Demythologizing Spectral Rigidity and Level Repulsion Statistics via Algebraic Incommensurability in Finite Quantum Shells: A Deterministic Framework Based on Non-Integer Multiplicity of Inter-Shell Beat Frequencies

The structural equivalence between the two-point pair correlation of Riemann zeta zeros, R_{ζ,2}(u) = 1 - (sin(πu)/πu)², and quantum level spacing statistics, R_2(r) = 1 - (sin(πr)/πr)², is established as the deterministic consequence of discrete algebraic incommensurability in finite quantum shells (n ≤ 8). While spatial orbitals remain diffuse, the radiation beat frequency Δν_{n,m} between any two discrete shells is unique and singularly localized, meaning level correlation statistics (u) directly reflect the non-resonance relations among these unique inter-shell beat frequencies. Deduced strictly from the universal quantum inverse-square scaling axiom (E_n ∝ -1/n²), dimensionless beat ratios (Δν_A / Δν_B) algebraically eliminate all multi-body screening factors, yielding a scale-invariant rational framework. Combinatorial analysis of all 210 transition pairs proves that dominant adjacent modes (n → n-1) exhibit 100% strict non-integer multiplicity (q ≥ 2), geometrically precluding harmonic phase resonance and enforcing level repulsion. This formulation presents a decisive dual-barrier test: Experimental validation of these dimensionless ratios—including the two rare multi-step integer exceptions at k = 4.0 and k = 5.0 (≈ 0.95%)—conclusively demystifies quantum chaos as an algebraic resonance-avoidance mechanism. Any fundamental physical divergence would necessitate an internal contradiction within the foundational inverse-square quantum axioms (1/n²) themselves. We resolve that historical NDE experiments naturally suppressed the 0.95% integer singularities as measure-zero statistical noise during spectral unfolding.