OpenAI and Anthropic Claim AI Refuted ETH, Shaking P Versus NP

Summary

On October 27, 2026, OpenAI released solutions to three longstanding math problems, including an unexpected refutation of the Exponential Time Hypothesis, or ETH, proposed in 1999 by UC San Diego computer scientists Russell Impagliazzo and Ramamohan Paturi. Anthropic mathematician Levent Alpöge argued that OpenAI relied on existing work; hours later, Anthropic published its own refutation. Both labs claim their models found subexponential algorithms for logic problems previously believed to require exponentially more time as variables increase.

ETH supports the prevailing P ≠ NP intuition that quickly verifiable problems, such as Sudoku or decrypting with a key, may not be quickly solvable. If validated, the algorithms weaken that intuition and could advance a path toward P = NP. MIT complexity theorist Ryan Williams says P = NP now deserves more weight, although OpenAI’s algorithm is unlikely to be optimal or final. Williams asked OpenAI science lead Dan Roberts about cryptography; the lab found no implied breaks with reasonable confidence, though improved algorithms could make systems using certain key sizes insecure.

OpenAI enlisted Williams, Impagliazzo and Paturi to interpret its proof. They opened community feedback and discussion of AI research norms, while computer scientists formed a working group, requested public proofs and sought an OpenAI-funded conference. Mathematicians received GitHub files, and physicists formed a group to determine their result’s release. Williams, who called an earlier October proof dump a DDOS attack on mathematicians, found two or three unfamiliar sections among routine work. A decade of ETH-dependent papers may require review; Williams sees AI encouraging greater scientific ambition but worries about training future researchers.