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

Abstract

In Part I, we proposed that quantum spectral level repulsion is dynamically sustained by continuous wave-node incommensurability, whereby non-divisible transition beat frequencies suppress harmonic phase locking. Here, we test this mechanism through a baseline-and-inverse-intervention analysis of the empirical (²³⁸U) neutron-resonance spectrum and an independent structural test of Riemann zeta-zero spacings. All 143 adjacent nuclear transition-beat pairs avoid harmonic integer divisibility within the resonance tolerance, with (δ_min = 0.2277% > δ_tol ≈ 0.10%). We then drive the sequence toward harmonic commensurability through (λ ∈ [0, 1]), compressing wave-node separation into the resonance-locking window within the prescribed physical tolerance. As incommensurability is removed, the level-repulsion barrier progressively dissolves, culminating in strong mode clustering at exact commensurability. We next apply the same criterion to 10,000 consecutive critical-line Riemann zeta zeros and a high-altitude window. This is not a numerical re-verification of the known zeta-zero correlation law, but an independent test of whether the arithmetic spectrum exhibits the same incommensurability signature identified in the nuclear spectrum. The normalized zeta-zero spacings likewise avoid integer harmonic ratios within the tolerance, maintaining (δ_min(ζ) ≈ 0.1842% > δ_tol). Thus, the nuclear intervention tests what happens when incommensurability is deliberately removed, whereas the zeta-zero scan asks whether the same resonance-avoidance structure independently appears in an arithmetic spectrum. Their shared signature provides complementary numerical evidence consistent with the hypothesis that the Montgomery–Dyson correspondence reflects a common incommensurability mechanism: harmonic phase-locking avoidance across physical and arithmetic spectra.