Generative Compression Across Scales: Independent Capacity and Recursive Effective Degrees of Freedom

Recursive cross-scale generation requires more than combinatorial growth. We formulate a Generative Compression Cycle in which relational possibilities expand, explicit independent constraints reduce the admissible configuration space, persistent closure selects candidate structures Γ, and an effective-existence gate determines whether those structures can function as higher-scale degrees of freedom. We distinguish individual admission from joint independence: for admitted maps Γᵢ with state counts qᵢ, the realized joint image satisfies Ω_joint = |Im(Γ₁,…,Γₘ)| ≤ ∏ᵢ qᵢ; equality requires every declared combination to be realizable. Conditional information and Jacobian rank provide complementary diagnostics for redundant labels, but neither alone establishes causal independence. For connected relational graphs, β₁ = E − V + 1 separates relation growth from independent loop capacity. The formal results establish combinatorial bounds and logical separation among growth, compression, persistence and admission. A complete recursive generativity certificate remains a candidate criterion until the full sequence is demonstrated in a lineage-preserving run family under fixed resolution, mapping and observer rules. Configuration-space growth is not identified with physical-space expansion or cosmological expansion.