Anthropic published the preprint “A 2^o(n) Time Algorithm for 3-SAT,” with the author listed as “Claude, Anthropic”: the company’s internal research model found a deterministic algorithm on October 16, 2026, that solves 3-SAT and the counting problem #3-SAT in subexponential time 2^O(n log log n / sqrt(log n)). If the construction is correct, this refutes the Exponential Time Hypothesis (ETH, Impagliazzo–Paturi). The proof is formalized in Lean 4 and verified by the Lean kernel and the independent checker nanoda, but no human has yet verified it in its entirety, and two sorry statements remain in the file Challenge.lean.

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What happened

The preprint hexagon:2610.00577v1 appeared on the mathematical platform Hexagon on October 27, 2026; version v1 was submitted by Jelani Nelson on behalf of Anthropic. It describes a deterministic algorithm for 3-SAT and the more complex counting problem #3-SAT, found by the internal research model Claude: the core of the solution was obtained without human hints in a session of approximately 14 million output tokens. The formalization consists of 1050 modules on top of Mathlib in Lean v4.33.1 and is published in the official repository anthropics/formal-math in the eth folder, commit 16a5ce4 dated October 27, 2026, 03:07 UTC; local correctness is confirmed by the Lean kernel and the independent checker nanoda. According to Anthropic, Claude Opus 5.5 is capable of re-deriving the result from a short technical sketch — this is the claimed partial reproducibility. OpenAI announced the day before that its model had also refuted ETH, but no published artifact supporting this claim exists.

Context

The Exponential Time Hypothesis (ETH, Impagliazzo–Paturi) states that 3-SAT cannot be solved significantly faster than full brute force; it and its strengthening SETH serve as starting points for hundreds of proven lower bounds in computational complexity theory, which are derived by reduction from these hypotheses. Therefore, a claim of a subexponential algorithm strikes at the foundation of the field, and the solution to the counting problem #3-SAT further strengthens the result: counting solutions is harder than establishing their existence. Machine verification in such work is two-tiered: the Lean kernel certifies each step of the formalization, but only what is actually formalized is verified, and the sorry marker indicates an unfinished fragment of the proof. Accordingly, “machine-verified” here is a strong guarantee of local correctness of the construction, but it does not replace independent expertise from the community. The publication format is also important: the preprint, code, and Lean formalization were released simultaneously and openly, so distributed auditing can begin immediately, rather than after peer review.

Why this matters for the industry

If the construction holds up, both ETH and its strengthening SETH will be false, and along with them, hundreds of works with lower bounds derived by reduction from these hypotheses will require re-verification — a large portion of complexity theory will have to be reconsidered. For engineers and product teams, the main asset is not the algorithm itself, but the reproducible workflow: a long autonomous agent session ends with machine verification, and the entire cycle of “agent generates — formalization in Lean 4 — independent checker verification” is open in the repository, where any team can dissect it and transfer it to their own domains: code, contracts, computations. A realistic scenario for the coming months is the auditing of the formalization by qualified theorists and formalizers: the sorry statements will either be closed or become the point of failure for the entire construction, while attempts to extract a human-readable proof are also likely. The race among labs for “autonomous discoveries” has entered the public sphere, and both companies are clearly using it in the positioning of their flagships, which drives the development of platforms for research agents and formal checkers as a service. There is no direct product impact: this is a preprint and a repository, not a product — no API, no pricing, and no data on latency or throughput.

Why this matters for users

For the reader, this is a clear example of how large language models are becoming a tool for mathematical discovery, but reading it requires caveats. The algorithm does not speed up practical SAT solvers: the result is purely asymptotic, and the advantage over brute force only appears at a variable count of approximately 10^10000, whereas full brute force is impractical already at around one hundred variables; it also does not bring us closer to solving the P vs NP problem, so the phrase “SAT is now solved quickly” would be a distortion of the news. The only practically available action today is to clone anthropics/formal-math at the specified commit and dissect the formalization pipeline as a reference: transferring the “agent generates — Lean 4 verifies” scheme to one’s own tasks is available now. The key skill when reading such claims is to check for the presence of a machine-verifiable artifact, not just the announcement text: an open formalization allows everyone to verify local correctness independently. The preprint itself changes nothing in the everyday tools and services you use.

What is still unknown / limitations

The formal proof is not complete: two sorry statements remain in Challenge.lean, meaning unproven assumptions are accepted at these points. The formalization was written and verified by AI models themselves, and at the time of publication, no human has verified it in its entirety; the claimed reproducibility via Claude Opus 5.5 relies on models of the same family and is not yet an independent human verification. OpenAI’s claim of a similar refutation of ETH is not supported by a published artifact, so it cannot be considered an independent confirmation of Anthropic’s result. Finally, this is a version v1 preprint that has not undergone peer review, so the final judgment on the correctness of the entire construction remains with the mathematical community.

Sources

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