Corollary: The one-tree defect is a truncation defect

Delete $T_{\mathrm{mix}}$ and $T_{\mathrm{self}}$ from (eq:alltree). The remaining matter-only expression has defect

It is the negative of the background variation of the omitted mixed tree. A nonzero witness for that truncation is not an obstruction to Theorem thm:main.

Proof. The matter and self terms are inert; the mixed variation is the cancelling term computed above.