Retained Resonance as a Non-Violation Class Invariant Withdrawal, Branch Persistence, and the Ψ-NVP Axiom

This paper introduces retained resonance as a necessary class-level invariant within the ψ-Architecture under the Non-Violation Protocol (Ψ-NVP). It is shown that, after complete withdrawal of external input, locally coupled systems may preserve non-collapsible class-level distinguishability that cannot be eliminated by any non-violating evolution. Retention is thereby characterized not as a dynamical quantity, energetic remainder, or memory trace, but as a structural invariant that persists after the disappearance of its carrier. Using a minimal double-well node model, the paper establishes a strict operational test based on withdrawal, mismatch monotonicity, and branch persistence. The result provides a necessary condition for genuine retention and sharply distinguishes retention from intervention-based persistence across physical, biological, and conceptual domains. This work constitutes the axiomatic core of Volume III (Ψ-NVP) within the ψ-Architecture research corpus.