Corollary: Route II cross-check, not the proof of record

The normalization to X-A's classical coefficient and its on-shell covariance provide an independent pairing cross-check.

Proof. If a compact test $d$ annihilates every smooth free solution, take the solution $u=({{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}})d^{\prime }$ for arbitrary compact $d^{\prime }$. Antisymmetry gives $({{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}})d=0$. The Green-operator exact sequence then gives $d\in P_m\mathcal{D}(M)$ BGP (Theorem 3.4.7).

By normalization and X-A's on-shell identity, the compact test $e_2[q^{\prime }]-e_2[q]$ annihilates every smooth free solution. It therefore lies in $P_m\mathcal{D}(M)$.