Theorem: Exact defect of the specified truncation

In $\mathcal{W}/\mathcal{J}$, modulo ${\kappa }^3$,

Thus the one-tree expression is invariant through order $\kappa$. The defect is independent of the classical backgrounds and of $\zeta$. The scalar-only tree $\tau$ contributes zero to this background variation.

Proof. Let $z=h+r$ first be a sum of two smooth tensors. For a full two-jet transform of $z$, the required second coefficient would be

Lemma lem:scalaridentity is a polynomial identity in $z$ and $k$. Polarization and fixed-width admission permit replacement of $r$ by $\mathbf{r}$ and total Wick ordering. Its right-hand sides are compact scalar-slot $P_m$-images, hence belong to $\mathcal{J}$ by Lemma lem:ideal.

For the full transformed second slot the completed bare multiplet is therefore invariant modulo $\mathcal{J}$. The actually frozen second slot is smaller by ${\mathcal{L}}_{\xi }\mathbf{r}$. The affine identity (eq:affine) subtracts exactly ${\hat{A}}_1[{\mathcal{L}}_{\xi }\mathbf{r}]$ from the second coefficient. Finally $\tau$ has no classical-background dependence.