The Logical Architecture of Persistence - Minimal Assumptions, Derived Necessities, Formal Proofs, and the Regime Structure of Persistent Identity

This paper reconstructs La Profilée in complete logical order, establishing the formal architecture of the persistence theory from three minimal assumptions to the full regime space. It serves two purposes: it clarifies the stratified architecture and contains all formal proofs in self-contained form — no proof is deferred to external papers. Starting from M1 (Distinguishability), M2 (Real Transformation), and M3 (Determinable Persistence Relation) — where M3 is shown to be derived from M1+M2, not independently assumed — all structural necessities follow without additional premises: restricted admissibility, structural asymmetry, SCC/DAG topology, finite integration capacity (with explicit M2 connection), the Frame–Module–Coupling decomposition, and the persistence boundary IR = R/(F·M·K) ≤ 1. The Frame Continuity Condition (FCC) is derived from M1–M3 independently of IR ≤ 1. Nine structural consequences are derived: time, information, irreversibility, causality, decision, transformation, agency, selection, and the intervention paradox. The Persistence Admissibility Theorem (PAT) is established in full: four lemmas with complete proofs (Total Preorder, Scalar Representability via Debreu, Invariant Bipartite Decomposition, Minimal Complete Role Decomposition), the PAT Theorem, and the Closure Theorem. A Fallback Theorem establishes representation-independence: the structural core — bipartite structure R ≤ IK and triadic decomposition IK = F·M·K — holds independently of Debreu-style scalar representation. The Unavoidability Theorem establishes that any theory defining global persistence under real transformation either abandons the persistence problem, fails global lawhood, or instantiates R ≤ F·M·K — whether or not it names it. Persistence theories are constrained projections of La Profilée, not alternatives to it. Seven falsification conditions are stated.