Classical response freedom

For any two harmonic response maps solving $G^{(1)}({\mathcal{R}}_{i}Q)=Q$, their difference is a homogeneous harmonic linearized vacuum solution. Equality of trace-reversed Cauchy data implies equality of the maps by wave uniqueness. Without equality of data, no equality follows from the sourced equation alone.

For the two actual prescriptions in Section sec:classical3, the comparison was executed: their difference is the explicit $K\xi$ in Proposition prop:mapcomparison. They therefore agree on compact conserved currents, including $J_{\psi }$, but not on arbitrary metric test tensors. This is the precise transfer available here.