# 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.
[^1]
[^1]: Website: [everythingequation.com](https://everythingequation.com)
# Introduction
## The problem inherited from Papers 1 and 2
This is the third paper of a sequence on source–readout non-equivalence. The first companion manuscript \[[1](/source-readout/canonical-minimal-source-completion/references#bib-Paper1)\] established that a readout map $p\colon S\to T$ need not admit descent for a given source-defined quantity: a function $r\colon S\to V$ factors as $r=\bar r\circ p$ if and only if $r$ is constant on every fiber $F_t=p^{-1}(t)$, and equivariant refinements of this statement produce genuine obstructions. The second companion manuscript \[[2](/source-readout/canonical-minimal-source-completion/references#bib-Paper2)\] sharpened the significance of that failure. Information destroyed by a readout is not by itself a defect; it becomes consequential only *relative to a nominated target*. Paper 2 characterized exactly when a target with correct-answer map $a\colon S\to A_Q$ is solvable from the readout alone (precisely when $a$ is $\sim_p$-invariant), and, when it is not, constructed the coarsest extension of $T$ sufficient to determine that single answer map.
What Paper 2 does not supply is the family-level construction. Physical practice almost never nominates a single answer map. One nominates a *family*: a collection of invariant source relations $r_\alpha$, indexed by a set $A$, each entering some independently specified mathematical or physical target. Assembling the single-target completions of Paper 2 one at a time does not answer the family question, because the completions do not simply compose: the object one wants is a single extension of the readout on which *all* the nominated relations become simultaneously well defined, and one wants it to be canonical, adding no distinctions beyond those the nominated family forces. Neither existence, nor minimality, nor uniqueness of such an object is formal from Paper 2.
## What this paper proves
Fix the reduced source $S$, the surjective readout $p\colon S\to T$, and a set-indexed family $r_\alpha\colon S\to V_\alpha$, $\alpha\in A$. Write $V_A=\prod_{\alpha\in A}V_\alpha$, $R_A=(r_\alpha)_{\alpha\in A}\colon S\to V_A$, and $$
j_A=(p,R_A)\colon S\longrightarrow T\times V_A,
\qquad
E_A:=\mathop{\mathrm{Im}}(j_A).\tag{1.1}$$ Let $e_A\colon S\twoheadrightarrow E_A$ be the induced surjection and let $u_A\colon E_A\to T$, $\rho_\alpha\colon E_A\to V_\alpha$ be the coordinate projections. Our results are as follows.
1. *Explicit construction and quotient realization* ([Section 3](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#sec:construction)). We have $p=u_A\circ e_A$ and $r_\alpha=\rho_\alpha\circ e_A$ for all $\alpha$, and $E_A\cong S/\!\sim_A$ where $s\sim_A s'$ iff $p(s)=p(s')$ and $r_\alpha(s)=r_\alpha(s')$ for every $\alpha\in A$. The information ordering $\Delta_S\subseteq\ \sim_A\ \subseteq\ \sim_p$ holds, and the fiber $u_A^{-1}(t)$ is canonically the set $R_A(F_t)$ of *realized relation signatures* over $t$.
2. *Terminality* ([Section 4](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#sec:category), [Section 5](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#sec:main)). We define the category of surjective relation-sufficient extensions and prove that $\mathcal{E}_A$ is *terminal* in it: for every sufficient extension $\mathcal E=(E,e,u,(\rho_\alpha^E))$ there is a *unique* morphism $\phi_{\mathcal E}\colon E\to E_A$, and it is surjective. The arrow runs $E\to E_A$, never $E_A\to E$. Elementarily, terminality is the statement $\sim_e\ \subseteq\ \sim_A$ for every sufficient extension: $E_A$ is the *coarsest* sufficient extension, not the finest.
3. *Coordinate economy* ([Section 6](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#sec:redundancy)). A nominated relation $r_\alpha$ is indispensable if and only if some pair $s,s'$ agrees on the readout and on all $r_\beta$, $\beta\neq\alpha$, yet is separated by $r_\alpha$; and deleting $r_\alpha$ strictly coarsens the completion exactly in that case.
4. *Functoriality in the family* ([Section 7](/source-readout/canonical-minimal-source-completion/nested-families-refinement-maps-and-reparameterization#sec:refinement)). For $A\subseteq B$ there is a unique surjection $\pi_{BA}\colon E_B\to E_A$ over $S$, and these satisfy $\pi_{CA}=\pi_{BA}\circ\pi_{CB}$. Faithful reparameterization of value spaces leaves the completion canonically unchanged over $T$.
5. *Target completion* ([Section 8](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#sec:targets)). The completion $E_{\mathcal Q}$ determined by a family of answer maps is constructed, and when the relations mediate the answers there is a canonical surjection $\chi\colon E_A\to E_{\mathcal Q}$, an isomorphism precisely when the two induced equivalence relations coincide. Target completion may be *strictly coarser* than relation completion.
6. *Gauge boundary and topology* ([Section 9](/source-readout/canonical-minimal-source-completion/the-gauge-reduction-boundary#sec:gauge), [Section 10](/source-readout/canonical-minimal-source-completion/the-topological-universal-property#sec:topology)). Completion occurs *after* gauge reduction and cannot undo it. In the topological setting, $E_A$ with the quotient topology is terminal among topological sufficient extensions; the canonical injection $E_A\hookrightarrow T\times V_A$ is an embedding exactly when quotient and subspace topologies agree, for which compactness of $S$ and Hausdorffness of $T\times V_A$ suffice.
7. *Worked example* ([Section 11](/source-readout/canonical-minimal-source-completion/worked-example-flat-u-1-connections-on-the-circle#sec:u1)) and *exact reconstruction scope* ([Section 12](/source-readout/canonical-minimal-source-completion/the-exact-scope-of-reconstruction#sec:reconstruction)): $e_A$ is injective iff $\{p\}\cup\{r_\alpha\}$ separates points of $S$.
## What this paper does not prove
The theorem is *model-relative*. Once a reduced source space, a readout, and a nominated family of invariant relations or targets are specified, there is a canonical coarsest sufficient quotient. Nothing here shows that the source space is empirically real, that observed physics is a projection of a hidden source, that every lost distinction must be restored, that the full source is in general recoverable, or that any particular relation family is physically privileged absent independent justification. In particular, “minimal completion” is not “reconstruction of the source”: the two coincide only under the separation criterion of [Proposition 12.1](/source-readout/canonical-minimal-source-completion/the-exact-scope-of-reconstruction#prop:reconstruction). We return to these limits in [Section 13](/source-readout/canonical-minimal-source-completion/discussion-and-scope#sec:discussion).
## Notation
We keep the frozen notation of the companion manuscripts. $S$ is the reduced source space, $T$ the readout space, $p\colon S\to T$ the surjective readout, and $F_t=p^{-1}(t)$ the readout fiber over $t\in T$. Throughout, $A$ denotes a *set* indexing the nominated relation family; products $V_A=\prod_{\alpha\in A}V_\alpha$ are ordinary set-theoretic products, so that a point of $V_A$ is a choice function $\alpha\mapsto v_\alpha\in V_\alpha$, and no cardinality restriction on $A$ is needed. We write $\Delta_S=\{(s,s):s\in S\}$ for the equality relation on $S$ and $\sim_p$ for equality of the readout. For equivalence relations, $\sim\ \subseteq\ \approx$ means that $\sim$ is *finer* (identifies fewer pairs, retains more information) and $\approx$ is *coarser*.
# Reduced source, readout, and the nominated relation family
## The reduced source
We assume, as in Paper 1, that $S$ has already been reduced: all declared gauge redundancy and all coordinate-labelling redundancy have been quotiented out. Concretely, one begins with a raw configuration space $\Omega$ carrying an equivalence relation $\simeq_{\mathrm{phys}}$ of physical indistinguishability, gauge equivalence, diffeomorphism relabelling, choice of chart, and forms $$
q\colon\Omega\longrightarrow S=\Omega/\!\simeq_{\mathrm{phys}}.\tag{2.1}$$ The order of operations matters and is not a convention we are free to reverse: *reduction precedes completion*. [Section 9](/source-readout/canonical-minimal-source-completion/the-gauge-reduction-boundary#sec:gauge) states the consequence precisely.
**Definition 2.1** (Eligible relation).
A raw quantity $\widetilde r\colon\Omega\to V$ is *eligible* for source completion when it is invariant on physical-equivalence classes, $$
\omega\simeq_{\mathrm{phys}}\omega'
\quad\Longrightarrow\quad
\widetilde r(\omega)=\widetilde r(\omega'),\tag{2.2}$$ so that there is a unique $r\colon S\to V$ with $\widetilde r=r\circ q$. All relations nominated below are assumed eligible, i.e. genuinely defined on the reduced source $S$.
**Definition 2.2** (The family completion problem).
An extension of the readout *sufficient* for the family [(2.3)](/source-readout/canonical-minimal-source-completion/reduced-source-readout-and-the-nominated-relation-family#eq:family) is a space $E$ through which $S$ maps onto, such that
1. the original readout is retained: $p$ factors through $E$; and
2. every nominated relation becomes a well-defined function of the completed state: each $r_\alpha$ factors through $E$.
The problem is to construct the *coarsest* such extension: one that makes no distinction between source states beyond those forced by requirements (1) and (2), and to prove it canonical.
**Definition 3.1** (Minimal relation completion).
Define the *joint map* $$
j_A\colon S\longrightarrow T\times V_A,
\qquad
j_A(s)=\bigl(p(s),R_A(s)\bigr),\tag{3.1}$$ and define the *minimal relation completion* $$
E_A:=\mathop{\mathrm{Im}}(j_A)\ \subseteq\ T\times V_A .\tag{3.2}$$ Let $e_A\colon S\twoheadrightarrow E_A$, $e_A(s):=j_A(s)$, be the induced surjection, and let $$
u_A\colon E_A\longrightarrow T,
\qquad
\rho_\alpha\colon E_A\longrightarrow V_\alpha
\qquad(\alpha\in A)\tag{3.3}$$ be the restrictions to $E_A$ of the coordinate projections of $T\times V_A$.
**Proposition 3.2** (Factorization).
*The maps of [Definition 3.1](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#def:EA) satisfy $$
p=u_A\circ e_A,
\qquad
r_\alpha=\rho_\alpha\circ e_A
\qquad(\alpha\in A),\tag{3.4}$$ and $e_A$ is surjective. In particular every nominated relation descends to $E_A$.*
*Proof.* Surjectivity of $e_A$ is immediate, since $E_A$ was defined as the image of $j_A$ and $e_A$ is $j_A$ with codomain restricted to that image. For $s\in S$ we have $e_A(s)=(p(s),(r_\alpha(s))_{\alpha\in A})$. Applying the first coordinate projection gives $u_A(e_A(s))=p(s)$, and applying the projection onto the $\alpha$-th factor of $V_A$ gives $\rho_\alpha(e_A(s))=r_\alpha(s)$. As $s$ was arbitrary, [(3.4)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:factorization) holds as an identity of functions on $S$. ◻
Thus $(E_A,e_A,u_A,(\rho_\alpha)_{\alpha\in A})$ solves the *existence* half of [Definition 2.2](/source-readout/canonical-minimal-source-completion/reduced-source-readout-and-the-nominated-relation-family#def:problem). The content of the paper is that it solves the *minimality* and *uniqueness* halves as well.
## The induced equivalence relation
**Definition 3.3**.
Define a relation $\sim_A$ on $S$ by $$
s\sim_A s'
\quad:\Longleftrightarrow\quad
\bigl[\,p(s)=p(s')\ \text{ and }\ r_\alpha(s)=r_\alpha(s')\ \text{ for every }\alpha\in A\,\bigr].\tag{3.5}$$
**Proposition 3.4** (Quotient realization).
*The assignment $$
\kappa\colon S/\!\sim_A\ \longrightarrow\ E_A,
\qquad
\kappa\bigl([s]_A\bigr):=j_A(s)\tag{3.6}$$ is a well-defined bijection, and it satisfies $\kappa\circ\pi_A=e_A$, where $\pi_A\colon S\to S/\!\sim_A$ is the canonical projection. Hence $$
E_A\ \cong\ S/\!\sim_A\tag{3.7}$$ canonically.*
*Proof.* We verify the three assertions separately, rather than invoking a generic isomorphism principle for quotients, since the well-definedness step is precisely where the definition of the relation $\sim_A$ earns its keep.
*Well-definedness.* Suppose $[s]_A=[s']_A$, i.e. $s\sim_A s'$. By [(3.5)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:simA) this means $p(s)=p(s')$ and $r_\alpha(s)=r_\alpha(s')$ for every $\alpha\in A$. Two elements of the product $T\times V_A$ are equal precisely when all of their coordinates agree; and here the coordinates of $j_A(s)$ and $j_A(s')$ are $p(s),p(s')$ in the $T$-factor and $r_\alpha(s),r_\alpha(s')$ in the $\alpha$-th factor of $V_A$, respectively. All agree, so $j_A(s)=j_A(s')$. Hence the value assigned by [(3.6)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:kappa) does not depend on the chosen representative, and $\kappa$ is a well-defined function.
*Injectivity.* Suppose $\kappa([s]_A)=\kappa([s']_A)$, i.e. $j_A(s)=j_A(s')$. Comparing coordinates in $T\times V_A$ gives $p(s)=p(s')$ and $r_\alpha(s)=r_\alpha(s')$ for every $\alpha\in
A$, which is exactly [(3.5)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:simA). Hence $s\sim_A s'$ and $[s]_A=[s']_A$.
*Surjectivity.* Let $x\in E_A$. By [Definition 3.1](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#def:EA), $E_A=\mathop{\mathrm{Im}}(j_A)$, so there is $s\in S$ with $x=j_A(s)=\kappa([s]_A)$.
Finally, for every $s\in S$ we have $\kappa(\pi_A(s))=\kappa([s]_A)=j_A(s)=e_A(s)$, so $\kappa\circ\pi_A=e_A$. ◻
So the joint image and the quotient by “agreement of the readout and of every nominated relation” are the same object, viewed extensionally and intensionally. The first description makes existence manifest; the second makes minimality manifest.
## The information ordering
**Proposition 3.5** (Information ordering).
*As relations on $S$, $$
\Delta_S\ \subseteq\ \sim_A\ \subseteq\ \sim_p ,\tag{3.8}$$ and consequently there is a factorization of the readout, $$
S\ \xrightarrow{\ e_A\ }\ E_A\ \xrightarrow{\ u_A\ }\ T .\tag{3.9}$$*
*Proof.* The first inclusion is reflexivity: $s\sim_A s$ for every $s$, since $p(s)=p(s)$ and $r_\alpha(s)=r_\alpha(s)$. The second holds because the defining condition [(3.5)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:simA) contains the clause $p(s)=p(s')$ as a conjunct; hence $s\sim_A s'$ implies $s\sim_p s'$. The factorization [(3.9)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:tower) is [(3.4)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:factorization). ◻
The reading of [(3.8)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:ordering) must be got right, because the direction of the inclusions encodes the direction of every universal arrow in the paper. A *smaller* equivalence relation identifies *fewer* pairs, hence corresponds to a *finer* partition and to *more* retained information. Thus [(3.8)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:ordering) says: equality on $S$ retains everything; $\sim_A$ retains at most that, and at least the readout; $\sim_p$ retains the least of the three. The completion $E_A$ refines the readout partition, but, this is the whole point of minimality, it need not refine it all the way down to individual source states.
## What completion adds above a readout value
**Proposition 3.6** (Completed fibers).
*For every $t\in T$ there is a canonical bijection $$
u_A^{-1}(t)\ \cong\ R_A(F_t)=\{\,R_A(s)\ :\ s\in F_t=p^{-1}(t)\,\}\ \subseteq\ V_A .\tag{3.10}$$*
*Proof.* Consider the second-coordinate projection $\mathrm{pr}_{V_A}\colon T\times V_A\to V_A$ and restrict it to $u_A^{-1}(t)$.
*Its image is $R_A(F_t)$.* An element $x\in u_A^{-1}(t)$ lies in $E_A=\mathop{\mathrm{Im}}(j_A)$, so $x=(p(s),R_A(s))$ for some $s\in S$; and $u_A(x)=t$ forces $p(s)=t$, i.e. $s\in F_t$. Hence $\mathrm{pr}_{V_A}(x)=R_A(s)\in R_A(F_t)$. Conversely, given $v=R_A(s)$ with $s\in F_t$, the point $x=(t,v)=j_A(s)$ lies in $E_A$ and satisfies $u_A(x)=t$, so $v$ is attained.
*It is injective on $u_A^{-1}(t)$.* If $x,x'\in u_A^{-1}(t)$ have the same $V_A$-coordinate, then they also share the $T$-coordinate, namely $t$; being elements of a product with all coordinates equal, $x=x'$.
Thus $\mathrm{pr}_{V_A}$ restricts to a bijection $u_A^{-1}(t)\to R_A(F_t)$, and this bijection was constructed without any choice. ◻
[Proposition 3.6](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#prop:fibers) answers exactly the question that “completion” invites one to ask badly. What sits above a readout value $t$ in $E_A$ is *not* a preferred source state in $F_t$, and not an arbitrary enrichment of $t$ by extra labels. It is the set of *realized relation signatures* over $t$: the distinct values that the nominated family actually takes on the fiber. Two source states in $F_t$ that agree on every $r_\alpha$ contribute the *same* completed state; nothing in the construction selects between them, and nothing in the construction can. We record this as a standing prohibition, because it is the failure mode that a naive completion invites.
# The category of relation-sufficient extensions
We now fix the data $(S,T,p,\{r_\alpha\}_{\alpha\in A})$ for the remainder of the paper and make [Definition 2.2](/source-readout/canonical-minimal-source-completion/reduced-source-readout-and-the-nominated-relation-family#def:problem) precise as a category.
**Definition 4.1** (Relation-sufficient extension).
A *relation-sufficient extension* of $p$ (relative to the nominated family) is a tuple $$
\mathcal E=\bigl(E,\ e,\ u,\ (\rho_\alpha^E)_{\alpha\in A}\bigr)\tag{4.1}$$ consisting of a set $E$, a **surjection** $e\colon S\twoheadrightarrow E$, and maps $u\colon E\to T$ and $\rho_\alpha^E\colon E\to V_\alpha$ ($\alpha\in A$) satisfying $$
p=u\circ e,
\qquad
r_\alpha=\rho_\alpha^E\circ e
\qquad(\alpha\in A).\tag{4.2}$$
**Definition 4.2** (Morphism).
Let $\mathcal E=(E,e,u,(\rho_\alpha^E))$ and $\mathcal E'=(E',e',u',(\rho_\alpha^{E'}))$ be relation-sufficient extensions. A *morphism* $f\colon\mathcal E\to\mathcal E'$ is a function $f\colon E\to E'$ satisfying $$
e'=f\circ e .\tag{4.3}$$
**Lemma 4.3** (Structure maps are automatically preserved).
*If $f\colon\mathcal E\to\mathcal E'$ is a morphism in the sense of [Definition 4.2](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#def:morphism), then $$
u'\circ f=u,
\qquad
\rho_\alpha^{E'}\circ f=\rho_\alpha^E
\qquad(\alpha\in A).\tag{4.4}$$*
*Proof.* Let $x\in E$. Since $e$ is surjective, choose $s\in S$ with $x=e(s)$. Then, using [(4.3)](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#eq:morphism-over-S) and the factorizations [(4.2)](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#eq:suff-factorization) for $\mathcal E'$ and then for $\mathcal E$, $$u'\bigl(f(x)\bigr)=u'\bigl(f(e(s))\bigr)=u'\bigl(e'(s)\bigr)=p(s)=u\bigl(e(s)\bigr)=u(x),$$ and likewise, for each $\alpha\in A$, $$\rho_\alpha^{E'}\bigl(f(x)\bigr)
=\rho_\alpha^{E'}\bigl(e'(s)\bigr)
=r_\alpha(s)
=\rho_\alpha^{E}\bigl(e(s)\bigr)
=\rho_\alpha^{E}(x).$$ Both computations are independent of the chosen preimage $s$, since their left-hand sides do not mention $s$. As $x$ was arbitrary, [(4.4)](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#eq:structure-preserved) holds. ◻
We display [(4.4)](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#eq:structure-preserved) rather than merely deducing it, because it records what a morphism of completions actually preserves: the readout and every nominated relation value. What is *not* imposed is any compatibility beyond that, no metric, no ordering, no choice of labels.
**Proposition 4.4** (This is a category).
*Relation-sufficient extensions of $p$ relative to $\{r_\alpha\}_{\alpha\in A}$, with the morphisms of [Definition 4.2](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#def:morphism), form a category, which we denote $\mathbf{Suff}(p;\{r_\alpha\})$.*
*Proof.* For $\mathcal E=(E,e,u,(\rho_\alpha^E))$, the identity function $\mathop{\mathrm{id}}_E$ satisfies $e=\mathop{\mathrm{id}}_E\circ e$, so it is a morphism $\mathcal E\to\mathcal E$. If $f\colon\mathcal E\to\mathcal E'$ and $g\colon\mathcal E'\to\mathcal E''$ are morphisms, then $(g\circ f)\circ e=g\circ(f\circ e)=g\circ e'=e''$, so $g\circ f\colon\mathcal E\to\mathcal E''$ is a morphism. Associativity of composition and the unit laws are inherited from composition of functions. ◻
We add no further categorical apparatus; none is needed.
# The canonical minimal-completion theorem
**Theorem 5.1** (Canonical minimal completion).
*The tuple $$
\mathcal E_A=\bigl(E_A,\ e_A,\ u_A,\ (\rho_\alpha)_{\alpha\in A}\bigr)\tag{5.1}$$ of [Definition 3.1](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#def:EA) is an object of $\mathbf{Suff}(p;\{r_\alpha\})$, and it is **terminal**: for every relation-sufficient extension $\mathcal E=(E,e,u,(\rho_\alpha^E))$ there exists a **unique** morphism $$
\phi_{\mathcal E}\colon\ \mathcal E\longrightarrow\mathcal E_A,
\qquad\text{that is,}\qquad
\phi_{\mathcal E}\colon E\longrightarrow E_A .\tag{5.2}$$ It is given by the explicit formula $$
\phi_{\mathcal E}(x)=\Bigl(u(x),\ \bigl(\rho_\alpha^E(x)\bigr)_{\alpha\in A}\Bigr),
\qquad x\in E,\tag{5.3}$$ and $\phi_{\mathcal E}$ is surjective.*
*Proof.* *$\mathcal E_A$ is an object.* $e_A$ is surjective and the factorizations [(3.4)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:factorization) hold, by [Proposition 3.2](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#prop:factorization).
Now fix a relation-sufficient extension $\mathcal E=(E,e,u,(\rho_\alpha^E))$ and define $\phi_{\mathcal E}$ by the formula [(5.3)](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#eq:phiE). We establish the required properties in turn.
*(i) The formula is well defined as a function $E\to T\times V_A$.* For each $x\in E$ the value $u(x)\in T$ is determined, and for each $\alpha\in A$ the value $\rho_\alpha^E(x)\in V_\alpha$ is determined; the assignment $\alpha\mapsto\rho_\alpha^E(x)$ is therefore an element of the set-theoretic product $V_A=\prod_{\alpha\in A}V_\alpha$. Hence [(5.3)](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#eq:phiE) assigns to each $x\in E$ exactly one element of $T\times V_A$. No choice is involved, and no representative of $x$ is used: the formula depends only on $x$ through the given structure maps of $\mathcal E$.
*(ii) The image lies in $E_A$, and $e_A=\phi_{\mathcal E}\circ e$.* Let $x\in E$. By surjectivity of $e$ there is $s\in S$ with $x=e(s)$. Then, using [(4.2)](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#eq:suff-factorization), $$
\phi_{\mathcal E}(x)
=\phi_{\mathcal E}\bigl(e(s)\bigr)
=\Bigl(u\bigl(e(s)\bigr),\ \bigl(\rho_\alpha^E(e(s))\bigr)_{\alpha\in A}\Bigr)
=\Bigl(p(s),\ \bigl(r_\alpha(s)\bigr)_{\alpha\in A}\Bigr)
=j_A(s)
=e_A(s).\tag{5.4}$$ The right-hand side lies in $\mathop{\mathrm{Im}}(j_A)=E_A$ by [Definition 3.1](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#def:EA). Since $x\in E$ was arbitrary, $\phi_{\mathcal E}(E)\subseteq E_A$, so $\phi_{\mathcal E}$ may be regarded as a map $E\to E_A$; and reading [(5.4)](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#eq:phiE-computation) from the ends, $\phi_{\mathcal E}(e(s))=e_A(s)$ for every $s\in
S$, i.e. $$
e_A=\phi_{\mathcal E}\circ e .\tag{5.5}$$ Note that the computation [(5.4)](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#eq:phiE-computation) is legitimate for *every* choice of preimage $s$ of $x$, and its left-hand side does not depend on $s$; so no consistency question arises.
*(iii) It is a morphism, and preserves all structure maps.* Equation [(5.5)](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#eq:phiE-over-S) is precisely condition [(4.3)](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#eq:morphism-over-S) of [Definition 4.2](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#def:morphism), so $\phi_{\mathcal E}$ is a morphism $\mathcal E\to\mathcal E_A$. Preservation of the structure maps then follows from [Lemma 4.3](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#lem:structure-preserved), but in the present case it is also visible directly from [(5.3)](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#eq:phiE): applying the coordinate projections $u_A$ and $\rho_\alpha$ to $\phi_{\mathcal E}(x)=\bigl(u(x),(\rho_\beta^E(x))_\beta\bigr)$ gives $$
u_A\circ\phi_{\mathcal E}=u,
\qquad
\rho_\alpha\circ\phi_{\mathcal E}=\rho_\alpha^E
\qquad(\alpha\in A).\tag{5.6}$$
*(iv) Surjectivity.* Let $y\in E_A$. By [Proposition 3.2](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#prop:factorization) the map $e_A$ is surjective, so $y=e_A(s)$ for some $s\in S$. By [(5.5)](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#eq:phiE-over-S), $y=\phi_{\mathcal E}(e(s))$, and $e(s)\in E$. Hence $\phi_{\mathcal E}$ is onto $E_A$.
*(v) Uniqueness.* Suppose $\psi\colon\mathcal E\to\mathcal E_A$ is any morphism, i.e. any function $\psi\colon E\to E_A$ with $e_A=\psi\circ e$. Let $x\in E$ and choose $s\in S$ with $x=e(s)$. Then $$\psi(x)=\psi\bigl(e(s)\bigr)=e_A(s)=\phi_{\mathcal E}\bigl(e(s)\bigr)=\phi_{\mathcal E}(x),$$ using $e_A=\psi\circ e$ for the second equality and [(5.5)](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#eq:phiE-over-S) for the third. Since $x$ was arbitrary, $\psi=\phi_{\mathcal E}$. (This is exactly where surjectivity of $e$ is consumed: it guarantees that the condition $e_A=\psi\circ e$ pins down $\psi$ at *every* point of $E$, not merely on a subset.) ◻
**Corollary 5.2** (Uniqueness up to unique isomorphism).
*Any two terminal objects of the category $\mathbf{Suff}(p;\{r_\alpha\})$ are isomorphic by a unique isomorphism, and this isomorphism commutes with the completion maps from $S$ and with all of the structure maps. In particular, any relation-sufficient extension that satisfies the universal property of [Theorem 5.1](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#thm:terminal) is canonically isomorphic to $\mathcal E_A$.*
*Proof.* Let $\mathcal E,\mathcal E'$ be terminal. Terminality of $\mathcal E'$ gives a unique morphism $f\colon\mathcal E\to\mathcal E'$; terminality of $\mathcal E$ gives a unique morphism $g\colon\mathcal E'\to\mathcal E$. Then $g\circ f\colon\mathcal E\to\mathcal E$ is a morphism, as is $\mathop{\mathrm{id}}_{\mathcal E}$; by terminality of $\mathcal E$ there is only one morphism $\mathcal E\to\mathcal E$, so $g\circ f=\mathop{\mathrm{id}}_E$. Symmetrically $f\circ g=\mathop{\mathrm{id}}_{E'}$. Hence $f$ is an isomorphism, and it is unique because morphisms into a terminal object are unique. Compatibility with the completion maps is [(4.3)](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#eq:morphism-over-S); compatibility with the structure maps is [Lemma 4.3](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#lem:structure-preserved). ◻
**Corollary 5.3** (Coarsest sufficiency, elementary form).
*Let $\mathcal E=(E,e,u,(\rho_\alpha^E))$ be any relation-sufficient extension, and define its induced equivalence relation by $$
s\sim_e s'\quad:\Longleftrightarrow\quad e(s)=e(s').\tag{5.7}$$ Then $$
\sim_e\ \subseteq\ \sim_A .\tag{5.8}$$ Equivalently: every sufficient extension separates at least every pair that $E_A$ separates, while $E_A$ separates no pair that sufficiency does not force it to separate.*
*Proof.* Suppose $e(s)=e(s')$. Applying $u$ and using $p=u\circ e$, $$p(s)=u\bigl(e(s)\bigr)=u\bigl(e(s')\bigr)=p(s') .$$ Applying $\rho_\alpha^E$ and using $r_\alpha=\rho_\alpha^E\circ e$, for each $\alpha\in A$, $$r_\alpha(s)=\rho_\alpha^E\bigl(e(s)\bigr)=\rho_\alpha^E\bigl(e(s')\bigr)=r_\alpha(s') .$$ By [(3.5)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:simA) this is precisely $s\sim_A s'$. ◻
[Corollary 5.3](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#cor:coarsest) and [Theorem 5.1](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#thm:terminal) are two faces of one statement. Terminality is the categorical form; the inclusion [(5.8)](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#eq:coarsest) is the elementary, checkable form. The inclusion also settles the direction question once and for all: since $\sim_e$ is contained in $\sim_A$, the partition induced by $e$ *refines* the partition induced by $e_A$, so there is a well-defined map from the finer quotient onto the coarser one, from $E$ onto $E_A$, and not back.
# Redundant and indispensable relation coordinates
The nominated family may be wasteful: one of its members may already be determined by the readout together with the others, in which case its presence in the list changes nothing. We characterize this exactly.
Throughout this section fix $\alpha\in A$ and write $$
A_{-\alpha}:=A\setminus\{\alpha\},
\qquad
R_{A_{-\alpha}}=(r_\beta)_{\beta\in A_{-\alpha}}\colon S\to V_{A_{-\alpha}} .\tag{6.1}$$
**Definition 6.1** (Redundancy and indispensability).
The relation $r_\alpha$ is *redundant relative to $p$ and the remaining relations* if it factors through their joint map, i.e. if there exists a function $f_\alpha\colon T\times V_{A_{-\alpha}}\to V_\alpha$ with $$
r_\alpha=f_\alpha\circ\bigl(p,\ R_{A_{-\alpha}}\bigr).\tag{6.2}$$ It is *indispensable* if no such factorization exists.
**Theorem 6.2** (Indispensability criterion).
*The following are equivalent.*
1.
*$r_\alpha$ is indispensable.*
2.
*There exist $s,s'\in S$ with $$
p(s)=p(s'),
\qquad
r_\beta(s)=r_\beta(s')\quad\text{for every }\beta\in A_{-\alpha},
\qquad
r_\alpha(s)\neq r_\alpha(s').\tag{6.3}$$*
*Proof.* We prove the contrapositive equivalence: $r_\alpha$ is redundant if and only if no pair [(6.3)](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#eq:witness) exists, i.e. if and only if $r_\alpha$ is constant on every fiber of the joint map $g:=(p,R_{A_{-\alpha}})\colon S\to T\times V_{A_{-\alpha}}$. This is the descent statement of Paper 1 \[[1](/source-readout/canonical-minimal-source-completion/references#bib-Paper1)\] applied to $g$; we give the argument in full.
*Redundant $\Rightarrow$ constant on fibers of $g$.* Suppose [(6.2)](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#eq:redundant) holds and suppose $g(s)=g(s')$, i.e. $p(s)=p(s')$ and $r_\beta(s)=r_\beta(s')$ for all $\beta\neq\alpha$. Applying $f_\alpha$ to the equal values $g(s)=g(s')$ gives $r_\alpha(s)=f_\alpha(g(s))=f_\alpha(g(s'))=r_\alpha(s')$. Hence no pair as in [(6.3)](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#eq:witness) can exist.
*Constant on fibers of $g$ $\Rightarrow$ redundant.* Suppose that $g(s)=g(s')$ implies $r_\alpha(s)=r_\alpha(s')$. Define $f_\alpha$ on the image of $g$ by $$
f_\alpha\bigl(g(s)\bigr):=r_\alpha(s)\qquad(s\in S),\tag{6.4}$$ The prescription [(6.4)](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#eq:falpha-def) is well defined on $\mathop{\mathrm{Im}}(g)$ precisely by the assumed fiber-constancy: if $g(s)=g(s')$ then the two prescribed values $r_\alpha(s)$ and $r_\alpha(s')$ agree. Every point of $\mathop{\mathrm{Im}}(g)$ is of the form $g(s)$, so $f_\alpha$ is defined on all of $\mathop{\mathrm{Im}}(g)$. It remains to extend $f_\alpha$ to the whole of $T\times V_{A_{-\alpha}}$, and we treat the two cases explicitly.
*Case $S\neq\varnothing$.* Fix any $s_0\in S$ and define $f_\alpha(y):=r_\alpha(s_0)$ for every $y\in\bigl(T\times V_{A_{-\alpha}}\bigr)\setminus\mathop{\mathrm{Im}}(g)$. This is legitimate because $r_\alpha(s_0)$ is an element of $V_\alpha$, so $V_\alpha\neq\varnothing$ and a value is available. The extension changes nothing on $\mathop{\mathrm{Im}}(g)$, where [(6.4)](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#eq:falpha-def) already fixes $f_\alpha$.
*Case $S=\varnothing$.* Then $T=\varnothing$: the readout $p\colon S\to T$ is surjective by standing hypothesis, and the image of the empty map is empty, so $T=p(S)=\varnothing$. Hence the product $T\times V_{A_{-\alpha}}=\varnothing$ as well, and the unique map $\varnothing\to V_\alpha$, the empty map, serves as $f_\alpha$. It satisfies $r_\alpha=f_\alpha\circ g$ vacuously as an identity of maps out of $S=\varnothing$, both sides being the empty map. So $r_\alpha$ is redundant in this case, which is consistent: no witness pair [(6.3)](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#eq:witness) can exist in an empty source, so $r_\alpha$ is not indispensable, exactly as the equivalence requires. (The two sides of the criterion agree here; the case is not merely set aside.)
In either case $f_\alpha$ is defined on all of $T\times V_{A_{-\alpha}}$ and satisfies $r_\alpha=f_\alpha\circ g$, which is [(6.2)](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#eq:redundant).
Finally, “no pair as in [(6.3)](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#eq:witness)” is by inspection the same statement as “$g(s)=g(s')$ implies $r_\alpha(s)=r_\alpha(s')$”, since the first two clauses of [(6.3)](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#eq:witness) say $g(s)=g(s')$ and the third says $r_\alpha(s)\neq r_\alpha(s')$. Negating both sides of the equivalence just proved yields $\Leftrightarrow$. ◻
The criterion is a genuine separation test, and it is the reason a long list of nominated relations is not thereby an economical one. A coordinate earns its place only by separating some pair of source states that the readout and all the *other* coordinates identify.
**Corollary 6.3** (Strict coarsening on deletion).
*Let $\sim_A$ and $\sim_{A_{-\alpha}}$ be the equivalence relations [(3.5)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:simA) induced by the full family and by the family with $r_\alpha$ deleted, and let $\pi\colon E_A\to E_{A_{-\alpha}}$ be the canonical surjection of [Proposition 7.1](/source-readout/canonical-minimal-source-completion/nested-families-refinement-maps-and-reparameterization#prop:refinement) below (with $B=A$). Then always $\sim_A\ \subseteq\ \sim_{A_{-\alpha}}$, and the following are equivalent:*
1. *the inclusion is *strict*: $\sim_A\ \subsetneq\ \sim_{A_{-\alpha}}$;*
2. *$\pi$ is not injective, i.e. deleting $r_\alpha$ *strictly coarsens* the completion;*
3. *$r_\alpha$ is indispensable.*
*Proof.* The inclusion $\sim_A\subseteq\sim_{A_{-\alpha}}$ is immediate from [(3.5)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:simA), since the defining condition for $\sim_A$ is the condition for $\sim_{A_{-\alpha}}$ conjoined with the extra clause $r_\alpha(s)=r_\alpha(s')$.
*(1) $\Leftrightarrow$ (3).* Strictness means: there exist $s,s'$ with $s\sim_{A_{-\alpha}}s'$ but not $s\sim_A s'$. Unwinding [(3.5)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:simA), this says $p(s)=p(s')$ and $r_\beta(s)=r_\beta(s')$ for all $\beta\neq\alpha$, while $r_\alpha(s)\neq r_\alpha(s')$, which is exactly the witness [(6.3)](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#eq:witness). By [Theorem 6.2](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#thm:indispensable) this holds iff $r_\alpha$ is indispensable.
*(1) $\Leftrightarrow$ (2).* By [Proposition 3.4](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#prop:quotient), $E_A\cong S/\!\sim_A$ and $E_{A_{-\alpha}}\cong S/\!\sim_{A_{-\alpha}}$, and under these identifications $\pi$ is the canonical map $S/\!\sim_A\ \to\ S/\!\sim_{A_{-\alpha}}$ sending $[s]_A\mapsto[s]_{A_{-\alpha}}$, which is well defined by the inclusion just proved. This map is injective iff $[s]_{A_{-\alpha}}=[s']_{A_{-\alpha}}$ implies $[s]_A=[s']_A$, i.e. iff $\sim_{A_{-\alpha}}\ \subseteq\ \sim_A$, i.e. (given the reverse inclusion) iff the two relations are equal. Its failure is precisely strictness. ◻
# Nested families, refinement maps, and reparameterization
## Nested families
Let $A\subseteq B$ be sets of indices, the $A$-relations being a subfamily of the $B$-relations (i.e. the relations indexed by $A$ are the same functions in both families). Then, directly from [(3.5)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:simA), $$
\sim_B\ \subseteq\ \sim_A ,\tag{7.1}$$ since agreement of *all* $B$-relations (and the readout) entails agreement of the sub-collection indexed by $A$ (and the readout). In words: a larger nomination is at least as demanding, hence yields an at least as fine completion.
**Proposition 7.1** (Canonical refinement maps).
*Let $A\subseteq B$. There is a unique map $$
\pi_{BA}\colon E_B\longrightarrow E_A
\qquad\text{with}\qquad
e_A=\pi_{BA}\circ e_B ,\tag{7.2}$$ and it is surjective. It satisfies $u_A\circ\pi_{BA}=u_B$ and $\rho_\alpha^{(A)}\circ\pi_{BA}=\rho_\alpha^{(B)}$ for every $\alpha\in A$, where the superscripts indicate the completion in which the structure map lives. Moreover $\pi_{AA}=\mathop{\mathrm{id}}_{E_A}$, and for $A\subseteq B\subseteq C$, $$
\pi_{CA}=\pi_{BA}\circ\pi_{CB}.\tag{7.3}$$*
*Proof.* *Existence and uniqueness.* We exhibit $E_B$ as a relation-sufficient extension *for the $A$-family* and invoke [Theorem 5.1](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#thm:terminal). Indeed, $e_B\colon S\twoheadrightarrow E_B$ is surjective; $u_B\colon E_B\to T$ satisfies $p=u_B\circ e_B$; and for each $\alpha\in A\subseteq B$ the structure map $\rho_\alpha^{(B)}\colon E_B\to V_\alpha$ satisfies $r_\alpha=\rho_\alpha^{(B)}\circ e_B$. Hence $\bigl(E_B,e_B,u_B,(\rho_\alpha^{(B)})_{\alpha\in A}\bigr)$ is an object of $\mathbf{Suff}(p;\{r_\alpha\}_{\alpha\in A})$. [Theorem 5.1](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#thm:terminal) supplies a unique morphism from it to the terminal object $\mathcal E_A$, i.e. a unique map $\pi_{BA}\colon E_B\to E_A$ with $e_A=\pi_{BA}\circ e_B$, and asserts that it is surjective. Compatibility with $u$ and with the $\rho_\alpha$, $\alpha\in A$, is [Lemma 4.3](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#lem:structure-preserved) (equivalently [(5.6)](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#eq:phiE-structure)).
*Identity.* For $B=A$, the identity $\mathop{\mathrm{id}}_{E_A}$ satisfies $e_A=\mathop{\mathrm{id}}_{E_A}\circ e_A$; by the uniqueness just established, $\pi_{AA}=\mathop{\mathrm{id}}_{E_A}$.
*Cocycle law.* Let $A\subseteq B\subseteq C$. Both $\pi_{CA}$ and $\pi_{BA}\circ\pi_{CB}$ are maps $E_C\to E_A$. Compose each with the surjection $e_C$: $$\pi_{CA}\circ e_C=e_A,
\qquad
(\pi_{BA}\circ\pi_{CB})\circ e_C
=\pi_{BA}\circ(\pi_{CB}\circ e_C)
=\pi_{BA}\circ e_B
=e_A .$$ Two maps out of $E_C$ that agree after precomposition with the surjection $e_C$ agree on all of $E_C$: given $x\in E_C$, write $x=e_C(s)$ and evaluate. Hence [(7.3)](/source-readout/canonical-minimal-source-completion/nested-families-refinement-maps-and-reparameterization#eq:cocycle). (Alternatively, both maps are morphisms $\mathcal E_C\to\mathcal E_A$ in $\mathbf{Suff}(p;\{r_\alpha\}_{\alpha\in
A})$, and [Theorem 5.1](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#thm:terminal) says there is only one.) ◻
Nested nominations therefore generate a coherent tower of completions, $$
S\ \longrightarrow\ E_C\ \xrightarrow{\ \pi_{CB}\ }\ E_B\ \xrightarrow{\ \pi_{BA}\ }\ E_A\
\xrightarrow{\ u_A\ }\ T
\qquad(A\subseteq B\subseteq C),\tag{7.4}$$ in which each arrow forgets exactly the distinctions that the smaller nomination does not require. This is a hierarchy of *information content*. Its arrows run from finer to coarser, and by [Theorem 5.1](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#thm:terminal) each is uniquely determined by the pair of nominations.
## Faithful reparameterization
The completion depends on the *distinctions* carried by the nominated relations, not on the labels in which their values happen to be recorded. We make this precise.
**Proposition 7.3** (Invariance under faithful reparameterization).
*For each $\alpha\in A$ let $f_\alpha\colon V_\alpha\to W_\alpha$ be a map whose restriction to the realized subset $r_\alpha(S)\subseteq V_\alpha$ is injective, and set $$
r'_\alpha:=f_\alpha\circ r_\alpha\colon S\to W_\alpha .\tag{7.5}$$ Let $\sim_A$ and $\sim'_A$ be the equivalence relations induced by $\{r_\alpha\}$ and by $\{r'_\alpha\}$ (with the same readout $p$), and let $E_A$, $E'_A$ be the corresponding completions. Then $$
\sim_A\ =\ \sim'_A ,\tag{7.6}$$ and consequently there is a canonical bijection $E_A\cong E'_A$ commuting with the completion maps from $S$ and with the projections to $T$: $$
u'_A\circ\bigl(E_A\xrightarrow{\ \cong\ }E'_A\bigr)=u_A .\tag{7.7}$$*
*Proof.* *$\sim_A\subseteq\sim'_A$.* Suppose $s\sim_A s'$. Then $p(s)=p(s')$, and for each $\alpha$, $r_\alpha(s)=r_\alpha(s')$; applying the function $f_\alpha$ to equal arguments gives $r'_\alpha(s)=f_\alpha(r_\alpha(s))=f_\alpha(r_\alpha(s'))=r'_\alpha(s')$. Hence $s\sim'_A s'$. (Only well-definedness of $f_\alpha$ is used here; injectivity is not.)
*$\sim'_A\subseteq\sim_A$.* Suppose $s\sim'_A s'$. Then $p(s)=p(s')$, and for each $\alpha$, $f_\alpha(r_\alpha(s))=f_\alpha(r_\alpha(s'))$. Both $r_\alpha(s)$ and $r_\alpha(s')$ lie in the realized subset $r_\alpha(S)$, on which $f_\alpha$ is injective by hypothesis; therefore $r_\alpha(s)=r_\alpha(s')$. As this holds for every $\alpha$, $s\sim_A s'$.
This proves [(7.6)](/source-readout/canonical-minimal-source-completion/nested-families-refinement-maps-and-reparameterization#eq:same-relation). By [Proposition 3.4](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#prop:quotient) applied to each family, $E_A\cong S/\!\sim_A\ =\ S/\!\sim'_A\ \cong E'_A$, and the composite bijection sends $e_A(s)\mapsto e'_A(s)$, so it commutes with the completion maps. Since $u_A(e_A(s))=p(s)=u'_A(e'_A(s))$ for every $s$, and $e_A$ is surjective, [(7.7)](/source-readout/canonical-minimal-source-completion/nested-families-refinement-maps-and-reparameterization#eq:iso-over-T) follows. ◻
# Completion determined by a family of target answers
Paper 2 \[[2](/source-readout/canonical-minimal-source-completion/references#bib-Paper2)\] works with a single target $Q$ and its correct-answer map $a\colon S\to A_Q$, and constructs the coarsest extension of the readout that determines that answer; we will denote that single-target object by $E_Q$, following Paper 2. We now treat a *family* of targets, and we choose notation that cannot collide with the single-target object.
Let $\mathcal Q$ be a set of targets, each equipped with a correct-answer map $$
a_Q\colon S\longrightarrow A_Q,\qquad Q\in\mathcal Q ,\tag{8.1}$$ and define $$
j_{\mathcal Q}\colon S\to T\times\prod_{Q\in\mathcal Q}A_Q,
\qquad
j_{\mathcal Q}(s)=\Bigl(p(s),\ \bigl(a_Q(s)\bigr)_{Q\in\mathcal Q}\Bigr),
\qquad
E_{\mathcal Q}:=\mathop{\mathrm{Im}}\bigl(j_{\mathcal Q}\bigr),\tag{8.2}$$ with induced surjection $e_{\mathcal Q}\colon S\twoheadrightarrow E_{\mathcal Q}$, projection $u_{\mathcal Q}\colon E_{\mathcal Q}\to T$, and structure maps $\sigma_Q\colon E_{\mathcal Q}\to A_Q$. Write $\sim_{\mathcal Q}$ for the induced equivalence relation: $s\sim_{\mathcal Q}s'$ iff $p(s)=p(s')$ and $a_Q(s)=a_Q(s')$ for every $Q\in\mathcal Q$.
**Proposition 8.1** (Target-family completion).
*The data $\bigl(E_{\mathcal Q},e_{\mathcal Q},u_{\mathcal Q},(\sigma_Q)_{Q\in\mathcal Q}\bigr)$ is the terminal object of the category $\mathbf{Suff}\bigl(p;\{a_Q\}_{Q\in\mathcal Q}\bigr)$ of surjective extensions of the readout on which every answer map $a_Q$ becomes well defined. In particular $E_{\mathcal Q}\cong S/\!\sim_{\mathcal Q}$, and $E_{\mathcal Q}$ is the coarsest extension of the readout determining every answer in the family.*
*Proof.* The answer maps $a_Q\colon S\to A_Q$ form a set-indexed family of functions on $S$, exactly as the $r_\alpha$ do. [Definition 3.1](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#def:EA), [Definition 4.1](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#def:sufficient), [Definition 4.2](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#def:morphism) and [Proposition 3.2](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#prop:factorization), [Proposition 3.4](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#prop:quotient), [Theorem 5.1](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#thm:terminal), [Corollary 5.3](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#cor:coarsest) were stated and proved for an arbitrary such family and use no property of the $r_\alpha$ beyond their being functions on $S$. Substituting $\mathcal Q$ for $A$, $a_Q$ for $r_\alpha$, and $A_Q$ for $V_\alpha$ throughout yields the assertions verbatim. ◻
## Mediation and the canonical surjection
Now suppose the nominated relation family is *sufficient for the targets*, in the precise sense that each answer is computable from the readout together with the relation values.
**Definition 8.2** (Mediation).
The family $\{r_\alpha\}_{\alpha\in A}$ *mediates* $\mathcal Q$ if for every $Q\in\mathcal Q$ there is a map $$
\Phi_Q\colon T\times V_A\longrightarrow A_Q
\qquad\text{with}\qquad
a_Q(s)=\Phi_Q\bigl(p(s),\ R_A(s)\bigr)\quad\text{for all }s\in S .\tag{8.3}$$
**Theorem 8.3** (Relation completion versus target completion).
*Suppose $\{r_\alpha\}_{\alpha\in A}$ mediates $\mathcal Q$. Then there is a *unique* map $$
\chi\colon E_A\longrightarrow E_{\mathcal Q}
\qquad\text{with}\qquad
e_{\mathcal Q}=\chi\circ e_A ,\tag{8.4}$$ and $\chi$ is surjective; explicitly, $\chi(x)=\bigl(u_A(x),(\Phi_Q(x))_{Q\in\mathcal Q}\bigr)$ for $x\in E_A\subseteq T\times V_A$. Moreover, $\chi$ is a bijection if and only if $$
\sim_{\mathcal Q}\ =\ \sim_A ,\tag{8.5}$$ that is, if and only if equality of the readout together with all target answers induces exactly the same equivalence relation on $S$ as equality of the readout together with all selected relation values. In general one has only $$
\sim_A\ \subseteq\ \sim_{\mathcal Q} ,\tag{8.6}$$ so the target completion may be **strictly coarser** than the relation completion.*
*Proof.* *$E_A$ is sufficient for $\mathcal Q$.* Define $\sigma_Q^{A}:=\Phi_Q|_{E_A}\colon E_A\to A_Q$ (the restriction of $\Phi_Q$ to $E_A\subseteq T\times V_A$). For $s\in S$, $$\sigma_Q^{A}\bigl(e_A(s)\bigr)=\Phi_Q\bigl(p(s),R_A(s)\bigr)=a_Q(s)$$ by [(8.3)](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#eq:Phi). Together with the identity $p=u_A\circ e_A$ and the surjectivity of the map $e_A$, this exhibits the tuple $\bigl(E_A,e_A,u_A,(\sigma_Q^A)_{Q\in\mathcal Q}\bigr)$ as an object of the category $\mathbf{Suff}(p;\{a_Q\})$.
*Existence, uniqueness, surjectivity of $\chi$.* By [Proposition 8.1](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#prop:target-completion), $E_{\mathcal Q}$ is terminal in that category, so [Theorem 5.1](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#thm:terminal) supplies a unique morphism $\chi\colon E_A\to E_{\mathcal Q}$, necessarily satisfying $e_{\mathcal Q}=\chi\circ e_A$ and necessarily surjective; and by the explicit formula [(5.3)](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#eq:phiE) it is given on $x\in E_A$ by $\chi(x)=\bigl(u_A(x),(\sigma_Q^A(x))_{Q\in\mathcal Q}\bigr)
=\bigl(u_A(x),(\Phi_Q(x))_{Q\in\mathcal Q}\bigr)$.
*The ordering [(8.6)](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#eq:chi-order).* Suppose $s\sim_A s'$. Then $p(s)=p(s')$ and $R_A(s)=R_A(s')$, so by [(8.3)](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#eq:Phi), $a_Q(s)=\Phi_Q(p(s),R_A(s))=\Phi_Q(p(s'),R_A(s'))=a_Q(s')$ for every $Q$. Hence $s\sim_{\mathcal Q}s'$. (This is [Corollary 5.3](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#cor:coarsest) applied to the sufficient extension $E_A$ of the $\mathcal Q$-problem.)
*Injectivity criterion.* By [Proposition 3.4](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#prop:quotient) applied to both families, $\chi$ is identified with the canonical map $S/\!\sim_A\ \to\ S/\!\sim_{\mathcal Q}$, $[s]_A\mapsto[s]_{\mathcal Q}$, which is well defined by [(8.6)](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#eq:chi-order). Such a map is injective iff $\sim_{\mathcal Q}\ \subseteq\ \sim_A$; combined with [(8.6)](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#eq:chi-order) this is exactly [(8.5)](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#eq:chi-iso). Since $\chi$ is always surjective, it is a bijection precisely under [(8.5)](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#eq:chi-iso). ◻
# The gauge-reduction boundary
We return to the raw configuration space $\Omega$ and the physical quotient [(2.1)](/source-readout/canonical-minimal-source-completion/reduced-source-readout-and-the-nominated-relation-family#eq:gauge-quotient), $q\colon\Omega\to S=\Omega/\!\simeq_{\mathrm{phys}}$. Recall [Definition 2.1](/source-readout/canonical-minimal-source-completion/reduced-source-readout-and-the-nominated-relation-family#def:eligible): only quantities invariant under $\simeq_{\mathrm{phys}}$ descend to $S$, and only such quantities are eligible for nomination.
**Corollary 9.1** (Completion does not restore gauge coordinates).
*For every nominated family $\{r_\alpha\}_{\alpha\in A}$ of eligible relations, the composite $e_A\circ q\colon\Omega\to E_A$ is constant on raw physical-equivalence classes: $$
\omega\simeq_{\mathrm{phys}}\omega'
\quad\Longrightarrow\quad
e_A\bigl(q(\omega)\bigr)=e_A\bigl(q(\omega')\bigr).\tag{9.1}$$*
*Proof.* If $\omega\simeq_{\mathrm{phys}}\omega'$ then $q(\omega)=q(\omega')$ by definition of the quotient map, and applying the function $e_A$ to a single element of $S$ yields a single value. ◻
Mathematically this is a triviality, and we present it as such: it is a corollary of the order of operations, not a theorem with content of its own. Its *physical* significance is nevertheless worth stating precisely, because completion is easily misread as a licence to reintroduce structure that was deliberately discarded. The precise content of [(9.1)](/source-readout/canonical-minimal-source-completion/the-gauge-reduction-boundary#eq:gauge-constancy) is that the composite $e_A\circ q$ is constant on physical-equivalence classes. Consequently, no completion, however large the nomination, can
1. distinguish two gauge-equivalent raw configurations $\omega\simeq_{\mathrm{phys}}\omega'$;
2. recover which raw representative $\omega$ was originally supplied, given only the completed state;
3. restore discarded gauge or coordinate data as *physical* information, that is, as an invariant of the completed state; or
4. canonically generate a gauge section $\Omega\leftarrow S$ from the completion theorem.
We are careful not to overstate this. We do *not* claim that a representative of a physical-equivalence class can never be selected in any sense whatever: a separately chosen gauge-fixing convention, or a section of $q$, may well assign a representative when one exists, and such devices are routine in practice. The point is about provenance and status. A conventional gauge-fixing section is *extra, noncanonical structure supplied from outside*. It is not produced by the completion, is not privileged by terminality, and, this is the substantive claim, is not a recovery of lost raw information: the information it appears to supply was chosen by the convention, not extracted from the physics. Two different conventions give two different representatives of the same physical state, and nothing in $E_A$ adjudicates between them.
Completion therefore refines the readout partition, but only *within* the reduced source $S$; it cannot resurrect a distinction that the reduction $q$ was defined to destroy. Reduction precedes completion, and the universal completion does not undo physical equivalence. If one wants such a distinction back as physics, one must argue that the reduction was wrong, i.e. that the quantity was physical after all, hence eligible in the sense of [Definition 2.1](/source-readout/canonical-minimal-source-completion/reduced-source-readout-and-the-nominated-relation-family#def:eligible), and that is an argument about physics, not a construction available in this paper.
# The topological universal property
Assume now that $S$, $T$, and each $V_\alpha$ are topological spaces, and that $p$ and every $r_\alpha$ are continuous. Give $V_A=\prod_{\alpha\in A}V_\alpha$ the product topology and $T\times V_A$ the product topology; then $j_A=(p,R_A)$ is continuous, by the universal property of the product.
**Definition 10.1** (Topological completion).
Equip $E_A$ with the *quotient topology* induced by the surjection $e_A\colon S\twoheadrightarrow E_A$: a subset $U\subseteq E_A$ is open iff $e_A^{-1}(U)$ is open in $S$. A *topological relation-sufficient extension* is a relation-sufficient extension $\mathcal E=(E,e,u,(\rho_\alpha^E))$ in which $E$ is a topological space, $e$ is a *quotient map*, and $u$ and every $\rho_\alpha^E$ are continuous. Morphisms are continuous maps satisfying [(4.3)](/source-readout/canonical-minimal-source-completion/the-category-of-relation-sufficient-extensions#eq:morphism-over-S).
**Proposition 10.2** (Continuity of the structure maps).
*With the quotient topology of [Definition 10.1](/source-readout/canonical-minimal-source-completion/the-topological-universal-property#def:top-EA), the maps $u_A\colon E_A\to T$ and $\rho_\alpha\colon E_A\to V_\alpha$ are continuous, so $\mathcal E_A$ is a topological relation-sufficient extension.*
*Proof.* By the characteristic property of the quotient topology, a map out of $E_A$ is continuous iff its composite with $e_A$ is continuous. Now $u_A\circ e_A=p$ and $\rho_\alpha\circ e_A=r_\alpha$ by [(3.4)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:factorization), and $p$ and $r_\alpha$ are continuous by hypothesis. Finally $e_A$ is a quotient map by construction. ◻
**Theorem 10.3** (Topological universal property).
*$\mathcal E_A$, with the quotient topology, is terminal in the category of topological relation-sufficient extensions: for every such $\mathcal E$, the unique set map $\phi_{\mathcal E}\colon E\to E_A$ of [Theorem 5.1](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#thm:terminal) is continuous, and it is the unique continuous morphism $\mathcal E\to\mathcal E_A$. It is a continuous surjection.*
*Proof.* [Theorem 5.1](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#thm:terminal) supplies the unique *function* $\phi_{\mathcal E}\colon E\to E_A$ satisfying $e_A=\phi_{\mathcal E}\circ e$, and asserts its surjectivity. For continuity: the composite $\phi_{\mathcal E}\circ e=e_A$ is continuous (it is the quotient map onto $E_A$), and $e$ is a quotient map by [Definition 10.1](/source-readout/canonical-minimal-source-completion/the-topological-universal-property#def:top-EA); by the characteristic property of quotient maps, a function out of $E$ whose composite with the quotient map $e$ is continuous is itself continuous. Hence $\phi_{\mathcal E}$ is continuous. Uniqueness among *continuous* morphisms is inherited from uniqueness among all set maps, proved in [Theorem 5.1](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#thm:terminal) (v). Finally $\mathcal E_A$ is itself an object of the category by [Proposition 10.2](/source-readout/canonical-minimal-source-completion/the-topological-universal-property#prop:top-structure). ◻
## Quotient topology versus subspace topology
$E_A$ carries two a priori distinct topologies: the quotient topology of [Definition 10.1](/source-readout/canonical-minimal-source-completion/the-topological-universal-property#def:top-EA), and the subspace topology inherited from $T\times V_A$ via the canonical injection $$
\iota\colon E_A\hookrightarrow T\times V_A .\tag{10.1}$$ These must not be conflated. The universal property of [Theorem 10.3](/source-readout/canonical-minimal-source-completion/the-topological-universal-property#thm:top-terminal) is a statement about the *quotient* topology, and it is the quotient topology that the completion carries by definition.
**Proposition 10.4** (Comparison, and the embedding criterion).
*Equip $E_A$ with the quotient topology.*
1.
*The injection $\iota$ of [(10.1)](/source-readout/canonical-minimal-source-completion/the-topological-universal-property#eq:iota) is continuous. Consequently the quotient topology is *finer than or equal to* the subspace topology on $\mathop{\mathrm{Im}}(j_A)$.*
2.
*$\iota$ is a topological embedding if and only if the quotient topology coincides with the subspace topology; equivalently, if and only if the corestriction $j_A\colon S\to\mathop{\mathrm{Im}}(j_A)$ is a quotient map when $\mathop{\mathrm{Im}}(j_A)$ carries the product subspace topology.*
*Proof.* We have $\iota\circ e_A=j_A$, which is continuous; since $e_A$ is a quotient map, $\iota$ is continuous by the characteristic property. Continuity of $\iota$ says exactly that every subspace-open subset of $E_A$ (i.e. every $\iota^{-1}(W)$ with $W$ open in $T\times V_A$) is quotient-open; hence the quotient topology contains the subspace topology.
$\iota$ is injective and continuous, so it is an embedding iff it is open onto its image, i.e. iff every quotient-open subset of $E_A$ is subspace-open. Combined with , this holds iff the two topologies are equal.
For the reformulation, write $I:=\mathop{\mathrm{Im}}(j_A)$ with the subspace topology and let $\bar j_A\colon S\to I$ be the corestriction of $j_A$, so that as *functions* $\bar j_A=\iota\circ e_A$ with $\iota$ now regarded as the identity of underlying sets $E_A\to I$. Suppose the topologies agree. Then $\iota$ is a homeomorphism and $e_A$ is a quotient map, so $\bar j_A$ is a quotient map. Conversely, suppose $\bar j_A$ is a quotient map. A subset $U\subseteq
E_A$ is quotient-open iff $e_A^{-1}(U)$ is open in $S$; and $e_A^{-1}(U)=\bar j_A^{-1}(U)$ as subsets of $S$, since $e_A$ and $\bar j_A$ agree as functions into the common underlying set. Because $\bar j_A$ is a quotient map, $\bar j_A^{-1}(U)$ open in $S$ implies $U$ open in $I$, i.e. subspace-open. Hence quotient-open implies subspace-open, and with the two topologies coincide. ◻
**Corollary 10.5** (Compact–Hausdorff sufficient condition).
*Suppose $S$ is compact and $T\times V_A$ is Hausdorff. Then the quotient and subspace topologies on $E_A$ agree, and $\iota\colon E_A\hookrightarrow T\times V_A$ is a topological embedding; in particular $E_A$ is a compact Hausdorff space.*
*Proof.* Write $I=\mathop{\mathrm{Im}}(j_A)\subseteq T\times V_A$ with the subspace topology, and let $\bar j_A\colon S\to I$ be the (continuous, surjective) corestriction of $j_A$. As a subspace of a Hausdorff space, $I$ is Hausdorff. We show $\bar j_A$ is a closed map, hence a quotient map, and then apply [Proposition 10.4](/source-readout/canonical-minimal-source-completion/the-topological-universal-property#prop:embedding) .
Let $C\subseteq S$ be closed. Since $S$ is compact, $C$ is compact; since $\bar j_A$ is continuous, $\bar j_A(C)$ is a compact subset of $I$; since $I$ is Hausdorff, the compact subset $\bar j_A(C)$ is closed in $I$. Thus $\bar j_A$ is a continuous closed surjection, and every continuous closed surjection is a quotient map: if $\bar j_A^{-1}(U)$ is open then $S\setminus\bar j_A^{-1}(U)$ is closed, its image $\bar j_A(S\setminus\bar j_A^{-1}(U))=I\setminus U$ is closed by surjectivity, so $U$ is open. By [Proposition 10.4](/source-readout/canonical-minimal-source-completion/the-topological-universal-property#prop:embedding), the topologies agree and $\iota$ is an embedding. Finally $E_A\cong I$ is compact (continuous image of a compact space) and Hausdorff (subspace of a Hausdorff space). ◻
# Worked example: flat $U(1)$ connections on the circle
We now exhibit every phenomenon of the general theory in a single, completely explicit gauge-theoretic example. It is the extreme case of readout loss: the readout is *constant*, so it distinguishes nothing at all, and the completion must supply everything.
## The reduced source and the readout
Consider $U(1)$ connections on the trivial bundle over the circle $S^1$, restricted to the flat ones. Gauge reduction [(2.1)](/source-readout/canonical-minimal-source-completion/reduced-source-readout-and-the-nominated-relation-family#eq:gauge-quotient) identifies gauge-equivalent connections; on the circle, the resulting reduced moduli space of flat connections is classified by the holonomy around the loop, so that $$
S_{\mathrm{flat}}\ \cong\ U(1),
\qquad
z=e^{i\theta},\quad\theta\in\mathbb{R}/2\pi\mathbb{Z} .\tag{11.1}$$ This is the standard classification of flat connections on the circle by their holonomy (see, e.g., \[[5](/source-readout/canonical-minimal-source-completion/references#bib-KobayashiNomizu), [7](/source-readout/canonical-minimal-source-completion/references#bib-WuYang)\]); we take it as given, and stress that $S_{\mathrm{flat}}$ is already the *reduced* source in the sense of [Section 2](/source-readout/canonical-minimal-source-completion/reduced-source-readout-and-the-nominated-relation-family#sec:setup): gauge potentials and gauge transformations have been quotiented away, and it is not available to us to reintroduce them.
The readout is the curvature. For flat connections the curvature vanishes identically, so $$
p_F\colon S_{\mathrm{flat}}\longrightarrow T=\{0\},
\qquad
p_F(z)=0 .\tag{11.2}$$ The readout fiber is the whole source: $F_0=S_{\mathrm{flat}}\cong U(1)$. Every nontrivial distinction on $S_{\mathrm{flat}}$ therefore fails to descend along $p_F$, and $\sim_{p_F}$ is the total relation. Note carefully what is *not* being claimed: we do not claim that curvature is the complete physical observable content of a gauge field. On the contrary, the example exists precisely because it is not, the Aharonov–Bohm effect \[[6](/source-readout/canonical-minimal-source-completion/references#bib-AharonovBohm)\] and the Wu–Yang analysis \[[7](/source-readout/canonical-minimal-source-completion/references#bib-WuYang)\] are the physical statement that holonomy carries information the field strength does not. The readout $p_F$ is a readout, not an ontology.
## Transport completion
Nominate the single relation $$
r_{\mathrm{hol}}\colon S_{\mathrm{flat}}\to U(1),
\qquad
r_{\mathrm{hol}}(z)=z ,\tag{11.3}$$ the oriented holonomy itself. It is eligible in the sense of [Definition 2.1](/source-readout/canonical-minimal-source-completion/reduced-source-readout-and-the-nominated-relation-family#def:eligible): holonomy around a fixed loop is gauge-invariant (for the abelian group $U(1)$ it is invariant outright, not merely up to conjugacy), so it is a function on the reduced source. It is nominated because oriented parallel transport is what enters targets involving directed transport of charge around the loop.
By [Definition 3.1](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#def:EA), the completion is $$
E_{\mathrm{hol}}
=\mathop{\mathrm{Im}}\bigl(p_F,\ r_{\mathrm{hol}}\bigr)
=\{(0,z):z\in U(1)\}
\ \cong\ U(1),\tag{11.4}$$ and $e_{\mathrm{hol}}\colon S_{\mathrm{flat}}\to E_{\mathrm{hol}}$ is a bijection. Here $S$ is compact and $T\times V=\{0\}\times U(1)$ is Hausdorff, so [Corollary 10.5](/source-readout/canonical-minimal-source-completion/the-topological-universal-property#cor:compact-hausdorff) applies: the quotient and subspace topologies agree and [(11.4)](/source-readout/canonical-minimal-source-completion/worked-example-flat-u-1-connections-on-the-circle#eq:Ehol) is a homeomorphism. By [Proposition 12.1](/source-readout/canonical-minimal-source-completion/the-exact-scope-of-reconstruction#prop:reconstruction) below, the nomination $\{p_F,r_{\mathrm{hol}}\}$ separates points of $S_{\mathrm{flat}}$, so this is a case in which completion happens to reconstruct the whole reduced source. It does *not* restore gauge potentials or gauge representatives, by [Corollary 9.1](/source-readout/canonical-minimal-source-completion/the-gauge-reduction-boundary#cor:gauge) it cannot, and it restores nothing beyond the reduced source.
## Spectral completion
Now nominate instead the charged-scalar spectral target of Paper 2 \[[2](/source-readout/canonical-minimal-source-completion/references#bib-Paper2)\]. For a scalar of unit charge on $S^1$ minimally coupled to the flat connection with holonomy $e^{i\theta}$, let $H_\theta$ denote the associated Laplace-type operator. The result we import from Paper 2 is $$
\mathop{\mathrm{Spec}}(H_\theta)=\mathop{\mathrm{Spec}}(H_{\theta'})
\qquad\Longleftrightarrow\qquad
\theta'\equiv\pm\theta \pmod{2\pi},\tag{11.5}$$ with multiplicities. We do not reprove [(11.5)](/source-readout/canonical-minimal-source-completion/worked-example-flat-u-1-connections-on-the-circle#eq:spec) here; it is exactly the Paper 2 input, and the new content of the present section is the family-level completion and refinement structure built on top of it.
Take $\mathcal Q=\{Q_{\mathrm{spec}}\}$ with answer map $a_{\mathrm{spec}}(z)=\mathop{\mathrm{Spec}}(H_\theta)$ for $z=e^{i\theta}$ (a well-defined function of $z$, since $\theta\mapsto H_\theta$ depends on $\theta$ only mod $2\pi$). By [(11.5)](/source-readout/canonical-minimal-source-completion/worked-example-flat-u-1-connections-on-the-circle#eq:spec), the induced equivalence relation on $S_{\mathrm{flat}}$ is $$
z\ \sim_{\mathrm{spec}}\ z'
\qquad\Longleftrightarrow\qquad
z'=z\ \text{ or }\ z'=z^{-1},\tag{11.6}$$ the readout clause $p_F(z)=p_F(z')$ being vacuous. Hence, by [Proposition 8.1](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#prop:target-completion), $$
E_{\mathrm{spec}}\ \cong\ S_{\mathrm{flat}}/\!\sim_{\mathrm{spec}}\ =\ U(1)/(z\sim z^{-1}).\tag{11.7}$$
**Proposition 11.1** (The spectral completion is an interval).
*Give $U(1)/(z\sim z^{-1})$ the quotient topology. Then the assignment $$
c\colon U(1)/(z\sim z^{-1})\longrightarrow[-1,1],
\qquad
c\bigl([z]\bigr)=\mathop{\mathrm{Re}}z=\cos\theta
\qquad(z=e^{i\theta}),\tag{11.8}$$ is a well-defined homeomorphism. Hence $E_{\mathrm{spec}}\cong[-1,1]$.*
*Proof.* *Well defined.* $\mathop{\mathrm{Re}}(z^{-1})=\mathop{\mathrm{Re}}(\bar z)=\mathop{\mathrm{Re}}z$ for $z\in U(1)$, so the formula does not depend on the representative of the class.
*Injective.* Suppose $\cos\theta=\cos\theta'$ with $\theta,\theta'\in\mathbb{R}/2\pi\mathbb{Z}$. Then $\theta'\equiv\pm\theta\pmod{2\pi}$, i.e. $e^{i\theta'}=e^{\pm i\theta}$, i.e. $z'\in\{z,z^{-1}\}$, i.e. $[z']=[z]$.
*Surjective.* For $x\in[-1,1]$ set $\theta=\arccos x\in[0,\pi]$; then $c([e^{i\theta}])=x$.
*Continuous.* The composite of $c$ with the quotient map $U(1)\to U(1)/(z\sim z^{-1})$ is $z\mapsto\mathop{\mathrm{Re}}z$, which is continuous. By the characteristic property of the quotient topology, $c$ is continuous.
*Homeomorphism.* $U(1)$ is compact, hence so is its continuous image $U(1)/(z\sim z^{-1})$; and $[-1,1]$ is Hausdorff. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism: it is a closed map, since closed subsets of a compact space are compact, their continuous images are compact, and compact subsets of a Hausdorff space are closed; a closed continuous bijection is a homeomorphism. ◻
## The refinement map, concretely
The relation family $\{r_{\mathrm{hol}}\}$ mediates the spectral target in the sense of [Definition 8.2](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#def:mediation): by [(11.5)](/source-readout/canonical-minimal-source-completion/worked-example-flat-u-1-connections-on-the-circle#eq:spec) the spectrum is a function of the holonomy, so there is $\Phi_{\mathrm{spec}}\colon T\times U(1)\to A_{Q_{\mathrm{spec}}}$ with $a_{\mathrm{spec}}(z)=\Phi_{\mathrm{spec}}(p_F(z),r_{\mathrm{hol}}(z))$. [Theorem 8.3](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#thm:chi) therefore supplies a unique surjection $\chi\colon E_{\mathrm{hol}}\to E_{\mathrm{spec}}$, and under the identifications [(11.4)](/source-readout/canonical-minimal-source-completion/worked-example-flat-u-1-connections-on-the-circle#eq:Ehol) and [Proposition 11.1](/source-readout/canonical-minimal-source-completion/worked-example-flat-u-1-connections-on-the-circle#prop:interval) it is exactly $$
\chi\colon U(1)\longrightarrow[-1,1],
\qquad
e^{i\theta}\longmapsto\cos\theta .\tag{11.9}$$ It is surjective, and it is *not* injective: $\chi(e^{i\theta})=\chi(e^{-i\theta})$ while $e^{i\theta}\neq e^{-i\theta}$ whenever $\theta\not\equiv0,\pi$. So this is a concrete realization of the strict inequality permitted by [(8.6)](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#eq:chi-order): here $\sim_A\ \subsetneq\ \sim_{\mathcal Q}$, and the target completion is strictly coarser than the relation completion.
The physics of the strictness is transparent. The full oriented holonomy retains an *orientation*: it distinguishes transport of charge around the loop in one sense from transport in the other, i.e. $z$ from $z^{-1}$. The charged-scalar spectrum does not require that information, by [(11.5)](/source-readout/canonical-minimal-source-completion/worked-example-flat-u-1-connections-on-the-circle#eq:spec) it is invariant under $\theta\mapsto-\theta$, and so the minimal completion *for that target* discards it. Each completion is minimal for its own nomination. Which is appropriate depends on which targets one has nominated, and that is a physical decision made before the mathematics begins, not one the mathematics can make.
# The exact scope of reconstruction
Completion always reconstructs the joint readout/relation signature; that is what it is. Whether it reconstructs the source state is a separate question with a sharp answer.
**Proposition 12.1** (Reconstruction criterion).
*The completion map $e_A\colon S\to E_A$ is injective, equivalently, $\sim_A\ =\ \Delta_S$; equivalently, $e_A$ is a bijection onto $E_A$, if and only if the family $$
\{p\}\cup\{r_\alpha:\alpha\in A\}\tag{12.1}$$ *separates the points of $S$*: for all $s\neq s'$ in $S$, either $p(s)\neq p(s')$ or $r_\alpha(s)\neq r_\alpha(s')$ for some $\alpha\in A$.*
*Proof.* By [Definition 3.1](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#def:EA) and [(3.5)](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#eq:simA), for all $s,s'\in S$, $$\begin{aligned}
e_A(s)=e_A(s')
&\iff j_A(s)=j_A(s')\\
&\iff \bigl[\,p(s)=p(s')\ \text{ and }\ r_\alpha(s)=r_\alpha(s')\ \ \forall\alpha\in A\,\bigr]\\
&\iff s\sim_A s' .
\end{aligned}$$ Hence $e_A$ is injective iff ($s\sim_A s'\Rightarrow s=s'$), iff $\sim_A\ =\ \Delta_S$ (the reverse inclusion always holding by reflexivity), iff no pair of distinct points is identified by the whole family, which is the stated separation property, in contrapositive form. Surjectivity of $e_A$ is automatic ([Proposition 3.2](/source-readout/canonical-minimal-source-completion/the-joint-image-construction-and-its-quotient-realization#prop:factorization)), so injectivity is equivalent to bijectivity. ◻
[Section 11](/source-readout/canonical-minimal-source-completion/worked-example-flat-u-1-connections-on-the-circle#sec:u1) contains an instance of each case: the transport nomination $\{p_F,r_{\mathrm{hol}}\}$ separates points of $S_{\mathrm{flat}}$, so $e_{\mathrm{hol}}$ is bijective and completion recovers the whole reduced source; the spectral nomination does not separate $z$ from $z^{-1}$, so the spectral completion is a proper quotient, and no amount of further analysis of the spectrum will recover the orientation, because the orientation is not there to be recovered.
# Discussion and scope
## What has been established
Given a reduced source space $S$, a surjective readout $p\colon S\to T$, and a set-indexed family of eligible source relations $\{r_\alpha\colon S\to V_\alpha\}_{\alpha\in A}$, there exists a completion $E_A=\mathop{\mathrm{Im}}(p,R_A)$ of the readout on which every nominated relation becomes well defined; it is canonically the quotient $S/\!\sim_A$; it is terminal among surjective relation-sufficient extensions and therefore unique up to unique isomorphism; it is the *coarsest* such extension, in the exact sense $\sim_e\ \subseteq\ \sim_A$ for every sufficient $e$; its fibers over $T$ are the realized relation signatures; it is functorial and cocyclic in the nomination; it is insensitive to faithful relabelling of relation values; it admits a canonical surjection onto the completion determined by any family of targets that the relations mediate; it restores no gauge redundancy; and in the topological setting it is terminal for the quotient topology, embedding into $T\times V_A$ exactly when the quotient and subspace topologies agree.
Together with \[[1](/source-readout/canonical-minimal-source-completion/references#bib-Paper1)\] (obstruction) and \[[2](/source-readout/canonical-minimal-source-completion/references#bib-Paper2)\] (target-relative necessity), this supplies the constructive step of the sequence: the object whose existence, uniqueness, and minimality were previously only desiderata.
## The three standing confusions
*Minimality is not smallness.* [Remark 5.5](/source-readout/canonical-minimal-source-completion/the-canonical-minimal-completion-theorem#rem:minimality): minimality is least retained distinction, an order-theoretic statement about equivalence relations, not a cardinality bound and not an optimality claim in any other metric.
*Completion is not reconstruction.* [Remark 12.2](/source-readout/canonical-minimal-source-completion/the-exact-scope-of-reconstruction#rem:two-reconstructions): the completed state always determines the nominated signature and determines the source state only under a separation hypothesis that must be checked, not assumed.
*Coordinates are relative to the nomination.* [Remark 6.4](/source-readout/canonical-minimal-source-completion/redundant-and-indispensable-relation-coordinates#rem:not-unique-coords) and [Proposition 7.3](/source-readout/canonical-minimal-source-completion/nested-families-refinement-maps-and-reparameterization#prop:reparam): the nominated relations are a presentation of the quotient, and irredundancy constrains that presentation without making it unique. Relative to a fixed nominated family, $E_A$ does have canonical structure maps $u_A$ and $(\rho_\alpha)_{\alpha\in A}$; what it lacks is a family-independent, intrinsically unique coordinate presentation. Different faithful reparameterizations, and different irredundant families, may present the same quotient.
## What has not been established
The theorem is model-relative, and the following are *not* consequences of it.
That the universe is a projection of a hidden source. Nothing here bears on that; $S$ is an assumed structure, and the theorem tells one what follows from assuming it, not that it exists.
That the source space is empirically real. The construction is available whether $S$ is a physical posit, a redundant description, or a bookkeeping device.
That every lost distinction ought to be restored. On the contrary: the entire point of [Theorem 8.3](/source-readout/canonical-minimal-source-completion/completion-determined-by-a-family-of-target-answers#thm:chi) and of [Section 11](/source-readout/canonical-minimal-source-completion/worked-example-flat-u-1-connections-on-the-circle#sec:u1) is that a distinction is worth restoring only relative to a nominated target, and that nominating more than the target requires yields a completion that is finer than necessary, correct, but not minimal for that target.
That the full source can in general be reconstructed. [Proposition 12.1](/source-readout/canonical-minimal-source-completion/the-exact-scope-of-reconstruction#prop:reconstruction) makes this a hypothesis, and the spectral example shows it can fail.
That the nominated relation family is physically privileged. This is the load-bearing external input, and the mathematics cannot supply it. A relation is not physical because it varies on a readout fiber; if it were, every gauge artifact would be an observable. The nomination must be justified by the physics of the targets, and this paper’s theorems are conditional on it in exactly that one respect, which is, we think, where the conditionality belongs.
## Outlook
The present construction is purely set-theoretic and, in [Section 10](/source-readout/canonical-minimal-source-completion/the-topological-universal-property#sec:topology), topological. It says nothing about when $E_A$ carries additional structure, smooth, symplectic, metric, or when the relation signatures arise as genuine geometric invariants rather than as formal coordinates. The companion manuscript in preparation on geometric realization takes up exactly that question, on restricted spaces of metrics, bundles, connections, and matter fields. We have deliberately imported none of that machinery here; the theorems above should stand or fall on their own, and they require nothing beyond images, quotients, and quotient topologies.
# 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.
# References
9
**\[1\]** J. Rodgers, *Source–Readout Non-Equivalence: Descent and Equivariant Reconstruction Obstructions*, Zenodo, 2026. [doi:10.5281/zenodo.21369967](https://doi.org/10.5281/zenodo.21369967).
**\[2\]** J. Rodgers, *Target-Relative Necessity of Completion: When Readout Loss Obstructs, and What a Sufficient Extension Must Retain*, Zenodo, 2026. [doi:10.5281/zenodo.21370501](https://doi.org/10.5281/zenodo.21370501).
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