An Arithmetic Operator on the Logarithmic Prime Lattice: GUE Statistics and a Robust Positive Spectral Gap

AbstractWe introduce a family of finite-dimensional Hermitian operators ˜KN constructed on thelogarithmic lattice xk = log pk of the first N prime numbers. The original operator ˜K(0)N withdiagonal entries log pk reproduces the Gaussian Unitary Ensemble (GUE) nearest-neighborspacing statistics with high fidelity (Kolmogorov–Smirnov test p-value ≈ 0.905).Its rank-one perturbed version ˜K(1)N admits an exact secular equation with residual ≈10−15. Both constructions exhibit strong anti-persistence in the sign sequence of the triple-gap discriminant Dj , consistent with GUE level repulsion.Despite chaotic bulk statistics, linear finite-size scaling reveals a robust positive spectralgap that converges to∆∞ = 0.9610 [0.9605, 0.9615] (95% CI)as N → ∞ (R2 ≈ 0.98).The proposed model provides a concrete, computationally accessible realization of theHilbert–P´olya philosophy, combining GUE-like local statistics with a stable mass gap in anarithmetic quantum system. We emphasize that this work does not constitute a proof of theRiemann hypothesis.