Theorem: First-order background Ward identity

For every classical background $h$ and every $\xi \in {\mathfrak{g}}_U$,

Thus $\hat{\varphi }(f)+\kappa {\hat{A}}_1[h]$ is invariant modulo ${\kappa }^2$, with the free field held fixed.

Proof. The first term of (eq:variation) vanishes in ${\mathfrak{A}}_0$. The copied preflight identity kills the second. On $\mathrm{supp}\xi \subset U$, $g=1$ and $\mathrm{d}g=0$; hence ${\Sigma }_{g,\xi }=0$. These are exactly the declared support conditions, not restrictions introduced after the defect calculation.