Canonical Minimal Source Completion: The Coarsest Readout Extension on Which a Nominated Family of Source Relations Becomes Well Defined

This is Paper III of the seven-paper Shadow Theory foundational series. The paper develops the canonical family-level completion problem. Given a reduced source space, a surjective readout, and a nominated family of physically invariant source relations, it constructs the coarsest extension of the readout on which every nominated relation is well defined. The construction records the original readout together with the realized values of the nominated relations, then identifies source states that agree on all of this data. The resulting completion is proved to be terminal among all surjective relation-sufficient extensions: every other extension that retains the readout and determines every nominated relation maps uniquely and surjectively onto it. This gives a rigorous minimality theorem, uniqueness up to unique isomorphism, an exact criterion for redundant and indispensable relation coordinates, and canonical refinement maps for nested relation families. The paper also distinguishes relation completion from target completion. A relation family may retain distinctions that none of the nominated target answers require; in that case the target completion is a strictly coarser quotient. The framework is extended to quotient topologies and illustrated using flat U(1) connections on a circle. Full oriented holonomy yields the circle, while the charged-scalar spectral target yields the inversion quotient, represented by the interval from minus one to one. The result is a formal set-theoretic source-quotient construction. It does not provide a procedure for recovering non-descending relations from the original readout, nor does it by itself establish that the completion is experimentally accessible or geometrically realized. Those realization questions are addressed in the subsequent paper. Paper III builds on the source–readout obstruction framework of Paper I and the target-relative completion analysis of Paper II.