The Formal Structure of LP: One Primitive · One Axiom · Complete Derivation · Uniqueness Theorem

This paper establishes the formal structure of La Profilée. Reading P000 — the canonical derivation — against the requirements of formal axiomatics reveals a precise result: LP has one genuine primitive (the definition of state as minimal distinguishability) and one genuine axiom (real transformation). Everything else — M1, M3, F/M/K, IR ≤ 1, structural time, and the regime architecture — is derived. M1 is a theorem, not an axiom. M3 is a theorem, not an axiom. From these two starting points the paper derives the complete LP structure in formal notation, constructs the minimal model proving consistency, and proves the Uniqueness Theorem: any globally coherent persistence theory that admits real transformation and non-trivial states is LP-equivalent. The formal structure of LP is therefore closed. This paper does not replace P000. It translates P000's derivation into a formal language and closes the gaps that informal derivation leaves open: it proves the two primitives are genuinely minimal, proves consistency by explicit model construction, and proves uniqueness by exhaustion over all possible structural alternatives.