Abstract

Abstract

Let $p\colon S\to T$ be a surjective source-to-readout map on a reduced source space, and let $\{r_\alpha\colon S\to V_\alpha\}_{\alpha\in A}$ be a set-indexed family of physically invariant source relations that need not descend along $p$. We construct the coarsest extension of the readout on which every $r_\alpha$ becomes well defined. The construction is explicit: form the joint map $j_A=(p,(r_\alpha)_{\alpha\in A})\colon S\to T\times\prod_{\alpha\in A}V_\alpha$ and set $E_A=\mathop{\mathrm{Im}}j_A$; equivalently, quotient $S$ by equality of the readout together with every selected relation value. We prove that $E_A$ is terminal in the category of surjective relation-sufficient extensions: every sufficient extension $E$ admits a unique surjection $E\to E_A$. This yields existence, uniqueness up to unique isomorphism, and a precise minimality theorem, without nominating a preferred representative of any readout fiber. We characterize redundant and indispensable relation coordinates by an exact separation criterion, construct canonical refinement maps for nested families and prove their cocycle law, prove invariance under faithful reparameterization of value spaces, and construct the completion determined directly by a family of target-answer maps, together with the canonical surjection from the relation completion onto it. We give the gauge-reduced form of the theorem (completion cannot resurrect redundancy already quotiented away) and the topological form, distinguishing carefully between the quotient topology on $E_A$ and the subspace topology inherited from the product. For flat $U(1)$ connections on a circle, whose curvature readout is identically zero, oriented transport completion returns $U(1)$, whereas the minimal completion for the charged-scalar spectral target is the inversion quotient $U(1)/(z\sim z^{-1})\cong[-1,1]$; the canonical refinement $e^{i\theta}\mapsto\cos\theta$ exhibits the general relation-to-target surjection concretely.

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