Arithmetic Renormalization Group and the Multiscale Statistics of Prime Gaps

This work introduces the Arithmetic Renormalization Group (ARG), a scale-dependent transformation acting on empirical distributions of normalized prime gaps. Prime gaps exhibit statistical signatures that closely resemble critical phenomena in statistical physics, including scale invariance, long-range anticorrelations, 1/f-type spectra, power-law tails, anti-persistent fluctuations, and weak multifractality. Classical probabilistic models such as Cramér’s model, as well as analytic frameworks based on Hardy–Littlewood heuristics, successfully describe certain local properties of the primes but do not account for these large-scale statistical features. The ARG framework defines a renormalization flow that maps the gap distribution at scale X to that at scale bX, followed by a canonical rescaling by log(bX). Using large-scale numerical computations, we show that this flow exhibits strong evidence of convergence toward a nontrivial fixed-point distribution. Across several decades in scale, the rescaled distributions remain nearly invariant, while effective exponents—tail exponents, spectral slopes, correlation decay exponents, and Hurst exponents—drift slowly toward stable limiting values. Motivated by these observations, we formulate a family of conjectures concerning the existence, uniqueness, and universality of an ARG fixed point. These conjectures connect the empirical phenomenology of prime gaps with the structural features of renormalization flows, including scaling directions, marginal stability, and universality classes. The ARG perspective suggests that prime gaps may realize a new universality class at the interface of analytic number theory, dynamical systems, and critical phenomena. This dataset and accompanying manuscript provide the empirical foundation, theoretical formulation, and numerical methods underlying the ARG framework, offering a new lens through which to understand the multiscale statistical organization of the primes.