Mathematicians disprove sum-product conjecture using OpenAI's class field tower method

A group of mathematicians published a counterexample to a longstanding sum-product conjecture for real numbers, adapting the algebraic technique behind OpenAI's recent Erdős unit-distance breakthrough. The preprint, posted to arXiv on May 27, 2026, explicitly credits the inspiration: the authors "were inspired to reconsider the possibility of disproving the conjecture thanks to the counterexample invented at OpenAI for the unit-distance problem." They adapted the same class field tower construction—the algebraic number theory tool OpenAI's internal model used to link geometry and algebra—to build a counterexample in a different domain. The team also consulted GPT-5.5 Pro during their work but completed the final proof independently.

The sum-product conjecture, a central problem in additive combinatorics, concerns the growth rate of sums and products of finite sets of real numbers. OpenAI's April 2026 announcement that its model had found an infinite family of configurations disproving the Erdős unit-distance problem—a problem mathematicians had believed solved for 80 years—hinged on connecting geometry to algebraic number theory via class field towers, a connection human mathematicians had overlooked. The new preprint demonstrates the technique's portability: the same algebraic machinery, applied to a different conjecture, yielded a second high-profile disproof within weeks of the original discovery.