Four corrections made before the proof

corrections made before the proof{ The full tree kernel is a compact distribution, not a smooth test in ${\mathcal{D}}_C$. Its smooth partial contractions do belong to ${\mathcal{D}}_C$.

The scalar-response part can have its first argument in $C\setminus K$, so the general two-point support is $C\times S$, not $K\times S$.

The coefficient of the tree is denoted $\tau$; its contribution is ${\kappa }^2\tau$, counted once.

The nonzero background defect is a defect of the specified one-tree truncation. It is not a no-go theorem for additional trees or a different gauge completion.}