The Necessary and Sufficient Structural Condition of Persistent Identity Under Real Transformation

This paper states, proves, and closes the Persistence Admissibility Theorem (PAT) and establishes the structural unavoidability of La Profilée. It supersedes Papers 81 v1/v2 and incorporates Paper 82 (Admissibility Necessity), closing all previously open precision points. Version 2 incorporates five reviewer-hardening additions to the original P103 argument: (1) explicit proof that F, M, K are not re-labelings but the irreducible logical invariants of any persistence verdict; (2) clarification that the classification into three transformation structures is structural and non-evaluative, not presupposing M3; (3) the functional class exhaustion proof establishing multiplicativity as the only form in the admissible class, not merely the unique representative; (4) the separation of universality from empirical coverage; (5) the residual condition argument establishing that IR ≤ 1 is not a tautology but the result of eliminative closure. Part I (Sections 1–5): PAT — within the full admissibility class C defined by Conditions 1–7, the unique global persistence condition is R ≤ F·M·K. Key improvements over v1: Sub-Lemma 2.1 formally derives countable order-density from Condition 4; Lemma 4 closes the triadic architecture through Q1–Q3 exhaustion and B1–B4 exclusion; the C₀→C₁ transition is established internally; C3 is proved as a structural consequence of C1+C2, not an independent axiom. Part II (Sections 6–9): Unavoidability — F, M, K are logical invariants of any possible persistence verdict, not variables introduced into a model. The admissibility class C is not assumed — it is induced. LP is not derived from persistence theories. Persistence theories are constrained projections of LP.