AI BriefingAnthropic's Claude Refutes Exponential Time Hypothesis with Lean 4 Formalization
·2026.10.28 09:00
Key point
The algorithm, discovered by an internal research Claude model, solves #3-SAT in 2^(O(n log log n / sqrt(log n))) time and is formalized in Lean 4.
Details
- An internal research Claude model discovered a deterministic algorithm that solves 3-SAT on n variables in 2^o(n) time, refuting the Exponential Time Hypothesis (ETH).
- The algorithm actually solves the harder #3-SAT problem (counting satisfying assignments) in 2^(O(n log log n / sqrt(log n))) time.
- The result is purely asymptotic and does not imply practical solvability for real-world formulas.
- The proof is formalized in Lean 4 (v4.33.1) using Mathlib, with a build time of approximately 8 minutes on 24 virtual CPUs and a peak memory usage of 7 GB.
- A condensed description of the key idea allows Claude Opus 5.5 at high effort to rederive the refutation, though it only proves 3-SAT is decidable in O(2^(delta n)) for any delta > 0, not the sharper bounds.
- The repository is a research artifact, licensed under Apache 2.0, and is not maintained or accepting contributions.
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