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Inter-level connections in science routinely require constructs that neither of the connected theories contains. Statistical mechanics requires assumptions such as the Stosszahlansatz to generate thermodynamic irreversibility - assumptions that Hamiltonian mechanics cannot provide. Quantum chemistry offers four incompatible analyses of chemical bonding for the same quantum state, none of which are selected by the Schrodinger dynamics. Molecular genetics has not converged on a stable definition of the gene despite decades of molecular detail. These are not isolated anomalies but instances of a common architectural pattern. The missing apparatus is the bridge theory: a third theoretical role that connects a dynamical theory to an observational theory through a many-to-one inter-level map. That map generates the contingent space - the set of dynamical states compatible with an observational description but not selected by it - whose geometry neither connected theory determines. Completing the bridge theory requires three conditions in order: a Partition that defines observational equivalence classes; a Magnitude that characterizes the geometry and scale of the contingent space; and a Closure that selects or weights its elements. The framework yields an objective distinction between closing and introducing rules, formalized by the Mirror Test, and supports a tripartite taxonomy of emergence. It explains why some inter-level disputes persist and what would be required to resolve them.