Proposition: Constraint-preserving realization of the device class

In the finite gauge net (eq:gaugenet), choose kinematic Kraus operators $v_{b,\mu }\in \mathfrak{F}(b)$ satisfying

Their physical restrictions satisfy (eq:internaldevice) and all conclusions of Lemma lem:devicesupport. Complete selected outcomes may be grouped and recorded as in Proposition prop:records.

Proof. Constraint preservation makes $q_{\mathrm{phys}}$ a $\ast$-homomorphism, so the kinematic completeness identity restricts to the physical completeness identity. The physical Kraus operators lie in $\mathfrak{A}(b)$ by definition. Apply Lemma lem:devicesupport and Proposition prop:records.