Theorem: Complete coefficient formula on an admitted domain

If (eq:expansion) is constructed on the same native domain and its classical symbol is (eq:O), then the retained breaking is

If, in fixed normalization coordinates,

where ${\Gamma }_f$ is a defined field-independent scalar Wick correction, then

Neither $R_{1,f}$ nor ${\mathcal{A}}_{\xi }{O}_f$ may be set to zero merely because the free scalar correction has zero ordinary field variation.

Proof. Multiply the formal expansions. The order-zero term is ${\mathcal{V}}_{\xi }{O}_f=0$ by Lemma lem:classical. The order-one term is exactly (eq:breaking). Field independence gives ${\mathcal{V}}_{\xi }{\Gamma }_f=0$ in the stated fixed coordinates. It says nothing about the other two terms.