Two cautions recorded on the sheet

1. The density representative is frozen before multiplication by $g$. The switched scalar density is ${\mathfrak{L}}_g=(1-g){\mathfrak{L}}_{\eta }+g{\mathfrak{L}}_q$ with ${\mathfrak{L}}_q$ the natural curved density (eq:Lq). An integration by parts performed before switching need not equal one performed afterwards; the multiplied-equation prescription $g\cdot (\text{curved operator})$ is not the Euler operator of the switched action (Appendix app:audit).

2. The convention is not $q=\eta +\kappa gh$. That substitution produces different powers of $g$ and different switch-derivative terms. It is not used anywhere below.

The coordinate prescription uses an auxiliary density different from the density of the switched action: the edge-regulated density ${\mathfrak{A}}_{\epsilon }[q]$ of (eq:Aeps), regulated by ${\sigma }_{\epsilon }$ and not by $g$. This split is declared here. It is the analogue of VII-A's split between switched scalar action and unswitched harmonic equation, with the unswitched equation replaced by the edge-regulated one.