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

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

Abstract:OpenAI's recent disproof of the Erdős unit distance conjecture marked a milestone for AI in mathematics. It also inspired another breakthrough: a human disproof of the Erdős--Szemerédi 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 regions of algebraic integers. The system used an average of 132.4k reasoning tokens per trial. We release the code, intermediate outputs, and generated proofs, providing a reproducible, data-contamination-free case study in autonomous proof generation.

Artificial Intelligence (cs.AI)

arXiv:2607.20525 [cs.AI]

(or arXiv:2607.20525v1 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2607.20525 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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