OpenAI has published a 67-page paper claiming that the company's AI disproved the Exponential Time Hypothesis (ETH). The conjecture dates to 1999 and has no connection to Ethereum, Ethereum's price or co-founder Vitalik Buterin. The claim prompted discussion among mathematicians and cryptocurrency users on X. X user Takumi joked, "Damn algo traders just slaughtered $ETH". Andrew Jiang wrote, "Vitalik has been real quiet since this came out."
Computer scientists Russell Impagliazzo, Ramamohan Paturi and Francis Zane proposed ETH. The conjecture holds that solving the logic problem 3-SAT cannot be much faster than testing every combination. The 3-SAT problem requires finding true-or-false assignments that satisfy rules involving three options at a time. OpenAI says hundreds of papers rely on ETH to establish computational limits in graph theory, computational biology, databases and geometry. OpenAI argues that disproving ETH renews questions about possible speed improvements and could direct research toward better algorithms.
The paper carries an October 17 date and an October 25 revision date. The paper presents a deterministic algorithm, which uses no randomness, for counting all satisfying assignments in subexponential time. Subexponential time means that the required work increases more slowly than any exponential function of the number of variables. Counting satisfying assignments is harder than determining whether any satisfying assignment exists, so the counting algorithm also addresses the decision problem. OpenAI says the AI discovered the claimed disproof on October 16. OpenAI says the result was formalized on October 17 using Lean, software that mechanically checks each logical step. The paper and formalization are available on GitHub. OpenAI also published 722 mathematics manuscripts through GitHub earlier this month, but only 162 included a computer-checked principal result.
Impagliazzo, Paturi and MIT's Ryan Williams are leading an independent examination of the claim. The three researchers warned in an open letter that participation should not be interpreted as certification. The researchers said a finding of such importance needs rigorous independent review and an explanation that other researchers can reconstruct, challenge and develop. OpenAI says the participants are unpaid and signed no non-disclosure agreements. OpenAI has agreed to release all raw data and finance a conference about the result.
OpenAI cautions that identifying useful algorithms and translating theoretical progress into practical gains will require substantial additional research. The paper does not establish that all NP problems admit subexponential-time solutions. P consists of problems that computers can solve efficiently, while NP consists of problems whose proposed solutions can be checked efficiently. P versus NP, which asks whether efficient verification always implies efficient solution, remains unresolved.
Anthropic subsequently published a paper attributed to Claude, titled "A 2^o(n) Time Algorithm for 3-SAT" and dated October 26. The accompanying repository says an internal research version of Claude discovered the algorithm, which also addresses ETH. The repository cautions that the result concerns asymptotic behavior and makes no claim about practical formula solving. Claude formalized Fermat's Last Theorem using Lean in September. Mathematician Terence Tao warned in September that even rumors of research on a problem can prompt extensive AI-assisted efforts before the original project fully develops.
OpenAI recently contributed $5 million to the Simons Foundation. The contribution will support workshops, conferences and special programs that help researchers scrutinize and extend results produced by AI.