Radiation Beat Frequency Incommensurability: Resolving Quantum Level Repulsion and Riemann-Type Spectral Rigidity via Finite-Shell (n ≤ 8) Orbit Ratios (1/n²)

Abstract

For over half a century since the historic encounter between Montgomery and Dyson, the exact correspondence between the pair correlation of non-trivial Riemann zeta zeros and the eigenvalue spacing statistics of complex quantum systems has remained an open foundational puzzle—frequently attributed to infinite-dimensional quantum chaos or transcendental prime distributions. In this work, we demonstrate that quantum spectral rigidity and level repulsion do not require infinite-dimensional complexity; rather, they emerge deterministically from the discrete algebraic geometry of finite quantum shells (n ≤ 8). By reformulating inter-shell transitions as singularly localized radiation beat frequencies Δνn,m = S · (1/m² - 1/n²) and constructing dimensionless pairwise frequency ratios R = ΔνA / ΔνB, all system-specific screening factors, effective nuclear charges, and isotope-dependent constants cancel out identically, yielding a universal, scale-invariant rational field (p/q). An exhaustive combinatorial evaluation across all 210 pairwise transition channels reveals that dominant adjacent dipole modes (n → n - 1) exhibit 100% strict non-integer multiplicity (q ≥ 2), geometrically precluding harmonic phase locking. Furthermore, when evaluated against empirical spectroscopic and nuclear data resolution thresholds (ε ∼ 0.01% - 0.1%), zero non-integer pairs fall within the 0.01% harmonic lock-in tolerance. The two exact multi-step integer exceptions (k = 4.0, 5.0) constitute only ≈0.95% of the combinatorial space and are suppressed as measure-zero noise under standard spectral unfolding. Derived strictly from the inverse-square quantum scaling axiom without adjustable parameters, this paper establishes that quantum level repulsion is the deterministic manifestation of discrete arithmetic incommensurability across physical orbits.