Dynamical Origin of Spectral Rigidity and the Montgomery–Dyson Correspondence: Part II. Controlled Commensurate Transition of Nuclear Level Statistics

In Part I, we proposed that quantum spectral level repulsion is dynamically sustained by continuous wave-node incommensurability—the non-divisibility of transition beat frequencies suppressing harmonic resonance locking. Here, we provide controlled numerical evidence through a two-stage baseline-and-inverse-intervention analysis of empirical neutron resonance spectra of compound nuclei (²³⁸U, N = 145) from Columbia University’s Nevis Laboratories. First, a baseline scan across all 143 adjacent transition beat pairs reveals that 100% avoid harmonic integer divisibility, with a minimum detuning δ_min = 0.2277% exceeding the experimental tolerance δ_tol ≈ 0.10%. Second, the entire sequence is driven toward harmonic commensurability through a global parameter λ ∈ [0, 1]. Advancing λ compresses the wave-node distance from its natural incommensurate margin into the resonant tolerance window, while displacements remain bounded within physical level tolerance. Each configuration thus represents a physically accessible state within the finite-interaction framework of Part I. Under this collective compression, the natural level-repulsion barrier systematically dissolves: entering the locking window triggers phase locking and ultimately mode clustering at exact commensurability. We then apply the same criterion to consecutive normalized Riemann zeta-zero spacings. Their local spacing ratios likewise avoid low-order integer harmonics and maintain a finite detuning margin, reproducing the structural condition associated with suppression of near-coincident spacings. Together, the controlled nuclear intervention and arithmetic spacing test provide complementary numerical evidence that the Montgomery–Dyson correspondence may reflect a shared incommensurability mechanism—the dynamic avoidance of harmonic phase locking across physical and arithmetic spectral domains.