OpenAI and Anthropic announce AI models discovered a sub-exponential algorithm for 3-SAT

Summary

OpenAI and Anthropic independently announced that their AI models discovered a sub-exponential algorithm for 3-SAT on the same day, October 16. This finding refutes the Exponential Time Hypothesis from 1999, which underpins over 800 theoretical computer science papers. Both proofs were machine-verified in Lean; however, human peer review, led by the hypothesis's original authors, has only just begun.

Why it matters

If the refutation holds, the lower-bound arguments supporting hundreds of complexity theory results (clique, vertex cover, planning, verification) lose their foundation. This would reshape a fundamental assumption of theoretical computer science. The event is also a striking milestone: an AI model producing a proof of this quality without human intervention during the discovery phase, with two labs reaching the same conclusion within hours. However, the practical impact is near zero; the algorithm only surpasses brute force beyond 10^10000 variables. Therefore, its significance is almost entirely foundational.

Highlights

• Both proofs were formalized in Lean; Anthropic's proof had zero human verification upon publication and was reviewed only by other instances of the same model.
• The algorithm achieves a time complexity of approximately 2^n / √(log n) and only overtakes brute force beyond 10^10000 variables. This does not affect SAT solvers or cryptography.
• OpenAI also claims a proof of the Kakeya conjecture and a proof of Yang-Mills existence, expanding the scope of mathematics discovered by AI.

Critical take

The article clearly points out a critical gap: A formal Lean proof guarantees logical consistency with axioms; however, it does not guarantee that the formalized statement matches the intended mathematical claim. The human review required to close this gap is explicitly non-definitive and is still in its early stages. Declaring ETH 'dead' (as Fortnow did) before the hypothesis's own authors have finished reading the proof is, at best, premature.