Reciprocity is a separate classical obstruction

A variational Hessian is formally symmetric. If it agreed with the frozen source-switched metric map on every free input, its antisymmetric pairing on the displayed free pair would vanish, contradicting (eq:reciprocity). This is the specified source-preserving exclusion, not a theorem against every changed action prescription.

The master residual has ghost number one; $D_w(u,v)$ has ghost number zero and different arguments. Equation (eq:reciprocity) is not a loop anomaly and is not identified with ${\mathcal{B}}_a$.

For laboratory interpretation of this onset witness, its compact source support and the required Cauchy shadows must fit the inherited regular reserve. That is the existing X-G-ST support certificate, not a consequence of $\mathrm{supp}a⋐U$.