Prime Resonance Theory: A Unified Framework for Frequency Set Optimisation via Number-Theoretic Structure

Why do prime-ratio frequency sets outperform composite-ratio sets in physical resonators? We present Prime Resonance Theory, a unified number-theoretic frame- work that answers this question from first principles. Starting from the experimen- tal observation that the six-element frequency divisor set 1, 2, 3, 5, 6, 7 produces 28% greater coupling amplitude and 22% higher coherence than composite-ratio sets in a torsion-coupled RC network, we develop a Five-Factor Theory decompos- ing resonance quality into: (1) structural resonance via the Tusk Series spectrum, (2) anchor frequency presence, (3) pairwise coprimality, (4) prime factor depth, and (5) scaffold inclusion of the 1, 2, 3, 6 core. These factors are unified into a single computable Resonance Quality Function R (S) = wiFi (S), whose sub-functions are derived from Fourier analysis of the sum-of-prime-factors arithmetic function and elementary number theory. Evaluated exhaustively over all 15 6 = 5005 six- element subsets of 1,. . . , 15, R (S) independently identifies 1, 2, 3, 5, 6, 7 as the unique optimum—reproducing the experimental discovery from pure mathematics (Spearman ρ = 0. 736, p = 0. 010). We establish a Spectral Honeycomb Theorem: co- prime frequency sets tile harmonic space with∼6% greater efficiency than random sets, with first harmonic overlap occurring at 1/ gcd—directly analogous to the ge- ometric honeycomb theorem’s∼5% perimeter advantage. We introduce the Prime Harmonic Transform, connecting our framework to Möbius inversion and Dirichlet series, and decompose the 10–15% performance gap between Riemann zeta zeros and primes into four quantifiable factors. The theory generates falsifiable predic- tions for forthcoming experiments and suggests applications in signal processing, cavity design, and neural oscillation architecture.