On the Structural Non-Equivalence of Admissibility and Regularization

This paper establishes the structural non-equivalence between admissibility-induced stability and common stabilization paradigms such as regularization, control, projection, and Lyapunov-based redesign. Whereas conventional stabilization mechanisms achieve stability by modifying evolution laws, introducing feedback, or tuning parameters, admissibility-based stabilization operates by restricting the set of allowed trajectories while preserving the underlying dynamics. Stability arises through exclusion rather than correction or intervention. The paper provides a structural taxonomy of stabilization mechanisms, identifies invariants unique to admissibility-based constructions—such as invariance of the evolution map, idempotence on admissible states, and independence from tunable parameters—and formalizes these distinctions through a diagnostic classification criterion. A concise proposition establishes that admissibility-induced stability cannot be realized as regularization or feedback control without changing mechanism class. No claims are made regarding superiority, universality, performance, or physical realization. The analysis is abstract and classificatory in nature, intended to clarify mechanism rather than outcome. This work serves as Paper 4 of a 12-paper series developing admissibility as a foundational stability principle and provides conceptual protection against misclassification in subsequent applications.