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

arXiv: https://arxiv.org/abs/2607.20525 · HTML: https://mlsanity.com/paper/2607.20525/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 10 references that are in the map.
- First posted on arXiv 9 Jul 2026.

## Built on by (most cited first, 1 of 1)

- [Learning to Discover Interesting Mathematics](https://mlsanity.com/paper/2609.28603/) (2026)

## Abstract (arXiv)

OpenAI's recent disproof of the ErdHos unit distance conjecture marked a milestone for AI in mathematics. It also inspired another breakthrough: a human disproof of the ErdHos--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 Lp-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.