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).