Effective Existence Across Scales: Operator-Invariant Admission, History Transport and Bidirectional Recovery Closure

When does a persistent relational organization become an effective dynamical degree of freedom at a higher scale? We formulate admission as a role-relative, non-compensatory test requiring predictive closure, intervention consistency, retained-history sufficiency, recovery preservation and, for propagation claims, causal-lineage continuity. The merged theorem layer adds four distinctions that passive prediction alone cannot establish: quotient closure does not guarantee realizable transition lifting; state sufficiency depends on the declared operator family; history transport must commute with cross-scale updates; and recovery equivalence requires bidirectional simulation together with safety and lineage preservation. These results are formal under their stated finite-state and task assumptions. The associated computational results remain simulation-level evidence, and the mapping from admitted relational structures to physical entities remains open. The framework therefore supplies a falsifiable admission and recovery gate without asserting that every admitted structure is a physical particle or a universal renormalization group.