Arithmetic Depth: A Diagnostic Framework for Prime Number Phenomena

We introduce Arithmetic Depth (AD), a diagnostic framework for classifying prime number phenomena by the level of arithmetic structure required to explain them. Rather than proposing new theorems, AD provides an operational method for separating generic baseline effects (such as sparsity and divisibility geometry) from genuinely arithmetic effects associated with analytic objects like zeta and L-function zeros. The framework is based on systematic baseline subtraction across a hierarchy of explanatory models: (0) density-matched randomness, (1) geometric/sieve structure, (2) analytic models informed by explicit formulas, and (3) currently unexplained residual structure. A phenomenon’s Arithmetic Depth is defined as the minimal level at which the residual becomes statistically negligible. We apply this framework to several canonical phenomena in number theory. We show that the Lemke Oliver–Soundararajan bias and Primorial Coverage Coherence are predominantly geometric (AD = 1), while the Chebyshev prime race and the growth of the Mertens function exhibit substantial residual structure that cannot be reproduced by any geometric or randomized baseline (AD = 2), consistent with analytic control via zeta zeros. Twin prime locations exhibit no explanatory power at levels 0–2 (AD = 3). Arithmetic Depth does not claim intrinsic classification of primes, but rather provides a model-relative diagnostic that predicts tractability, clarifies the scope of existing methods, and prevents misinterpretation of projection-induced structure as deep arithmetic. The framework is reproducible, falsifiable, and intended as a methodological tool for experimental and analytic number theory.