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 Profílée. It supersedes Papers 81 v1/v2 and incorporates Paper 82 (Admissibility Necessity), closing all previously open precision points. Part I (Sections 1–5) establishes PAT: within the full admissibility class C defined by Conditions 1–7, the unique global persistence condition is R ≤ F·M·K. Three precision improvements over PAT v2: (1) Sub-Lemma 2.1 formally derives countable order-density from Condition 4 rather than assuming it; (2) Lemma 4 closes the exhaustivity of the triadic architecture through Q1–Q3 functional exhaustion and B1–B4 systematic exclusion of fourth roles; (3) the C₀→C₁ transition is established internally rather than deferred to Paper 82. Part II (Sections 6–9) establishes the Unavoidability of the Persistence Structure: F, M, K are not variables chosen to model persistence — they are the logical invariants of any possible persistence verdict. Any theory forming persistence verdicts satisfying Conditions 1–3 already contains F, M, K in the structure of its judgments. The admissibility class C is not assumed — it is induced. LP is not derived from persistence theories. Persistence theories are constrained projections of LP.