Three new discoveries in mathematics, computer science, and physics

The fundamental assumption about the minimum time required to solve certain computational problems has been broken.

Researchers at Anthropic, using an internal Claude model, have discovered a deterministic algorithm that solves the 3-SAT problem in 2^o(n) time. This finding refutes the Exponential Time Hypothesis (ETH), which posited that 3-SAT would require at least 2^Ω(n) time. The algorithm also addresses the #3-SAT problem, counting satisfying assignments, with a time bound of 2^O(n log log n / √log n). The paper and its formalization in Lean 4 are publicly available on GitHub as a research artifact. While the result is asymptotic and does not immediately apply to practical problem sizes, it has been rigorously verified by Lean and a secondary kernel, nanoda. The formalization process itself was significantly aided by AI models, demonstrating a new avenue for mathematical proof and verification.

The refutation of the Exponential Time Hypothesis (ETH) by Anthropic's AI marks a significant theoretical advancement, suggesting that some problems previously thought to be computationally intractable in polynomial or sub-exponential time might have more efficient algorithmic solutions than anticipated. This breakthrough could eventually influence the design of algorithms for a wide range of NP-complete problems, potentially impacting fields reliant on optimization and constraint satisfaction.

The contribution extends beyond theory by providing a Lean 4 formalization of the proof, generated and assisted by AI models. This demonstrates a growing synergy between AI and formal verification, potentially accelerating the pace of discovery and validation in theoretical computer science and mathematics. The accessibility of the formal proof and paper invites further scrutiny and development within the academic community, heralding a new era where AI actively participates in generating and verifying complex mathematical results.