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.