The Structural Law of Persistence

This paper consolidates a series of results concerning the structural conditions under which identity can persist under real transformation. Starting from a minimal representation of a system as a state space equipped with admissible transformations and an identity assignment, it is shown that unrestricted transformation collapses all non-trivial identity distinctions. Persistent identity therefore requires the admissible transformation structure to be selectively restricted. This structural selectivity induces asymmetry in the reachability structure of states. The resulting transformation topology yields a directed acyclic condensation structure whose induced ordering constitutes the minimal structural form of temporal direction. These results together form the Structural Law of Persistence, describing the minimal structural architecture required for persistent identity under real transformation. The architecture corresponds to the structural principle termed La Profilée, consisting of stable identity carriers, transformable elements, and regulated coupling between them. An operational diagnostic layer, LP becoming, applies this structural law to the analysis of real systems.