Lemma: Variational reciprocity on free inputs

Let an even variational interaction have reduced mixed metric Hessian $L_w$, with compact coefficients. If its classical response agrees with $-{{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{\mathsf{N}}_{w}u$ on all free metric solutions, then

for all such solution pairs.

Proof. Symmetry of second directional derivatives gives

Equality of the retarded responses, followed by application of $P_{\mathrm{gr}}$, gives $L_{w}u={\mathsf{N}}_{w}u$ on every free input. Substitute into the symmetric pairing. Compactness of the coefficients removes all boundary terms.