Reciprocity is a separate classical obstruction

Let ${\mathsf{Q}}_w$ denote the first background coefficient of the natural gauge-fixed metric Hessian, with

The two Lie derivatives have different tensor types. Define the frozen source-switched operator

For free metric solutions $u,v$, its reciprocity defect is

These are the X-G/X-G-ST diagnostic objects XG,XGST, not the Euler operator of (eq:newaction).

Choose a nonnegative compact spatial bump ${\chi }_0$ with $M_{\chi }=\int {\chi }_0{\mathrm{d}}^{3}x>0$, and choose $\zeta \in {C_c}^{\infty }((0,\epsilon ))$, nonnegative, such that

Set

with all other components zero. The background is compact in the onset layer. The vector ${\xi }_{\mathrm{on}}$ is not being declared a member of the relative plateau algebra ${\mathfrak{g}}_U$.