La Profilée - The Persistence Admissibility Theorem - A Self-Contained Formal Statement

This paper states and proves the Persistence Admissibility Theorem (PAT): within the full admissibility class C defined by Conditions 1–7 (three minimal meaning conditions and four representation admissibility conditions) — distinguishable states, real transformation, determinate persistence verdicts — any admissible persistence condition is structurally equivalent to R ≤ F·M·K. v2 replaces the three previously weakest steps with formally rigorous lemmas: Lemma 2 grounds scalar representability in explicit empirical-topological admissibility conditions (Conditions 4–7) rather than countable order-density; Lemma 3 derives bipartite decomposition as an invariance necessity rather than a structural argument; Lemma 4 establishes the triadic role architecture as a minimal basis proof; and S5 closes the multiplicativity argument with an explicit separability condition. All alternatives trivialize the problem, fragment it, or reduce to R ≤ F·M·K. A fourth class does not exist. The admissibility conditions themselves are shown to be minimal: weakening any of Conditions 4–7 eliminates the possibility of a global persistence law rather than yielding an alternative formulation.