Admissibility Depth in Ordered State Spaces

This paper introduces a scalar depth functional associated with admissibility when admissibility is formulated as a restriction on realizable trajectories. Building on earlier results showing that admissibility induces a partial order and lattice structure on admissible-closed subsets of a universe of states, a notion of depth is defined using finite strictly descending admissibility chains. The resulting depth functional is shown to be monotone with respect to set inclusion and invariant under closure equivalence. The construction depends only on the order-theoretic structure induced by admissibility and introduces no additional assumptions. No physical interpretation is assumed, and no claim is made beyond what follows from the stated definitions. This work is intended as a structural component of a broader admissibility-based framework. It isolates a scalar quantity derived solely from admissibility-induced order, prior to any application or interpretation.