When Does a System Remain the Same System? - The Complete Structure of Persistence and Identity under Transformation

Every system undergoes change. Yet some systems remain the same. This paper addresses the fundamental question: under what conditions does identity persist under real transformation? We show that this question admits a single structural answer. From three minimal assumptions — distinguishability, real transformation, and decidability — a unique structure follows: identity-bearing structure, transformation-processing capacity, and their coupling. These roles combine into a single capacity measure, yielding a necessary and sufficient condition for persistence. This condition is not a modeling choice. It is the only admissible form consistent with the persistence problem itself. This paper derives LP from minimal axioms and establishes the dynamic persistence equation. Parts I–XIII constitute the formally closed core (Level A): derived from M1–M3 alone. Parts XIV–XXIII constitute formally derived extensions (Level B), applying M1–M3 to multi-system environments and higher-order structures. All structural hypotheses are formally closed. LP is precisely falsifiable. Physical stability results in thermodynamics, fluid dynamics, and quantum mechanics are special cases of IR ≤ 1.