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

Yichen Huang

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

Artificial Intelligence 2026-07-09 v1

Authors: Yichen Huang

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 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.

Keywords

GPT-5.5 Pro sum-product conjecture Erdős autonomous proof generation mathematics

Cite

@article{arxiv.2607.20525,
  title  = {Autonomous disproofs of the sum-product conjecture over $\mathbb R$ with GPT-5.5 Pro},
  author = {Yichen Huang},
  journal= {arXiv preprint arXiv:2607.20525},
  year   = {2026}
}