Numerical Consolidation and Stability of the Prime–Resonance Operator

This paper provides the numerical stability capstone of the Prime–Resonance Operator (PRO) program. It verifies that the core analytic properties established across PRO I–VII—boundedness, uniform positivity, and localization—persist under discretization and finite-resolution numerical realization. Representative numerical schemes consistent with analytic admissibility constraints are employed to test robustness under perturbation and resolution refinement. The results confirm that PRO behavior is not an artifact of idealized functional settings and survives contact with numerical implementation. Together with the equivalence and closure synthesis of the PRO program, this work completes the Prime–Resonance Operator framework by addressing standard numerical objections and demonstrating stability beyond purely analytic regimes.