Computational study reveals quantum chaotic level statistics in deterministic one-dimensional L-function Hamiltonians, indicating an arithmetic origin of spectral universality without classical chaos.
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, 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 collapse of the signal under coefficient shuffling and under a percent-to-ten-percent hardware-noise threshold. v2 additions: pure-zeta finite-size scaling extended to N=100 via a 1300-digit construction (b_n^2 > 0 on all 99 off-diagonals; 55 locked zeros at relative error ~1e-15..1e-16; locked gap ratio 0.61-0.62 from N=22 up with zero sub-0.3 gaps); and an Eisenstein E4 discriminative control L(E4,s)=zeta(s)zeta(s-3), whose zeros lie on Re(s)=1/2 and Re(s)=7/2 off the symmetry center: the identical pipeline reproduces Xi_E4(2)=1/72 to ~15 digits but the moment sequence turns negative at order 8 and the Jacobi chain fails at order 4, showing long positive-definite chains co-vary with on-line zeros. The bundle also ships the Quantum Arithmetic Benchmark Dataset (five tridiagonal Hamiltonians with metadata and a loader) as a turnkey quantum-simulation target. All results are finite-truncation numerical evidence; the spectra are computer-generated, not measured; 'quantum chaos of arithmetic origin' is a physical hypothesis, not an established fact. The construction assumes neither RH nor GRH.
No takes yet. Share an insight, caveat, or question.
Zhuo Chen (2026) studied this question.
Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context: