The relative master equation

The rule (eq:kin) preserves ${\mathfrak{F}}_W$: $sc$ is supported in $W$ because ${\mathfrak{g}}_W$ is closed under the bracket, and $s\gamma$, $s\varphi$ are supported in $W$ apart from the field-independent term absent at this order. So $S_{\mathrm{ext}}$ is a well-defined functional on ${\mathfrak{F}}_W$.