OpenAI 10 26 announced three results in mathematics, computer science, and physics produced by its internal models, one of which proves that the Exponential Time Hypothesis (ETH) in computational complexity theory is false; hours later, Anthropic also published a refutation of ETH derived from its internal Claude research model, accompanied by a Lean 4 formal proof written by AI. Both results are still awaiting peer review.
Confidence: High (100) · Event time 2026-10-25
Reasoning
- ETH asserts that 3-SAT must take exponential time in the worst case. The OpenAI paper presents a deterministic algorithm that exactly counts the number of satisfying assignments for any k-CNF formula in sub-exponential time; the Anthropic repository similarly provides a 2^o(n) time algorithm for 3-SAT. Both parties state that this is an asymptotic result, and Anthropic explicitly notes that it is unrelated to the speed of solving formulas in practice.
- The Decoder reports that hundreds of papers in graph theory, computational biology, databases, and computational geometry rely on ETH as a premise, proving that if ETH were true, these conclusions would lose their foundation (which does not mean the conclusions are incorrect). MIT Professor Ryan Williams, interviewed by TechCrunch, said that OpenAI claims to have verified this and found no cryptographic systems that would be compromised as a result.
- Two other results (according to The Decoder and The Verge): In mathematics, they claim to have proven the local smoothing conjecture in all dimensions, which implies the Kakeya conjecture and the Stein restriction conjecture, and this has been made directly public; in physics, they claim to have proven the mathematical existence of four-dimensional Yang-Mills and other quantum field theories (without the "mass gap," meaning the Millennium Prize problem is not fully resolved), reviewed by a team led by Nobel laureate David Gross, with materials not yet public.
- The Anthropic repository also states that the publicly released Claude Opus 5.5, under high reasoning settings, can re-derive this refutation after being given a compressed key insight (reaching only a weaker bound, not the strongest result in the paper).
Usage
- The PDF of OpenAI's ETH paper is in the preprints directory of the GitHub openai/math repository; Anthropic's paper and Lean proof are in the eth directory of GitHub anthropics/formal-math, with the README providing lake build verification steps (officially stated to take about 8 minutes and require about 7GB of memory for the maximum process).
- The original proposers of ETH, Russell Impagliazzo, Ramamohan Paturi, and Ryan Williams, have issued an open letter soliciting peer opinions, and review progress will be released gradually (according to The Decoder).
Risks
- The original announcement on openai.com is inaccessible (403); the release method for the three results, the review team, and OpenAI's donation of 500 ten thousand dollars to the Simons Foundation to host a workshop are all reported by The Decoder and The Verge.
- Lean only guarantees the proof of the proposition written in Lean; Anthropic also states that there is no machine check for consistency between the Lean statement and the paper. Williams and others emphasize that participating in the review does not constitute endorsement.
- Anthropic mathematician Levent Alpöge believes that OpenAI's solution builds on existing work (TechCrunch); the Mathematics and AI Advisory Group AGMAI criticizes OpenAI for showing the work to designated experts first and not making the physics part public (The Verge).
- OpenAI's announcement predates this reporting window (10 26); the basis for inclusion in this issue is the primary papers readable on GitHub and Anthropic's new results.
Open · OpenAI paper "The Exponential Time Hypothesis is False" (GitHub openai/math) · Anthropic paper and Lean proof (GitHub anthropics/formal-math/eth) · The Decoder · The Verge