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

Abstract

For over half a century following the 1973 encounter between Hugh Montgomery and Freeman Dyson, the matching pair-correlation statistics governing the non-trivial zeros of the Riemann zeta function and the eigenvalues of random matrix ensembles have presented a profound, unresolved question in mathematical physics. In the preceding study (Part I), we introduced a physical framework demonstrating that 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 present a controlled numerical investigation of this mechanism applied directly to empirical neutron resonance spectra of compound nuclei (U-238) obtained from Columbia University's Nevis Laboratories. By systematically controlling a commensurability parameter alpha in [0, 1] that shifts empirical transition beat ratios from natural incommensurability (delta_min > delta_tol) toward exact harmonic multiplicity (R in Q, delta_min <= delta_tol), we track the continuous deformation of the pair-correlation function R_2(s). The computation demonstrates that within the controlled transformation, the zero-spacing repulsion barrier (R_2(0) -> 0) is strongly associated with the condition of spectral incommensurability: as harmonic integer divisibility is progressively introduced, the characteristic level-repulsion barrier steadily diminishes, rapidly giving way to resonant phase-locked clustering (R_2(0) >> 1). These findings provide strong quantitative evidence that quantum level repulsion and spectral rigidity are intimately connected to continuous wave incommensurability. By framing arithmetic non-divisibility as the discrete integer-lattice manifestation of continuous resonance avoidance, this work provides a concrete physical basis for interpreting the 50-year-old Montgomery–Dyson correspondence.