Autonomous disproofs of the sum-product conjecture over $\mathbb R$ with GPT-5.5 Pro

תקציר מקורי באנגליתarXiv:2607.20525v1 Announce Type: new Abstract: OpenAI's recent disproof of the Erd\H{o}s unit distance conjecture marked a milestone for AI in mathematics. It also inspired another breakthrough: a human disproof of the Erd\H{o}s--Szemer\'edi sum-product conjecture over $\mathbb R$. In this paper, we present a simple agent built on GPT-5.5 Pro. Using a problem-agnostic, three-stage prompting pipeline -- proof-plan proposal, proof construction, and review -- the agent autonomously generated correct proofs that the sum-product conjecture is false over $\mathbb R$ in 7 of 8 independent trials; in the remaining trial, it identified an unresolved gap in its argument. The seven proofs are diverse: some are close to existing unit-based constructions, while others avoid units by using $L^p$-type r