The exact source-preserving upgrade obstruction

Let $Q_w$ be the first background coefficient of the natural metric Hessian including background de Donder gauge:

Its tensor types give the identity

It follows by differentiating naturality under simultaneous background and fluctuation pullback XF,XG.

X-F's mixed equation-level source map is

For compact $w$, $Q_w$ is formally symmetric, but ${N_w}^{\mathrm{T}}=Q_w{\sigma }_{\epsilon }$. A symmetric action Hessian $L_w$ reproducing $k_w[u]$ on every smooth free metric input would necessarily satisfy

Indeed, equality of retarded fields and application of ${\mathsf{P}}_{\mathrm{gr}}$ imply $L_{w}u=N_{w}u$; symmetry then gives (eq:reciprocity). Compact background coefficients make these pairings well defined.

Choose nonnegative $\zeta \in {C_c}^{\infty }((0,\epsilon ))$, nonzero where ${\sigma }_{\epsilon }^{\prime \prime }\ne 0$, and

Put