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 can naturally emerge from continuous wave-node incommensurability—the dynamical non-divisibility of transition beat frequencies that suppresses harmonic resonance locking. In this companion paper (Part II), we provide controlled numerical evidence for this mechanism by implementing a two-stage baseline-and-inverse-intervention analysis on empirical neutron resonance spectra of compound nuclei (²³⁸U) obtained from Columbia University’s Nevis Laboratories (N = 145). First, an empirical baseline consistency scan across all 143 adjacent resonance intervals mapped to effective transition beat pairs (Δ₁ / Δ₂) reveals that 100% of the analyzed adjacent nuclear resonance pairs avoid harmonic integer divisibility, maintaining a minimum detuning offset of δ_min = 0.2277% that exceeds the experimental resonance tolerance boundary (δ_tol ≈ 0.10%). Second, we conduct an inverse intervention test by systematically driving these empirical transition beat ratios into the tolerance window (δ_min ≤ δ_tol) via a control parameter α ∈ [0, 1] through direct microscopic three-level wave-node adjustment. Under this implementation, the natural level repulsion barrier (R₂(s ∈ [0, 0.1]) ≈ 0.18) steadily diminishes as harmonic commensurability is introduced (α = 0.35 ⇒ R₂ ≈ 0.34), culminating in resonant phase-locked mode clustering (R₂(0) ≈ 2.68 ≫ 1 at α = 1.0). This demonstration confirms that spectral level repulsion in compound nuclear data is consistently coupled to wave incommensurability. By connecting arithmetic non-divisibility with continuous resonance avoidance, these results provide a concrete physical framework for interpreting the 50-year-old Montgomery–Dyson correspondence.