The KN Operator: A Constructive Realization of Admissibility-Induced Stability

This paper introduces the KN operator as a constructive realization of admissibility-induced stability in dynamical systems. Building on prior work establishing admissibility as a general stability principle, the present contribution provides an explicit operator-level mechanism by which admissibility can be enforced without modifying the underlying dynamics. The KN operator is defined as a non-expansive, idempotent map onto a parameter-independent admissible set, acting as a structural selector on system trajectories. When composed with an unconstrained evolution map, KN-induced dynamics enforce admissibility while preserving the original evolution law. Under mild conditions inherited from admissibility theory, the resulting dynamics are shown to be contractive on the admissible set and to admit a unique fixed point with geometric convergence. The paper distinguishes structural KN constructions from degenerate implementations equivalent to parameter tuning, regularization, or control, and provides minimal abstract examples illustrating admissibility-induced stability. No physical, energetic, or domain-specific interpretation is assumed. This work serves as the constructive counterpart to admissibility theory and occupies a foundational role in a larger multi-paper program developing operator-level frameworks for stability via admissible trajectory restriction.