Proposition: Coefficientwise causal dependence

For fixed free input ${\varphi }_0$, the coefficients through order ${\kappa }^2$ of ${\tilde{\mathcal{O}}}_f[g]$ depend on $h$ and $k$ only through their values on every neighbourhood of ${\mathcal{D}}_f$. The switch dependence has the same causal bound.

In particular, the functional support in the metric variables is contained in ${\mathcal{D}}_f$, not necessarily in $\mathrm{supp}f$.

Proof. The Cauchy operator ${\mathsf{E}}_m$ has retarded support. Therefore ${\varphi }_{1,g}(y)$ and $w(y)$ depend only on sources in $J_{\eta }^{-}(y)\cap M_{+}$.

The local differential operators ${\mathsf{V}}_g$, ${\mathsf{W}}_g$, ${{\mathsf{Q}}_{\mathsf{h}}}^{(0)}$, and $C$ do not enlarge support. For a second-order nested term, an intermediate point $z$ lies in $J_{\eta }^{-}(y)$, and every point affecting its first-order coefficient lies in $J_{\eta }^{-}(z)\cap M_{+}$. Transitivity gives

This proves the asserted bound for ${\varphi }_{2,g}$ and $v$, and hence for every term in (eq:A2). Taking the union over $y\in \mathrm{supp}f$ gives ${\mathcal{D}}_f$.