OpenAI and Anthropic have separately claimed that their artificial intelligence (AI) systems have produced mathematical results challenging a 27-year-old hypothesis that has shaped theoretical computer science for decades. The findings concern the Exponential Time Hypothesis (ETH), a long-standing assumption about the computational difficulty of solving certain mathematical problems. If independently verified, the results could reshape researchers' understanding of computational complexity and prompt a review of hundreds of existing studies.

The Exponential Time Hypothesis was developed in the late 1990s and is widely used in theoretical computer science. It suggests that certain problems, particularly those involving Boolean satisfiability, cannot be solved without computational effort increasing exponentially as the problem grows.

Boolean satisfiability, commonly known as SAT, is a mathematical problem in which a computer must determine whether a set of logical conditions can be made true simultaneously. One important version, called 3-SAT, involves logical formulas containing clauses with three variables. Researchers have long considered these problems fundamentally difficult to solve efficiently.

OpenAI said its internal AI system generated a mathematical argument challenging ETH on October 16. The company subsequently formalised the result using Lean, a proof-verification system designed to check whether mathematical arguments follow from their stated assumptions.

Anthropic also announced an AI-generated research paper presenting an algorithm that it says can solve a related satisfiability problem in subexponential time. The company published supporting formal verification materials alongside its research, presenting another challenge to the long-standing hypothesis.

Subexponential time refers to computational growth that is slower than exponential growth as the size of a problem increases. If the proposed algorithms and their mathematical arguments are correct, they would challenge the belief that certain satisfiability problems necessarily require exponential computational time.

The use of Lean is an important part of both announcements. It is a formal mathematical system that can verify whether a proof follows logically from its definitions and assumptions. However, formal verification does not automatically establish that every statement in a research paper accurately matches the result checked by the software. Independent examination remains essential.

The claims have attracted considerable attention because ETH has been used as a foundation for numerous theoretical results. Researchers have relied on the hypothesis to establish the computational difficulty of problems in areas including graph theory, databases, algorithms and computational biology.

A successful refutation could therefore require scientists to revisit hundreds of findings that depend on ETH. It could also open new directions for research into algorithms and the limits of computation.

However, the announcements do not mean that computers can suddenly solve every difficult problem faster or that the famous P versus NP question has been resolved. The proposed algorithms are theoretical results, and their practical usefulness for solving real-world problems remains uncertain.

The mathematical claims must also withstand detailed scrutiny from independent researchers. Until experts have thoroughly examined the proofs and confirmed their validity, the findings should be treated as significant but unverified research claims.

If the results hold up, the development could mark an important moment in computer science, demonstrating how AI systems may contribute not only to solving established problems but also to challenging assumptions that have guided research for decades.