search until the number of variables is far larger than anything found in real use. Nor does it bear directly on the famous P versus NP problem.

Anthropic also points to an important gap in the machine check: Lean verifies the proofs, but no machine checks that the formal statement says the same thing as the paper. That has to be read and judged by people. The result has not been peer-reviewed, and we have found no independent confirmation of the algorithm.

What it means for leaders

The story has no immediate practical consequence for businesses. No systems get faster, and Anthropic does not mention cryptography in the paper. What matters is that two of the largest AI vendors are now competing to show that their models can produce new basic research on their own, with machine-checked proofs as documentation.

Leaders assessing AI vendors should separate a company's claim from independent verification. Both companies present the results as their own, and the outside review is not yet finished. If the results hold, they show that AI models can solve problems researchers have worked on for decades. If they do not, they are a reminder that even machine-checked proofs need human scrutiny.

Sources and media

• Anthropic: A 2^o(n) Time Algorithm for 3-SAT (paper and Lean proof on GitHub)
• Hexagon: A 2^o(n) Time Algorithm for 3-SAT, Contributed by Anthropic
• OpenAI: Building scientific understanding with academic communities

Anthropic's contribution was added to the GitHub repository on 27 October 2026 at 03:07 UTC, and the Hexagon version is dated the same day. All information in this story comes from these sources.

Illustration: AI-generated with Grok Bot / hogby.ai