LA PROFILÉE Foundations: Axiomatic Core of Persistent Identity under Transformation

This paper presents the axiomatic core of La Profilée (LP) in compact form. It derives, from minimal admissibility conditions, the necessary structural architecture of any persistence system under real transformation: the Frame–Module–Coupling decomposition, the persistence law IR = R/(F·M·K) ≤ 1, the identity-continuity condition FCC, the Q1/Q2 independence theorem, the primary regime-space, and the universal sub-regime structure. LP begins from four primitive terms and three minimal conditions. From these it derives six theorems establishing: the necessity of the F·M·K architecture, the multiplicative form of integration capacity, the necessity of the Frame-deviation threshold δ_F, the exhaustion of the primary regime-space into three real regimes, the reduction of all persistence theories to LP conditions, and LP as the minimal structural law of persistence. The exhaustion of alternative coherent persistence structures — trivialization, indeterminacy, structural equivalence — is derived in Section 10.1. Six universal sub-regimes are derived: Flourishing, Fragile Persistence, Directed Transmutation, Drift Transmutation, Collapse, and Dissolution. These constitute the minimal universal structural partition currently derivable within the LP admissibility architecture. Full formal proofs — including the Persistence Admissibility Theorem (PAT) with Lemmas 1–4 and the Closure Theorem — are in P162. Domain-specific extensions build on these foundations.