Dynamical Origin of Spectral Rigidity and the Montgomery–Dyson Correspondence: Part III. Semiclassical Orbit Selection Hypothesis and Wave-Dynamical Interference in the Gutzwiller Trace Formula

While the Montgomery–Dyson correspondence revealed a profound statistical equivalence between the non-trivial zeros of the Riemann zeta function and complex quantum spectra, its microscopic dynamical origin remains an open question. In semiclassical physics, the Gutzwiller trace formula relates quantum spectral density to a sum over classical periodic orbits, paralleling prime-number explicit formulas (Riemann–von Mangoldt–Weil) that express prime distributions through zeta zeros. However, both formalisms treat the underlying orbits kinematically, without addressing how wave-dynamical interference across adjacent orbits influences long-term spectral rigidity. Following the foundational wave-node incommensurability framework of Part I and the empirical controlled consistency test of Part II, this paper (Part III) proposes a semiclassical orbit-selection hypothesis grounded in resonance avoidance. Paralleling macroscopic orbital dynamics where mean-motion resonances destabilize trajectories (e.g., celestial Kirkwood gaps and Cassini divisions), commensurate periodic orbits with rational period ratios permit stationary phase alignment, facilitating multi-channel phase locking that can induce destructive parametric dephasing across adjacent semiclassical contributions. Conversely, incommensurate orbits exhibit persistent wave-node slipping (δ_min > δ_tol), suppressing secular phase accumulation and providing an intrinsic resonance-avoidance barrier. We formalize this mechanism via an effective cross-orbit filtering ansatz inside the trace formula, proposing that mutually incommensurate, prime-like periodic orbits are dynamically favored in sustaining coherent spectral rigidity. By connecting arithmetic non-divisibility with continuous resonance avoidance, this work presents a wave-dynamical perspective linking periodic-orbit expansions and the Montgomery–Dyson correspondence.