Physical Number Theory: Quantum Integrability from Modular Arithmetic Sequences

Abstract

Cephas Lem Baguot

We establish Physical Number Theory as a new discipline connecting modular arithmetic with quantum integrability. We demonstrate that sequences of integers avoiding specific residue classes—particularly Tq = {n ∈ N : n ̸≡ 0 (mod q)} for prime q—form genuine integrable quantum systems. Through rigorous mathematical proofs, we show these sequences possess: (1) �?(q) conserved quantities (modular projections), (2) Poisson spectral statistics in prime distributions (p-value = 0.201 for T7), (3) exact Hamiltonian decomposition into �?(q) independent subsystems, and (4) T�?(q) phase space structure. We provide experimentally testable predictions for heptagonal quantum billiards, 7-site optical lattices, and quantum dots with 7-fold symmetry. The work resolves the paradox between Poisson-distributed prime gaps and GUE-distributed zeta zeros by identifying them as different quantum dynamical regimes. This establishes a systematic design principle for integrable quantum systems and creates a fundamental bridge between number theory and quantum physics.

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