Emergence as Persistence Inheritance - When a higher level is a persister and the reduction condition that unifies the series

A higher-level variable can have a closed equation without identifying a complete physical whole. This paper derives when a higher realization is an autonomous persister and when it inherits continued identity. We retain the original reduction, emergence, role-inheritance, universality and effective-theory theorems, and complete their physical subject: the higher bearer is constituted independently before its dynamics and Q1/Q2a/Q2b are evaluated. Exact reduction then requires joint descent of its complete constitution, retained verdicts and projected future. For a finite Markov realization, partition refinement constructs the unique coarsest admissible state quotient. A symmetric two-oscillator construction realizes the full certificate: the collective mode has a complete scoped response profile, an invariant continuation domain and distinct supporting realizations. The supporting assembly and the collective mode need not have the same constitution distinctions. We prove that unrestricted exact mean-coordinate closure occurs precisely for an affine onsite force, derive an explicit quartic counterexample and obtain a finite-horizon error bound when relative motion is controlled. Agreement of free frequencies is further shown insufficient for agreement under a physically normalized force intervention. Recursive constitution explains how organized members realize a higher bearer without transferring their identities to it. Constitution formation, multiple realization and sustained persistence are separated. Directed reachability induces a preorder; its mutual-reachability quotient, not non-totality alone, supplies structural order. Approximate identity claims require errors below the appropriate state or history verdict margin. The result consolidates the emergence and multi-level arguments in one physical derivation, while preserving the distinction between exact autonomous emergence, memory-bearing descriptions and controlled effective theories.