Prime Number Lattice Operator and Spectral Invariants: A Fredholm Regularization of the Riemann Zeta Function

We present Version 6.0 of the spectral realization of Riemann zeros on the prime numberlattice L = {log p : p ∈ P}. This paper introduces a self-adjoint operator ˜K whose spectralstatistics strictly follow the Gaussian Unitary Ensemble (GUE) with a KS p-value of 0.905.We demonstrate that the Fredholm determinant det(I − e−s ˜K ) acts as a spectral detectorfor the non-trivial zeros γn. A fundamental discovery in this version is the existence of aninvariant scattering phase Φ(s) = Re log det(I−e−s ˜K ) ≈ 0.345, which remains stationaryon the critical line with sharp resonances at t = γn. We prove that ˜K effects a Fredholmregularization of the Hadamard product for ζ(s). Numerical evidence for a non-vanishingspectral gap Δ∞ ≈ 0.961 is provided, suggesting a structural link to the Yang-Mills massgap analogy.