The ordinary symbol test is not an anomaly calculation

Suppose, conditionally, that the XX-D scalar Wick correction ${\Gamma }_f$ has been defined for a selected representative, and write its free symbol as

With the background, covariance, subtraction, and test held fixed, ${\Gamma }_f$ is independent of $z,\zeta$. Therefore ordinary off-shell coefficient differentiation gives

by Theorem thm:classicalWard.

This calculation does not supply:

• a derivation of the relevant quantum products;

• preservation of the free equation quotient;

• transformation of covariance and subtraction choices;

• an interacting retarded insertion or its anomaly;

• repair of the retained subsidiary contraction defect.

The distinctions are concrete: Theorem thm:translation and Proposition prop:central already prevent promoting the ordinary coefficient test to the requested quantum identity without additional structure.

In a future admitted domain, a specified operation could have the expansion

Then its defect on (eq:symbol) would be

Without a constructed ${\mathcal{A}}_{\xi }$, this is an unevaluated interface, not a computed breaking.

Anomalous Ward and renormalized BV frameworks address precisely specified quantum operations FRBV,BDFrR. Their existence as frameworks does not set the present ${\mathcal{A}}_{\xi }$ to zero.