We extend the fully deterministic operator–trace framework from The Path of Light (Paltooet al., 2025) to the Generalized Riemann Hypothesis (GRH).For any Dirichlet character χ modulo q, we construct a canonical self-adjoint HamiltonianHcan,χ and a trace functional Tcan,χ that deterministically encode all arithmetic information ofthe primes. Standing/Sitting Band (SSB) intervals propagate primes without material clumpingacross residue classes, ensuring exact correspondence between the spectrum of Hcan,χ andnontrivial zeros of L(s, χ) on the critical line Re(s) = 1/2.All logical steps are fully deterministic, trace-class convergence is established, and diagonaldominance enforces spectrum on the critical line. Optional Gaussian test functions illustratepositivity but are not essential, due to structural rigidity of the operator-trace construction.