Geometric Realization of Completed Source Relations: Descent, Orbit Spaces, Invariant Relations, and Variational Response in Shadow Theory

This is Paper IV of the seven-paper Shadow Theory foundational series. The paper develops the geometric realization problem for the canonical completions constructed in the earlier papers. It specifies when declared source relations arise from compatible global metrics, principal bundles, gauge connections, matter fields, boundary data, and invariant observables, rather than merely as abstract coordinates of a quotient. Compatible local geometric data are shown to descend to global fields, uniquely up to the appropriate bundle isomorphism. Physical configurations are then reduced by a groupoid of allowed gauge and diffeomorphism transformations. Gauge- and diffeomorphism-invariant relations, including holonomy classes, Wilson observables, characteristic quantities, boundary data, and suitably specified spectra, become well-defined functions on the resulting orbit space. Their joint image with a declared readout realizes the canonical minimal completion on the restricted geometric domain. A fully explicit example classifies U(1) connections on an oriented circle. Although curvature vanishes identically, holonomy completely classifies gauge orbits and realizes the full circle as a completion. The charged-scalar spectrum instead identifies inverse holonomies, giving a strictly coarser interval-valued completion. The paper also gives a variational realization framework. Invariant relation-dependent actions produce stress-energy, gauge-current, and matter-response terms through functional variation. The resulting Einstein, Yang–Mills, and matter equations have the standard normalizations, while diffeomorphism and gauge invariance yield the corresponding Ward–Noether conservation identities under explicitly stated hypotheses. The results are model-relative. They do not claim that every abstract completion is geometrically realizable, that a finite collection of curvature invariants separates arbitrary geometries, or that our universe has been shown to be a source projection. Paper IV provides the geometric bridge to the later analysis of observable quotients, projected dynamics, identifiability, and the RS2 physical witness.