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 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 classical periodic orbits, paralleling prime-number explicit formulas that express prime distributions through zeta zeros. However, both formalisms treat the underlying structures kinematically, without addressing how wave-dynamical interference across orbits influences spectral rigidity. Following the wave-node incommensurability framework of Part I and the controlled consistency test of Part II, this paper proposes a semiclassical orbit-selection hypothesis grounded in resonance avoidance. Commensurate periodic orbits with rational period ratios permit recurrent phase alignment and multi-channel phase locking, whereas incommensurate orbits exhibit persistent wave-node slipping that suppresses secular phase accumulation. We formalize this mechanism through a first-principles cross-orbit dephasing formulation within the trace formula, proposing that mutually incommensurate, prime-like periodic orbits are dynamically favored in sustaining spectral rigidity. Prime logarithmic periods provide a canonical arithmetic realization because distinct log p are linearly independent over the rationals, suppressing rational commensurability. Their incommensurate interference therefore provides a dynamical basis for the spectral-spacing incommensurability and minimum detuning margin identified in Part II. The Montgomery–Dyson correspondence is thus interpreted as a dual manifestation of incommensurability across semiclassical orbit space and spectral-frequency space, providing a wave-dynamical bridge between arithmetic non-divisibility and quantum spectral rigidity.