Admissibility Entropy in Ordered State Spaces

This paper introduces an entropy functional associated with admissibility when admissibility is formulated as a restriction on realizable trajectories. Building on prior results showing that admissibility induces a lattice structure on admissible-closed subsets of a universe of states and admits a scalar depth functional, entropy is defined by counting admissible refinements within that lattice. The resulting entropy functional is monotone with respect to inclusion, invariant under closure equivalence, and bounded in terms of admissibility depth where finite. The construction is order-theoretic and does not rely on probability, thermodynamics, or physical interpretation. No claim is made beyond what follows directly from the stated definitions. This work serves as a structural component within an admissibility-based framework, establishing entropy as an intrinsic property of admissibility-induced order prior to any application or interpretation.