Emergence and Persistence - The Structural Conditions under which Emergence Leads to Persistence

Every theory of emergence describes systems that arise and persist. None has formally derived the structural conditions under which emergence leads to persistence rather than immediate dissolution. This paper establishes those conditions within the framework of La Profilée. We prove that any emergent system capable of persisting as a non-trivial identity under real transformation must satisfy three structural conditions: distinguishable states, real transformation, and non-trivially invariant identity. From these conditions, three results follow by necessity. First, a fully undifferentiated origin state is structurally excluded — not empirically unlikely, but formally inadmissible as the basis of persistent identity. Second, identity is structurally induced as the maximal transformation-invariant equivalence relation, not assigned or assumed. Third, the persistence condition IR ≤ 1 constitutes the upper structural bound: any emergent system that violates it cannot persist as itself. Together these results show that emergence is not structurally unconstrained. LP does not explain how systems arise. It specifies which emergent systems can remain. Persistence is not a consequence of emergence — it is its admissibility condition.