Level Repulsion Matching the Riemann Zeros from a Time-Reversal-Broken Synaptic Ising Model

Abstract

We construct a transverse-field Ising Hamiltonian with synaptic couplings on a genuine spin lattice and show that its spectral statistics can be driven into the GUE universality class by a single, physically motivated time-reversal-symmetry-breaking term of ZiYjZ_i Y_jZiYj type. Numerical exact diagonalization on systems of N=8N = 8N=8–111111 spins reveals a level-repulsion exponent β≈1.76\beta \approx 1.76β≈1.76, statistically indistinguishable from the value βζ≈1.78\beta_\zeta \approx 1.78βζ≈1.78 measured on the first 1000 nontrivial zeros of the Riemann zeta function. Kolmogorov–Smirnov distances and qualitative number variance further support the match to GUE statistics. This provides a concrete lattice-based Hamiltonian whose short-range spectral statistics closely track those of the Riemann zeros — in sharp contrast to purely multiplicative constructions that lack any critical line. While no functional equation or claim about the Riemann Hypothesis is made, the numerical agreement raises a natural question: how much of the zeta-zero statistics can be reproduced by simple local spin models with broken time-reversal symmetry?