LLM Framework for Discovering Major Mathematical Conjectures: AI’s Quest for the Next Riemann Hypothesis

用于发现重大数学猜想的大语言模型框架：人工智能对下一个黎曼猜想的探索

Abstract: Major mathematical conjectures still depend heavily on expert intuition, so a unified method for the systematic generation and validation of conjectures with substantial mathematical potential remains unavailable. 摘要： 重大数学猜想的发现目前仍高度依赖专家直觉，因此，目前尚缺乏一种能够系统性生成并验证具有重大数学潜力猜想的统一方法。

We present a three stage pipeline for major conjecture discovery, with region search from explicit local evidence modules, reflective validation for foundationality, novelty, and potential significance, and formal validation in Lean 4 and Mathlib. 我们提出了一种用于发现重大猜想的三阶段流水线：首先通过显式局部证据模块进行区域搜索，接着进行关于基础性、创新性和潜在意义的反射式验证，最后在 Lean 4 和 Mathlib 中进行形式化验证。

The objective is the discovery of mathematical problems with high problem taste, namely problems whose proofs could reorganize the language of a research area and provide durable help to human mathematical research. 该研究的目标是发现具有高“问题品味”（problem taste）的数学问题，即那些其证明过程能够重构特定研究领域语言，并为人类数学研究提供持久帮助的问题。

Experiments on twenty candidates show stable passage from natural language to formal checks, with twenty out of twenty candidates passing Lean parsing and type checking, twenty out of twenty candidates not directly absorbed by exact?, twenty out of twenty candidates not automatically discharged by aesop, and no explicit duplicates or near duplicates. 针对 20 个候选猜想的实验表明，从自然语言到形式化检查的过程表现稳定：所有 20 个候选猜想均通过了 Lean 的解析与类型检查，且均未被 exact? 直接吸收，未被 aesop 自动解决，同时不存在明显的重复或近似重复问题。