Remark: Support and scope of the repair

The construction gives compact support, but it does not assert that $q_{\mathrm{sm}}$ has support in the original ${\mathcal{D}}_{\mathrm{supp}a}$. More importantly, (eq:on-shell-equality) is an equality on the free mass shell, not on the full configuration space.

For the gauge-fixed graviton in (eq:free-action), the free equation is $\mathsf{T}\square h=0$, equivalent to $\square h=0$. Thus Theorem thm:onshell applies to the linear graviton observable. It does not construct an off-shell polynomial representative of the bilinear insertion, and it does not control contact terms in interacting retarded products.