Anthropic and OpenAI Report ETH-Challenging Algorithms

Anthropic said on the 27th local time that one of its AI models had found an algorithm challenging the Exponential Time Hypothesis (ETH), a central computer science conjecture. OpenAI disclosed a similar result, and the companies released their research about six hours apart. Both said they found their algorithms on October 16.

Anthropic researcher Levent Alpoege wrote on X that an internal Claude model for research found the core algorithm on October 16 without human intervention. The AI generated 14 million tokens of output. Researchers then worked with Claude to prepare the results as a paper. Anthropic published the paper, “A Subexponential-Time Algorithm for 3-SAT,” and code written in Lean, a tool for checking mathematical proofs, on GitHub. The paper lists “Claude (Anthropic)” rather than a person as an author.

ETH is the conjecture that the time needed to solve logical problems grows exponentially with the number of variables. Proposed in 1999, it has been used in computer science to analyze the limits of algorithms. OpenAI and Anthropic proposed algorithms that solve 3-SAT, a prominent Boolean satisfiability problem, in subexponential time—a time bound ETH is generally taken to rule out. This would overturn ETH’s basic premise, but it concerns theoretical running time and does not mean computer programs will immediately become faster.

OpenAI said its model found a proof challenging ETH on October 16 and that the proof was verified with Lean the next day. The companies independently reached similar conclusions on the same day, but their detailed algorithmic approaches differ. Neither claim has received final academic verification. If confirmed, the results could prompt a review of existing research that relies on ETH. The hypothesis has been used in graph theory, computational biology and databases; OpenAI said hundreds of research results rely on it.

MIT professor Ryan Williams, who is involved in verifying ETH, said researchers may need to revisit the foundations of existing work. He added that the algorithm has not been shown to directly break a specific cryptographic system, and that challenging ETH does not solve the P versus NP problem.

OpenAI and Anthropic presented the results as evidence of AI’s research capabilities. Advanced AI models have demonstrated performance through scores on mathematics, coding and reasoning benchmarks; the companies said their models produced proofs and algorithms for problems that academics had not solved for decades. Williams told TechCrunch that AI encourages researchers to ask higher-level scientific questions.

Anthropic said on the 27th local time that one of its AI models had found an algorithm challenging the Exponential Time Hypothesis (ETH), a central computer science conjecture. OpenAI disclosed a similar result, and the companies released their research about six hours apart. Both said they found their algorithms on October 16.

Anthropic researcher Levent Alpoege wrote on X that an internal Claude model for research found the core algorithm on October 16 without human intervention. The AI generated 14 million tokens of output. Researchers then worked with Claude to prepare the results as a paper. Anthropic published the paper, “A Subexponential-Time Algorithm for 3-SAT,” and code written in Lean, a tool for checking mathematical proofs, on GitHub. The paper lists “Claude (Anthropic)” rather than a person as an author.

ETH is the conjecture that the time needed to solve logical problems grows exponentially with the number of variables. Proposed in 1999, it has been used in computer science to analyze the limits of algorithms. OpenAI and Anthropic proposed algorithms that solve 3-SAT, a prominent Boolean satisfiability problem, in subexponential time—a time bound ETH is generally taken to rule out. This would overturn ETH’s basic premise, but it concerns theoretical running time and does not mean computer programs will immediately become faster.

OpenAI said its model found a proof challenging ETH on October 16 and that the proof was verified with Lean the next day. The companies independently reached similar conclusions on the same day, but their detailed algorithmic approaches differ. Neither claim has received final academic verification. If confirmed, the results could prompt a review of existing research that relies on ETH. The hypothesis has been used in graph theory, computational biology and databases; OpenAI said hundreds of research results rely on it.

MIT professor Ryan Williams, who is involved in verifying ETH, said researchers may need to revisit the foundations of existing work. He added that the algorithm has not been shown to directly break a specific cryptographic system, and that challenging ETH does not solve the P versus NP problem.

OpenAI and Anthropic presented the results as evidence of AI’s research capabilities. Advanced AI models have demonstrated performance through scores on mathematics, coding and reasoning benchmarks; the companies said their models produced proofs and algorithms for problems that academics had not solved for decades. Williams told TechCrunch that AI encourages researchers to ask higher-level scientific questions.