算术重正化群:素数间距的数学框架 Arithmetic Renormalization Group: A Mathematical Framework for Prime Gaps

This preprint introduces the Arithmetic Renormalization Group (ARG), a mathematical framework for describing the multiscale statistical structure of prime gaps. Instead of treating normalized gaps as static objects, the ARG formalizes how the empirical distributions of raw prime gaps evolve across scales through a renormalization operator consisting of coarse-graining and rescaling. Large-scale numerical evidence shows that prime gaps exhibit approximate scale invariance, heavy-tailed decay, long-range correlations, and finite-size scaling. These features cannot be fully explained by classical models such as Cramér’s random primes or the GUE statistics of zeta zeros. Within the ARG framework, these phenomena correspond naturally to RG fixed points, critical directions, and slow RG flows. The paper formulates conjectures on the existence of a nontrivial fixed-point distribution, the RG invariance of correlation functions, and the convergence of the tail exponent along the RG flow. The results suggest that prime gaps may belong to a new universality class, distinct from both Poisson and GUE statistics. The ARG framework provides a unified perspective on the multiscale behavior of prime gaps and opens several directions for future mathematical investigation. 本文提出“算术重正化群”(Arithmetic RG, ARG)框架,用于描述素数间距在不同尺度下的统计结构。ARG 将原始素数间距的经验测度视为随尺度演化的动力系统,通过“粗粒化—重标度”的算符刻画其跨尺度行为。 大量数值证据显示,素数间距具有跨尺度稳定性、重尾分布、长程相关性以及有限尺度漂移等特征,而这些现象无法由传统模型(如 Cramér 随机素数模型或 GUE 零点统计)解释。在 ARG 框架中,这些现象自然对应于 RG 固定点、临界方向与 RG 流。 本文提出若干核心猜想,包括固定点分布的存在性、关联函数的 RG 不变性,以及尾部指数沿 RG 流的收敛性。结果表明,素数间距可能属于一个新的普适类,介于 Poisson 与 GUE 之间。ARG 为理解素数间距的跨尺度结构提供了统一视角,并开启了多个未来研究方向。