Remark

The retained identity is a statement, not a verification. For the finite one-loop, order-${\kappa }^2$, bulk two-point sector on $S_{\mathrm{inv}}$, with antifields and sources retained only through vertex-derived ghost-number $0$ seeds, the organization is

at one loop and order ${\kappa }^2$ in the retained sector, followed by a projection and a counterterm equation. Here $(\cdot ,\cdot )$ is the classical antibracket of XIV-A and XVI-N, and $A_1$ is the one-loop defect. Vanishing of the projected defect, up to admitted finite counterterms and BV-exact changes, is the goal of the symmetry restoration problem. It is not a claim that restoration has occurred.

This identity is not the classical statement $(S_{\mathrm{inv}},S_{\mathrm{inv}})=O({\kappa }^2)$. Classical consistency is infrastructure, not a substitute. It is not an automatic identification of an effective-action identity with a time-ordered-product identity. It is not a prepared $S_D$ identity. It is not XVI-M selected-image matching. Free equations of motion are not used: the free-EOM ledger is Unresolved. Verification of (eq:identity) is not Constructed. No claim is made that $A_1=0$ after counterterms. Bulk symmetry_restored stays false.