Harmonic–Differential Equivalence in Structured Resonance Dynamics

CODES Framework Abstract This paper demonstrates the formal equivalence between the harmonic phase-alignment law and the differential coherence law that together define deterministic persistence across physical, computational, and biological systems. The harmonic form PAS_h ≥ θ_L ∧ ΔPAS_ζ ≤ ε_drift and the differential form dC/dt = Γ Cⁿ − λ C + η (∇S · ∇Ω) are shown to be coordinate transforms of a single persistence condition under a monotone mapping C = g(PAS_h) and a bounded-drift (Lipschitz) constraint. This establishes the first calculus bridge between phase-coherence dynamics and continuous field equations. Chirality (Δχ ≠ 0) and prime indexing are identified as structural requirements for lawful emission. A falsifiable prediction (ε_drift ∝ V⁻¹⁄²) is proposed for multi-platform experimental verification. Structured Resonance Dynamics thereby unifies harmonic and differential coherence models within one lawful substrate.