Quantum Chaos of Arithmetic Origin

Abstract

v3 changelog: Added §7.4 OTOC (out-of-time-order correlator) dynamical non-integrability evidence; added §6 Discussion: three conceptual implications (third route to quantum chaos; correlational structure of matrix elements; structural correspondence beyond statistical analogy). Total 22 pages. Core result: A one-dimensional tight-binding Hamiltonian whose hopping amplitudes and on-site potentials are constructed solely from the arithmetic recursion coefficients (Jacobi αn, bn) of entire-function Taylor expansions derived from L-function Hadamard products. Without any stochastic input, the spectrum exhibits Wigner–Dyson level repulsion (matching GUE/GOE statistics), sub-diffusive eigenstate structure, and slow non-integrable scrambling dynamics (confirmed by OTOC). Six independent evidence lines: spectral statistics, wave-packet dynamics, OTOC, shuffle control, Eisenstein-series discriminant, and precision budget. Conceptual message: Quantum-chaotic spectral fingerprints do not require semiclassical chaos or stochastic disorder; deterministic ordered arithmetic recurrence is an independent route. Companion numerical-algorithms paper: DOI 10.5281/zenodo.22182156. Bundle includes LaTeX source, compiled PDF, all experiment scripts, data files, and figures. Code repository: github.com/maomao8412/hilbert-polya-jacobi