Determinate Existence Under Real Transformation - A Formal Structural Necessity Result

This paper formalizes the structural necessity result established across the preceding papers in this series. Starting from three minimal conditions — distinguishability, real transformation, and identity continuity — it derives through a sequence of lemmata and theorems that any system admitting determinate existence under real transformation must instantiate exactly three non-substitutable structural functions (Frame, Module, Coupling), a multiplicative integration capacity, and a bounded persistence ratio IR ≤ 1. The Frame Continuity Condition (FCC) is shown to be independent of IR, generating a closed five-regime phase space. No alternative structure is compatible with the stated conditions of determinate identity under real transformation. The result is not a model of system behavior. It is a structural admissibility condition on determinate existence.