Proposition: Fixed host under catalogue replacement

Replacing a regulator label ${\rho }_a$ by ${\rho }_b$, or a scale label ${\mu }_1$ by ${\mu }_2$, in the comparison catalogue leaves $M_{{\delta }_{\mathrm{c}}},g_{{\delta }_{\mathrm{c}}},\mathcal{W},\Sigma$, the lapse, and the installed causal relation unchanged.

Proof. A comparison command acts on the catalogue and finite programme registers. The host data in (eq:host) are held fixed and are not outputs of that command. No metric-update map is included.