Application to the constructed background

By Proposition prop:holonomy, the main conical metric has $G_{\mu \nu }=0$ on its regular part. This says nothing about a distributional defect source on $\mathcal{W}$, which is not constructed here. Its conical gluing is prescribed rather than sourced by the Enter residual.

More generally, if a cosmological Einstein–scalar background is adjoined, its equations must be imposed on the geometric side. A quotient or relational scalar construction does not insert a missing metric source equation. This is the equation-space discipline illustrated by III and VI CheungIII,CheungVI.