The Modular Resonance Principle: A Fourier-Analytic Heuristic for Short-Interval Prime Density Fluctuations

This repository contains the computational framework, empirical data, and manuscript investigating the Modular Resonance Principle (MRP) as a method for predicting prime number densities in short intervals. The MRP hypothesis models the distribution of primes by treating their multiples as interfering continuous waves (Fourier series). This research establishes a rigorous mathematical boundary for the MRP. We empirically demonstrate that pure continuous wave models hit a hard predictive ceiling: even when integrating all second-order (pairwise) wave interactions, the MRP recovers a maximum of only 6.5% of the available combinatorial variance. This limitation is driven by the Gibbs phenomenon, where continuous waves fail to accurately capture the sharp, high-frequency density shifts caused by the smallest primes.