Finite-Horizon Structures V: Dynamics Compatible with the Homogeneous Invariant

This article develops a purely structural notion of admissible dynamics for abstract objects equipped with a homogeneous invariant. Building on the categorical framework introduced in the Finite-Horizon Structures series, it defines admissible flows as one-parameter families of structure-preserving morphisms and characterises their infinitesimal behaviour through differential inclusions in a closed convex cone of logarithmic velocities. The framework yields a natural class of monotone and homogeneous flows, together with a complete classification of homogeneous regimes in terms of scaling exponents. Induced evolution inequalities for the invariant are established, allowing the identification of monotone, invariant, and persistent regimes. Internal gauge transformations preserving the invariant are introduced, showing that structural viability is tied to the existence of compensatory reparametrisations of the underlying triplet rather than to any specific mechanism. The theory is further extended to measured spaces by introducing invariant-weighted measures and compatible flows satisfying multiplicative cocycle relations. All constructions are purely mathematical and independent of any physical interpretation.