Quantum Chaos of Arithmetic Origin: Deterministic L-function Jacobi Hamiltonians exhibit GUE/GOE level statistics without random disorder or classical chaos

Abstract

We construct explicit one-dimensional tight-binding (Jacobi) Hamiltonians whose recurrence coefficients are generated purely from the arithmetic of completed L-functions (Riemann zeta, Dirichlet beta, Ramanujan delta), via circle sampling, Hausdorff moments of the log-derivative Taylor coefficients, and the orthogonal-polynomial (Gram-Schmidt) construction. No zero location, prime table, random number, or fitted parameter enters the construction. Despite being deterministic, one-dimensional, disorder-free, and having no classical chaotic limit, the locked spectra display Wigner-Dyson (GUE/GOE) level repulsion, a spectral form-factor ramp, extended eigenstates, and ballistic-to-diffusive wavepacket transport, while shuffled-coefficient controls revert to Poisson statistics and localization. We propose quantum chaos of arithmetic origin as a third origin mechanism of quantum chaos, distinct from classically chaotic quantization (BGS) and random disorder (Anderson). The paper states four graded physical conjectures (arithmetic GUE/GOE universality; spectral-superposition Poisson crossover; extended-localized eigenstate boundary; critical hardware-noise threshold epsilon_c ~ 0.2-0.3) and three open problems, with a turnkey experimental protocol (Appendix D): coefficient tables, predicted observables, control runs, noise budget, and pass/fail criteria for superconducting-qubit or trapped-ion quantum simulators. All results are finite-truncation numerical evidence; they neither prove nor assume the Riemann hypothesis. The companion ZIP is the reproducibility bundle (LaTeX source, exp1-exp4 scripts and result JSONs, coefficient tables for zeta/beta J50 and delta J22, construction pipeline, figures). Code repository: github.com/maomao8412/hilbert-polya-jacobi. Companion mathematics preprints: zenodo.22167882 and zenodo.22167742.