GPT-5.5 Pro自主证明实数域上和积猜想为假，7次成功
Autonomous disproofs of the sum-product conjecture over $\mathbb R$ with GPT-5.5 Pro
- 三阶段提示框架：构想、构造、审查，无须人类干预
- 8次试验中7次生成正确反例，平均使用13.2万推理令牌
- 结果多样且可复现，适合对自动化数学证明感兴趣者
OpenAI此前对埃拉多斯单位距离猜想的反证标志着人工智能在数学领域的里程碑。这启发了另一项突破：人类对实数域上埃拉多斯-塞迈雷迪和积猜想的反证。本文展示一个基于GPT-5.5 Pro构建的简单智能体。采用问题无关的三阶段提示流程——证明方案提出、证明构造、审查——该智能体在8次独立试验中，有7次自主生成了正确的反证明，表明和积猜想在ℝ上不成立；另一次发现论证中的未解漏洞。七份证明各具特色：部分接近已有基于单位的构造，另一些则通过代数整数的L^p型区域避开单位。系统平均每轮使用132.4千个推理令牌。我们公开代码、中间输出及生成的证明，提供一个可复现、无数据污染的自主证明生成案例。
原文摘要 · Abstract (English)
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.
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