Theorem: Completed background Ward identity

For every $F\in {\mathcal{G}}_U$, with the free field held fixed,

Equivalently, ${\hat{A}}_1[h^{\prime }]={\hat{A}}_1[h]$ and ${\hat{A}}_2[h^{\prime },k^{\prime }]={\hat{A}}_2[h,k]$.

Proof. The coordinate plateau gives ${\Theta }_F=0$ by Lemma lem:theta. The interaction plateau gives ${\Sigma }_F=0$ by Lemma lem:sigma: $g=1$ and $\mathrm{d}g=0$ on a neighbourhood of both generator supports.

Equation (eq:variation) reduces to

with a compact function in each coefficient. Every difference therefore lies in the specified equation ideal. Passing to free-field classes proves the theorem.