OpenAI on Monday released three AI-generated proofs spanning mathematics, physics and computer science, including one its internal model produced on 16 October that it says refutes the Exponential Time Hypothesis, a conjecture that underpins hundreds of computer-science results. Within hours, Anthropic posted an independent disproof of the same hypothesis, produced by one of its own models. The dueling results are the clearest sign yet that the two leading AI labs are now racing to settle open problems that mathematicians have worked on for decades.
The Exponential Time Hypothesis, first proposed in 1999, holds that the 3-SAT satisfiability problem cannot be solved faster than exponential time. OpenAI says its model found an algorithm that beats that bound by a small margin, and that it verified the result the following day in Lean, software that checks proofs by machine. Russell Impagliazzo and Ramamohan Paturi of UC San Diego, who first proposed the hypothesis, and Ryan Williams of MIT have agreed to organise an independent examination. In an open letter they stressed that their involvement "should not be taken as certification of the claim," and said they signed no non-disclosure agreements and will not be paid.
The two other results are no less sweeping. OpenAI says its models proved the local smoothing conjecture in every dimension, which would also settle the Kakeya conjecture in all dimensions; mathematicians had proved Kakeya only in three dimensions last year, a feat that won Hong Wang a Fields Medal. In physics, OpenAI claims a rigorous proof that pure Yang-Mills theory exists in four dimensions, part of one of the Clay Mathematics Institute's million-dollar Millennium Prize Problems. The quantitative "mass gap" half of that problem remains open.
The announcements continue a pattern that has unsettled the mathematics community since OpenAI published 722 machine-generated proofs on 6 October. The model behind the new work remains proprietary, and the prompts, compute time and reasoning traces have not been released. Scientific American reported that the results violate recommendations from the Advisory Group on Mathematics and Artificial Intelligence, the panel OpenAI itself convened after earlier criticism; the group had asked that designated experts shepherd such results into public view, and has called on the company to release its Yang-Mills paper without further delay.
The Yang-Mills claim also cuts across work already under way. Northwestern mathematician Ethan Sussman and Anthropic employee Levent Alpoge had built a public repository to finish a decades-old program sketched by Tadeusz Balaban toward the same existence result, and believed they were close. OpenAI's claim, with no public proof, appears to have reached the finish line first.
That raises the question the three computer scientists posed directly in their letter: how credit should work when AI finds a result first and humans are left to check and explain it. Mathematicians voiced the same worry in June's Leiden declaration. OpenAI says it is not seeking academic credit and has given five million dollars to the Simons Foundation for workshops on AI-produced results. For now, the proofs are claims under review. As Impagliazzo, Paturi and Williams put it, a result of this significance "requires careful, independent scrutiny."