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

The exact scope of reconstruction

Jeremy RodgersSection 13 of 16Original PDF ↗Publication DOI ↗

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 and (3.5), 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), so injectivity is equivalent to bijectivity. ◻

Remark 42 (Two senses of “reconstruction”; only one is automatic).

Distinguish carefully:

Reconstruction of the joint signature. The completed state determines the readout value and every nominated relation value: this is (3.4), and it holds always, by construction. It is the entire purpose of the object.
Reconstruction of the reduced source state. The completed state determines $s\in S$ itself. This holds if and only if the nominated family separates points, by Proposition 12.1, and in general it fails, necessarily so, since by Corollary 5.3 $E_A$ is the coarsest sufficient extension and therefore retains nothing the nomination does not force it to retain.

Neither sense licenses a third: recovery of raw configuration data in $\Omega$ , or selection of a preferred representative of a physical equivalence class or of a readout fiber. Corollary 9.1 forbids the former and Remark 3.7 the latter. The word “reconstruction” should not be used in this subject without saying which of (1) and (2) is meant.

Section 11 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.