Anthropic Says Claude Also Refuted ETH, Using 14M Tokens

Anthropic has published a paper, dated 26 October 2026, that says an internal Claude research model refuted the Exponential Time Hypothesis (ETH), the same conjecture OpenAI said on 26 October its own system had proved false. The paper says the core algorithm was found on 16 October “with 14M output tokens and no human input.”

The paper, “A 2o(n) Time Algorithm for 3-SAT”, is credited to Claude and Anthropic. It was added with a Lean 4 formalization to Anthropic’s public formal-math repository on GitHub on 27 October. It names OpenAI’s work directly: “Independently and concurrently, an internal model at OpenAI claims to have proven a similar result”.

Video explainer

A Business Engineer explainer on the two labs’ ETH claims and the thread behind them, narrated from our published pieces.

What Anthropic’s Paper Says

The paper says it presents a deterministic algorithm that decides 3-SAT on n variables in time that grows more slowly than any fixed exponential, written 2 to the o(n). It says the algorithm also counts the satisfying assignments, the harder #3-SAT problem.

3-SAT asks whether a formula of true/false variables can be satisfied. The ETH, proposed by Impagliazzo and Paturi, holds that 3-SAT needs exponential time. The paper puts the previous best deterministic algorithm at O(1.328ⁿ) time and the best randomized one at O(1.308ⁿ).

The paper says its result also refutes the stronger SETH, but not the Orthogonal Vectors hypothesis, on which many lower bounds in fine-grained complexity rest. In a literature search carried out by Claude, it says, the authors found more than 800 papers that prove at least one result of their own only under ETH or a hypothesis that implies it.

What Was Posted on X

Levent Alpöge, whose X profile lists Anthropic, posted on 27 October 2026 at 04:00 UTC the paper’s provenance passage: the core algorithm “was found on October 16 by an internal Claude research model with 14M output tokens and no human input”, with a link to the eth folder of Anthropic’s formal-math repository. This piece quotes the same passage from the paper itself.

Chubby (@kimmonismus) posted 29 minutes later: “An internal Anthropic model also refutes the ETH. With only 14m Output tokens and already on October 16th.” The line “Suddenly everything at once” is the poster’s own view. The 16 October date is the same one OpenAI gives for its disproof.

Andrew Curran posted at 04:20 UTC: “This is extremely confusing, but an internal Claude research model also independently refuted the Exponential Time Hypothesis on October 16, with 14M output tokens and no human input.” Curran called it “extremely confusing”; the post does not say why.

@choblin29 posted at 05:54 UTC a summary of the release, including that the formalization spans 1,050 Lean modules checked by Lean and the independent nanoda kernel, and that “no person is yet known to have checked the full formalization.” The paper’s Appendix E says all 1,050 Lean modules were built together, that the proofs are accepted by Lean’s kernel and by nanoda, and that “at the time of writing no person is known to have checked it”.

How Claude Found It, per Anthropic

“The core algorithm was found on October 16 by an internal Claude research model with 14M output tokens and no human input,” the paper says. “Anthropic researchers worked with Claude to improve the presentation, resulting in this paper.”

The repository README says the Lean 4 formalization “was written by AI models”. It says Lean checks the proofs against Lean’s three standard axioms, and it adds a caution: “no machine checks that this file says what the paper says”. The statement file is kept short so a reader can check it.

The README also says Claude Opus 5.5 at high effort can rederive the refutation from a condensed description of the key idea, though only the weaker bound, not the sharper one at the top of the page. Levent Alpöge, whose X profile lists Anthropic, posted the provenance passage on X on 27 October.

What It Does Not Change in Practice

The paper is blunt about practical use: “Thus our result says nothing about solving formulas in practice.” For 3-SAT formulas with five clauses per variable, it says its bound does not beat the 2ⁿ of exhaustive search unless n is greater than 10 to the power 10,000.

It adds that brute force is already impractical for n = 100, and that its result “does not bear on the P vs NP question directly, nor does it yield any practical algorithms for NP-complete problems.”

Connecting the Coverage

On 26 October we reported that OpenAI says its AI refuted the Exponential Time Hypothesis, with a disproof it dates to 16 October and a Lean formalization on 17 October. Anthropic’s paper puts its own discovery on the same date. In September we covered how Claude formalized Fermat’s Last Theorem in Lean, and on 6 October how OpenAI’s internal model generated 722 math papers.

The S-1 Anthropic filed this month also said Claude wrote over 90% of Anthropic’s new code. For how the company earns its revenue, see our Anthropic business model breakdown.

The Structural Read

As we read it, the 14M output tokens figure gives the claim a scale: the paper attributes the core algorithm to a model run measured in tokens, with Anthropic researchers entering at the write-up stage.

As we read it, Anthropic’s README puts its own limit up front: Lean checks the proofs, but whether the formal statement matches the paper is left to human readers, which is why the statement file is kept short.

As we read it, the practical caveat matters for business readers: the paper says its speed-up over exhaustive search appears only for formulas far larger than any used in practice.

Three Implications

FOR RESEARCHERS Anthropic’s paper and Lean files are public under the Apache 2.0 license, and the README gives the commands to rebuild and check the proofs, a build it says took about 8 minutes on 24 virtual CPUs.

FOR COMPLEXITY THEORY The paper counts more than 800 papers that prove at least one result only under ETH or a hypothesis that implies it, and says its theorem implies faster algorithms for problems such as Independent Set, Clique and Vertex Cover.

FOR THE AI LAB RACE OpenAI and Anthropic each say an internal model reached the result on 16 October; Anthropic’s paper calls the two results independent and concurrent.

The Business Engineer Lens

This story maps onto the Business Engineer framework The Product Overhang Doctrine.

The framework opens: “In any product cycle, there are two curves to follow. The first is what your underlying technology can do. The second is what your users can actually do with it.”

As we read it, this result sits at the far end of that gap: Anthropic’s paper says its algorithm improves on exhaustive search only for formulas far larger than any used in practice. The framework was published in June 2026 and does not discuss this result.

What Is Not Established

No peer-reviewed venue is named in the paper or README; the README says a Hexagon link is “to appear”. The Lean proofs check the formal statements, and the README itself says no machine checks that those statements match the paper. The paper refers to “an internal research Claude model” and does not name a model version for the discovery. OpenAI says outside groups will examine its claim; in the paper and README we read, Anthropic does not describe an outside review of its own.

The Bottom Line

Anthropic’s paper says an internal Claude model refuted the Exponential Time Hypothesis on 16 October, using 14 million output tokens and no human input, the same date OpenAI gives for its own refutation. Both labs pair the claim with a Lean formalization, and Anthropic’s paper says the result has no practical use for solving formulas.