Non-Source Projection and Internal Identifiability

The source-to-readout architecture

Jeremy RodgersSection 3 of 14Original PDF ↗Publication DOI ↗

The source-to-readout architecture

Reduction and readout

Definition 2.1 (Reduced source).

Let $\Omega$ be a raw realization space carrying an equivalence relation $\simeq_{\mathrm{phys}}$ , the declared physical equivalence of the model: $\omega\simeq_{\mathrm{phys}}\omega'$ means that the model regards $\omega$ and $\omega'$ as descriptions of the same physical situation, differing only by gauge choice, coordinate choice, or relabelling. Define the reduced source $S:=\Omega/\!\simeq_{\mathrm{phys}},\qquad q\colon\Omega\twoheadrightarrow S\tag{1}$ with $q$ the canonical surjection.

The order of operations in (1) is the crux of the entire paper and we pause on it. If one did not quotient first, then a noninjective readout would be uninformative: its fibers could consist entirely of gauge copies of a single physical state, and their nontriviality would be an artifact of description. By constructing $S$ first, we ensure that $s\neq s'\ \text{in}\ S\quad\Longrightarrow\quad s,s'\ \text{are \emph{not} related by any redundancy the model declares.}\tag{2}$ Consequently a nontrivial $p$ -fiber in $S$ is a genuine multiplicity of physically inequivalent states by the model’s own standard. This does not make it physically real , the model may be wrong , but it removes the cheapest objection.

Definition 2.2 (Exact readout).

Let $\Pi_{T_1}\colon\Omega\to T_1$ be a surjection that is invariant under $\simeq_{\mathrm{phys}}$ , i.e. $\omega\simeq_{\mathrm{phys}}\omega'\Rightarrow\Pi_{T_1}(\omega)=\Pi_{T_1}(\omega')$ . We call $T_1$ the readout or Tier 1.

Lemma 2.3 (Induced readout).

There is a unique map $p\colon S\to T_1$ with $\Pi_{T_1}=p\circ q,\tag{3}$ and $p$ is surjective. Moreover, writing $s\sim_p s'\iff p(s)=p(s')$ , the map $p$ induces a bijection $S/\!\sim_p\;\xrightarrow{\ \cong\ }\;T_1 .\tag{4}$

Proof. Existence and uniqueness of $p$ are the universal property of the quotient. Define $p(q(\omega)):=\Pi_{T_1}(\omega).$ This is well defined precisely by the $\simeq_{\mathrm{phys}}$ -invariance of $\Pi_{T_1}$ , and it is forced by (3), which gives uniqueness. Surjectivity of $p$ follows from surjectivity of $\Pi_{T_1}$ , since $\Pi_{T_1}=p\circ q$ and hence $\mathop{\mathrm{Im}}p\supseteq\mathop{\mathrm{Im}}\Pi_{T_1}=T_1$ . Finally, $\sim_p$ is by construction the kernel pair of $p$ , so the induced map $[s]\mapsto p(s)$ on $S/\!\sim_p$ is well defined and injective; it is surjective because $p$ is. This is (4). ◻

We record the terminology that the rest of the paper turns on.

Definition 2.4 (Exactness).

We call $p$ an exact readout to mean precisely (4): $T_1$ is, up to canonical bijection, the quotient of $S$ by the relation “has the same readout”. That is, “exact” means exact as the quotient by the readout’s own kernel relation $\sim_p$ , and nothing more: it does not mean that $p$ is injective, that the readout is source-complete, or that information is preserved at the source level. Exactness is a statement about no information being lost within a fiber’s label; it is emphatically not a statement that fibers are singletons. The whole content of this paper lies in that gap.

Relations and the minimal completion

Definition 2.5 (Invariant relations).

Let $\mathcal R=\{r_\alpha\colon S\to V_\alpha\}_{\alpha\in A}$ be a family of maps, the physically invariant source relations. (Invariance is already built in: they are defined on $S$ , hence constant on $\simeq_{\mathrm{phys}}$ -classes.) Write $R_A:=(r_\alpha)_{\alpha\in A}\colon S\longrightarrow \prod_{\alpha\in A}V_\alpha .$

Definition 2.6 (Minimal completion).

Define $E_A:=\mathop{\mathrm{Im}}\bigl(p,R_A\bigr)\;=\;\Bigl\{\bigl(p(s),R_A(s)\bigr)\;:\;s\in S\Bigr\}\ \subseteq\ T_1\times\prod_{\alpha}V_\alpha,\tag{5}$ with the two canonical maps $e_A\colon S\to E_A,\quad e_A(s)=\bigl(p(s),R_A(s)\bigr), \qquad u_A\colon E_A\to T_1,\quad u_A(t,v)=t .\tag{6}$ The coordinate maps on the completion are $\rho_\alpha\colon E_A\to V_\alpha$ , $\rho_\alpha(t,v)=v_\alpha$ .

Lemma 2.7 (Elementary properties of the completion).

$e_A$ is surjective; $u_A$ is surjective; $p=u_A\circ e_A$ ; and $r_\alpha=\rho_\alpha\circ e_A$ for every $\alpha$ .

Proof. $e_A$ is surjective onto $E_A$ by the definition (5) of $E_A$ as an image. Then $u_A(e_A(s))=p(s)$ gives $p=u_A e_A$ , and since $p$ is surjective (Lemma 2.3) so is $u_A$ . The identity $\rho_\alpha(e_A(s))=r_\alpha(s)$ is immediate. ◻

Thus $E_A$ sits between $S$ and $T_1$ : it is the readout augmented by exactly the values of the nominated relations, and nothing more. Its universal character is proved in §4 in the form we actually need (Theorem 4.6).

Specializations

The abstract triple $(S,p,\mathcal R)$ has three standard realizations, which we record to fix the reader’s picture. They are specializations of one architecture, not competing definitions.

Example 2.8 (Geometric orbit space and geometric realization).

Let $\mathcal C_{\mathrm{geo}}$ be a space of field configurations and $\simeq_{\mathrm{geo}}$ the action of the gauge and diffeomorphism group. Then $S_{\mathrm{geo}}=\mathcal C_{\mathrm{geo}}/\!\simeq_{\mathrm{geo}}$ is the orbit space, and the readout is a map $p_{\mathrm{geo}}\colon S_{\mathrm{geo}}\to T_1$ recording whatever subset of invariants it resolves. The relations are now geometric: write $J_A=(J_\alpha)_{\alpha\in A},\qquad J_\alpha\colon S_{\mathrm{geo}}\to V_\alpha ,$ for a family of gauge- and diffeomorphism-invariant functionals: integrated, spectral, or relational curvature invariants; holonomy conjugacy classes; and Wilson-loop observables based on invariantly specified loops. Specializing Definition 2.6 gives the geometric realization of the completion, $E_A^{\mathrm{geo}}:=\mathop{\mathrm{Im}}\bigl(p_{\mathrm{geo}},J_A\bigr) =\bigl\{\bigl(p_{\mathrm{geo}}(s),\,J_A(s)\bigr)\;:\;s\in S_{\mathrm{geo}}\bigr\},\tag{7}$ the set of completed signatures actually attained by configurations.

The word “realization” carries the entire content of (7), and it is a restriction, not a licence. $E_A^{\mathrm{geo}}$ is by construction an image: it contains only those pairs (readout value, invariant values) that arise from an admissible, globally compatible field configuration in $\mathcal C_{\mathrm{geo}}$ . It is emphatically not the full product $T_1\times\prod_\alpha V_\alpha$ . An abstract signature , an arbitrary assignment of values to the invariants, consistent with no constraint one has bothered to impose , need not be realized by any global field at all: field equations, Bianchi identities, integrality and holonomy constraints, and global topological obstructions all cut the attainable set down. Determining which completed signatures are realized by compatible global geometric fields is thus a substantive geometric question about $\mathcal C_{\mathrm{geo}}$ , to be settled by geometry, and none of the theorems below settles it or presumes an answer.

We make no smoothness, Hausdorffness, or slice claims about $S_{\mathrm{geo}}$ : nothing below requires them, and orbit spaces are frequently badly behaved.

Example 2.9 (Bounded linear observation).

Let $K\colon\mathcal{H}_{\mathrm{src}}\to\mathcal{H}_{\mathrm{obs}}$ be a bounded linear map with closed range. Taking $S=\mathcal{H}_{\mathrm{src}}$ (with $\simeq_{\mathrm{phys}}$ trivial) and $p$ the corestriction of $K$ to its range, the closed-range theorem gives $T_1\;\cong\;\mathcal{H}_{\mathrm{src}}/\ker K\;\cong\;\mathop{\mathrm{Ran}}K .\tag{8}$ Fibers of $p$ are the cosets $\Psi+\ker K$ . Here the “essential distinction” of §4 is a vector $n\in\ker K$ , $n\neq 0$ , that nevertheless changes some quantity of interest. Closedness of the range is used only for (8) to be an isomorphism of Banach spaces; the set-theoretic quotient statement (4) needs no such hypothesis.

Example 2.10 (Coarse-grained state).

Let $S$ be a finite set of microstates modulo relabelling and $p$ a partition into macrostates. This is the setting of §9.