Unified Dielectric Mechanics and Resonance Well Architecture: Formulating Macro-Scale Planetary Systems and Discrete Lattice Invariants

Abstract This paper presents a rigorous mathematical framework that unifies classical gravitational potential fields and electromagnetic resonance systems by modeling mass as a macroscopic volume-bound atomic dielectric displacement. By introducing a universal coupling constant, \chi = \sqrt{G/k_e}, we re-evaluate celestial mechanics—specifically mean-motion resonances (such as the Jupiter-Saturn 5:2 ratio and the Galilean Laplace resonance)—as active capacitor networks governed by nonlinear phase-trapping pendulum dynamics. Furthermore, we establish the necessity of a discrete hexagonal lattice stabilized by a Mod 9 invariant, where a 7-cycle periodic break prevents thermal runaway at the 5184 \text{ Hz} (72^2) threshold.