The copied partial-diagonal class

This class includes $y=z$ with $x\ne z$. It is not replaced by ${\mathfrak{C}}_{\mathrm{tot}}$. A singular spectator is not declared smooth merely because its external weight is smooth.

The extension domain is the configuration space of the selected graphs, not ${\mathcal{C}}_W$. The four obligations are exactly Input statement input:four.