OpenAI and Anthropic refute Exponential Time Hypothesis with AI proofs
OpenAI’s model identified a proof on October 16, which the company rewrote in the formal language Lean and published on GitHub. The claim demonstrates that counting solutions to SAT problems with fixed-condition sizes can be done in sub-exponential time, undermining ETH’s core assumption that exponential growth is unavoidable. Anthropic followed with its own refutation, also verified via Lean,…
Key points
- Both labs invited original ETH proposers Impagliazzo and Paturi to verify results independently, with no compensation or NDAs
- OpenAI donated $5 million to the Simons Foundation for peer review conferences and research on AI-generated proofs
The breakthroughs extend beyond ETH: OpenAI’s model also proved a 30-year-old conjecture in harmonic analysis and validated the existence of a specific quantum field theory linked to the Clay Mathematics Institute’s $1 million Millennium Prize Problems. However, OpenAI clarified that the P vs NP problem remains unresolved, and practical applications are still distant. Researchers like MIT’s Ryan Williams emphasize the need for community scrutiny, framing this as a collaborative effort rather than a solitary AI achievement. The dual verification process—machine checks via Lean and human review by ETH’s proposers—aims to address skepticism about AI-generated proofs, setting a precedent for future claims.
The story so far
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AI Punches a Hole in a Wall Thought 'Unsolvable'—OpenAI's Refutation of the Exponential Time Hypothesis (ETH) and the Era of Human-AI Verification
note.com · 4 November 2026
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