Paper 03Canonical2026-07-15

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

Existence, uniqueness, and exact minimality of the completion, without selecting any fiber representative

DOI: 10.5281/zenodo.21370763 View on Zenodo

Authority role

Constructs the canonical minimal completion: for any nominated family of invariant source relations, the joint image of readout and relations is terminal among all relation-sufficient extensions — the coarsest enrichment on which every nominated relation becomes well defined.

Summary

Given a lossy readout and a nominated family of physically invariant relations that the readout fails to determine, this paper constructs the coarsest extension on which all of them become well defined — the joint image of readout and relations, equivalently the quotient by agreement of readout and every nominated relation. The completion is terminal: every sufficient extension maps uniquely onto it, giving existence, uniqueness up to unique isomorphism, and exact minimality with no fiber representative ever selected. Redundant and indispensable relation coordinates, refinement towers for nested families, the gauge-reduced form, and the topological form are all characterized.

Notes

Reading notes

Papers 1 and 2 posed the completion problem; this paper supplies the constructive step. Given the reduced source $S$ , the surjective readout $p: S \to T$ , and an externally nominated set-indexed family of eligible (physically invariant) relations $\{r_\alpha: S \to V_\alpha\}_{\alpha \in A}$ , form the joint map $j_A = (p, (r_\alpha)_\alpha)$ and set

$E_A \;=\; \mathrm{Im}\, j_A \;\cong\; S/\!\sim_A,$

the quotient by agreement of the readout and every nominated relation.

Theorem — Canonical minimal completion (terminality)

$E_A$ is terminal in the category of surjective relation-sufficient extensions: every extension on which the readout and all nominated relations are well defined admits a unique surjection onto $E_A$ . This yields existence, uniqueness up to unique isomorphism, and exact minimality — the coarsest sufficient extension — without nominating a preferred representative of any readout fiber.

The supporting structure is worked out in full: an exact separation criterion for redundant versus indispensable relation coordinates; canonical refinement maps for nested families with a cocycle law; invariance under faithful reparameterization of value spaces (the completion depends on distinctions carried, not on labels); the completion determined directly by a family of target-answer maps, with a canonical surjection from the relation completion onto it (the target completion may be strictly coarser); the gauge-reduced form (completion cannot resurrect redundancy already quotiented away); and the topological form, distinguishing quotient from subspace topology.

The flat $U(1)$ circle example makes it concrete: oriented transport completion returns the full holonomy circle $U(1)$ , while the charged-scalar spectral target needs only the strictly coarser inversion quotient $U(1)/(z \sim z^{-1}) \cong [-1,1]$ , with the canonical refinement $e^{i\theta} \mapsto \cos\theta$ exhibiting the general relation-to-target surjection.

Cite this paper

Rodgers, Jeremy. (2026). Canonical Minimal Source Completion: The Coarsest Readout Extension on Which a Nominated Family of Source Relations Becomes Well Defined. https://doi.org/10.5281/zenodo.21370763