An Arithmetic Operator on the Logarithmic Prime Lattice: GUE Statistics, a Robust Positive Spectral Gap, and a Spectral Circle Method for Goldbach's Conjecture

AbstractWe construct a family of finite-dimensional Hermitian operators ˜KN on thelogarithmic lattice xk = log pk of the first N prime numbers. The original operator˜K(0)N with diagonal entries log pk reproduces the Gaussian Unitary Ensemble (GUE)nearest-neighbor spacing statistics with high fidelity (Kolmogorov–Smirnov p-value≈ 0.905). Its rank-one perturbed version ˜K (1)N admits an exact secular equationwith numerical residual ≈ 10−15. Both constructions exhibit strong anti-persistencein the sign sequence of the triple-gap discriminant Dj , consistent with GUE levelrepulsion.Despite the chaotic bulk statistics, linear finite-size scaling reveals a robust positivespectral gap that converges toΔ∞ = 0.9610 0.9605, 0.9615 (95% CI)as N → ∞ (R2 ≈ 0.98). The positivity of the gap acts as a spectral stabilizerfor exponential sums over the arithmetic eigenvalues, leading to a Spectral CircleMethod that connects the operator to additive prime number theory. Numericalexperiments for even numbers up to 3000 demonstrate a statistically significantcorrelation (p < 0.002) between the gap ΔN and the fluctuations of Goldbachrepresentations G(2M) around the Hardy–Littlewood prediction. We formulate aspectral hypothesis that the non-vanishing gap implies a uniform lower bound forG(2M), thereby providing a spectral mechanism for the binary Goldbach conjecture.Finally, we develop a renormalization group (RG) analysis of the correlation kerneland prove, using Bochner’s theorem and the Prime Number Theorem unfolding,that the limiting kernel satisfies 0 ≤ K ≤ I and coincides with the sine kernel.This establishes GUE universality as the unique RG fixed point of the arithmeticoperator.1