La Profilée - The Structural Law of Persistent Identity - A Complete and Self-Contained Derivation

La Profilée does not describe how systems evolve — it determines whether they remain the same system at all. La Profilée (LP) establishes a structural law governing identity under real transformation. Its central question is unavoidable: under which conditions does a system undergoing real transformation remain the same system? LP does not describe how systems evolve. It determines whether evolution preserves identity. Central condition: A system persists as itself if and only if IR ≤ 1 and FCC holds. IR ≤ 1 determines whether the system continues to exist as a structured entity. FCC determines whether it remains this system. The empirical accessibility of the structural variables follows from the admissibility conditions of the persistence problem itself; proprietary diagnostic implementations are secondary realizations of this forced accessibility, not preconditions of it. The core derivation is self-contained with respect to all indispensable LP-specific claims required for the three laws. Standard mathematical structures are used in their conventional form. Selected extensions (Scale Invariance, Vertical IR Aggregation, Frame Collision Direction, Structural Time Invariance) are stated in compressed form; full derivations appear in the cited LP series papers. The paper establishes three structural laws from minimal conditions and specifies falsification conditions for each. The three laws: (I) IR = R/ (F·M·K) ≤ 1 is the necessary and sufficient condition for the persistence of the system as a structured entity (IR ≤ 1) ; persistence as this system additionally requires FCC. (II) Tcol θ. No geometric or metric structure is required — the scalar function represents the order structure alone. Any representation that does not admit such scalarization fails to preserve global comparability and therefore cannot represent a determinate persistence boundary. Any non-scalar representation either (i) fails to preserve total comparability, or (ii) contains redundant structure that collapses to the same scalar ordering under equivalence. Therefore every admissible persistence boundary admits a scalar representation. ✓ □ Lemma 3 (Forced Bipartite Decomposition of Persistence Boundaries). A persistence boundary that does not distinguish between transformation pressure and absorption capacity cannot determine persistence: it either collapses into triviality or becomes representation-dependent. Therefore any scalar persistence boundary necessarily decomposes into a comparison between transformation pressure and directed integration capacity: R ≤ IK. Proof. Let f (T) be the scalar representation from Lemma 2, with threshold θ: persistence ⇔ f (T) ≤ θ. A persistence boundary distinguishes two structurally distinct roles: (1) transformation-induced deviation from the identity-defining structure; (2) structural capacity to absorb such deviation without identity loss. This distinction is forced by the nature of the persistence problem. If no distinction is made: Case (i) — f measures only deviation: identical deviations across systems must yield identical persistence verdicts. But systems with different capacities respond differently to the same transformation. Persistence cannot be determined from deviation alone. Case (ii) — f measures only capacity: persistence verdicts are independent of transformation magnitude. All transformations would be always admissible or always inadmissible. The persistence boundary collapses into triviality. Case (iii) — f mixes both without structural separation: the contribution of deviation and absorption cannot be independently evaluated. Persistence verdicts become non-invariant under decomposition of transformations: the same total transformation may yield different results depending on internal representation. This violates admissible representation invariance (Theorem DI2). Therefore any admissible scalar persistence boundary must distinguish: R — a quantity measuring identity-challenging transformation; IK — a quantity measuring identity-preserving integration capacity. Persistence is determined by whether deviation exceeds absorption capacity: R ≤ IK, up to strictly monotone transformation. This decomposition is not a modeling choice but the minimal structure required to distinguish transformation from its admissible absorption. A scalar boundary that does not admit a decomposition into a monotone-increasing transformation term and a monotone-increasing capacity term cannot remain invariant under admissible decomposition of transformations: splitt