Proposition: Retained positive reciprocity witness

For these data,

The additional scalar interaction contributes zero to this metric diagnostic. In particular, it supplies no repair of the X-G exact source-preserving variational-upgrade obstruction.

Proof. Since ${\mathsf{P}}_{\mathrm{gr}}u={\mathsf{P}}_{\mathrm{gr}}v=0$, equation (eq:typedcommutator) gives ${\mathsf{Q}}_{w}u=-{\mathsf{P}}_{\mathrm{gr}}{\mathcal{L}}_{{\xi }_{\mathrm{on}}}u$ and likewise for $v$. The $00$ Lie components are

Their $0i$ components do not change the $00$ trace-reversed output used in the pairing. For such a tensor $z$, $({\mathsf{P}}_{\mathrm{gr}}z{)}_{00}=-\square z_{00}/8$. The spatial Laplacians integrate to zero, leaving

Two compact integrations by parts prove (eq:reciprocity).

The added potential is independent of $r$, so its metric first and second derivatives vanish identically. It changes neither ${\mathsf{N}}_w$ nor this pairing.