Conclusion

Conclusion

Source completion is neither arbitrary enrichment of the readout nor selection of a preferred state from each readout fiber. It is the canonical quotient of the reduced source that preserves the original readout while separating exactly those source states that the nominated family of invariant relations distinguishes, and no others. The joint-image construction proves existence; the terminal universal property, with its arrow running $E\to E_A$ from every sufficient extension onto the completion, proves minimality and uniqueness up to unique isomorphism; the indispensability criterion identifies which nominated coordinates do genuine work; the refinement tower organizes nominations by information content; the target-completion surjection separates “what the relations say” from “what the answers need”; the gauge boundary confirms that completion cannot undo reduction; and the topological theorem carries the whole construction into continuous settings without conflating the quotient and subspace topologies. What remains outside the theorem, whether a given source is real, and whether a given nomination is physical, remains outside it deliberately.