Proposition: Regulator-thickened metric dependence

Through order ${\kappa }^2$, the observable depends on $h,k$ only in

and on $\varphi$ only in $\mathrm{supp}f$, in the usual sense that agreement on neighborhoods of these sets gives agreement of the coefficients.

Proof. A coordinate source point $u$ affecting an observation point $z$ lies in $J_{\eta }^{-}(z)\cap \{t\ge 0\}$. A convolution input $y$ affecting $H(u)$ or $J(u)$ satisfies

For a nested second-order term, its intermediate wave source points remain in ${\mathcal{D}}_f$ by transitivity of the causal relation. Each metric input is within one convolution radius of such a point. The scalar argument and its derivatives are evaluated only on $\mathrm{supp}f$.