Spectral Gap Operators on the Logarithmic Prime Lattice: Theorems, Mass Gap, and Applications to Goldbach Conjecture

AbstractWe construct a family of Hermitian operators A(N )K on the logarithmic lattice {xk =log pk} of the first N primes, with kernelA(N )K,ij = v|i−j|, vk = 1log pke−k/K .The operators exhibit Gaussian Unitary Ensemble (GUE) bulk statistics while possessinga robust positive spectral gap ∆(N )K = λ(N )2,K − λ(N )1,K ≥ c(K) > 0. We prove exponentialdecay of higher eigenvalues λ(N )n,K ≲ exp(−αn/K) and establish an explicit lower bound forthe spectral gap c(K) ≥ C exp(−K/ζ). The positive gap acts as a spectral stabilizer forexponential sums over prime-related eigenvalues, leading to a Spectral Circle Method thatconnects the operator to additive prime number theory. Numerical experiments confirmstatistically significant correlation (p < 0.002) between the gap and Goldbach representationfluctuations, suggesting the non-vanishing gap implies a uniform lower bound for G(2M ).