Remark: Action-level regulation

Each ${{\mathsf{X}}_{\mathsf{q}}}^{\mu }$ is a critical point of $\frac{1}{2}\int {\mathfrak{A}}_{\epsilon }[q{]}^{\alpha \beta }{\partial }_{\alpha }{\mathsf{X}}^{\mu }{\partial }_{\beta }{\mathsf{X}}^{\mu }$ under compactly supported variations. The regulated prescription is therefore the action-level switching of VII-A with $g$ replaced by ${\sigma }_{\epsilon }$, not the multiplied equation $\square {\mathsf{X}}_1={\sigma }_{\epsilon }C[h]$. The two differ by the ${\sigma }_{\epsilon }^{\prime }$ term in (eq:V); see Lemma lem:kernel. The density ${\mathfrak{A}}_{\epsilon }[q]$ is the density of a formal auxiliary metric, since in four dimensions $\det \mathfrak{A}=\det q$ and $q^{\mu \nu }={\mathfrak{A}}^{\mu \nu }/\sqrt{-\det \mathfrak{A}}$; this fact is not used.