Abstract

Abstract

The minimal completion constructed in Paper 3 is an abstract quotient determined by a declared readout together with a set-indexed family of invariant source relations. This paper identifies the conditions under which such a completion possesses a genuine realization in differential geometry and classical field theory. The realization domain is restricted to compatible local metric, bundle, connection, matter, and boundary data: Čech cocycle and overlap equations are used to construct global fields, and uniqueness is asserted only up to bundle isomorphism, never up to a choice of coordinate chart or gauge representative. The physical geometric source is then defined as the orbit space of admissible global configurations under the groupoid of allowed principal-bundle isomorphisms covering allowed diffeomorphisms.

Gauge- and diffeomorphism-invariant global relations, holonomy conjugacy classes, Wilson observables, characteristic numbers, boundary and asymptotic data, and spectra of geometrically defined operators with specified domains, descend to functions on this orbit space. Their joint image with a declared Tier-1 readout realizes, on the restricted domain, the canonical minimal completion of Paper 3. We prove a restricted geometric realization theorem, together with an orbit reconstruction criterion characterizing when the completion map is injective. For flat $U(1)$ connections on an oriented circle we prove that holonomy classifies gauge orbits completely, yielding the exact realization $E_{\mathrm{hol}}\cong U(1)$, and we exhibit a strictly coarser spectral completion $E_{\mathrm{spec}}\cong[-1,1]$ arising from a charged scalar relation.

The final sections embed completed relation data into variational dynamics. A gauge- and diffeomorphism-invariant relation-dependent action yields correctly normalized stress-energy, gauge-current, and matter responses; the Einstein, Yang–Mills, and matter equations follow by functional variation, and the Ward–Noether identities associated with diffeomorphism and gauge invariance enforce covariant conservation. Completed source relations therefore enter Tier-1 field equations through invariant geometric structure and derived response, not through a choice of coordinates or gauge. No claim is made that every abstract completion is geometrically realizable; the realization boundary is precisely the content of the results below.