Proposition: Coefficientwise causal dependence

For fixed free input ${\varphi }_0$, the coefficients of ${\mathcal{O}}_f[g]$ through order two depend on $h,k,g$ only through their restrictions to every neighborhood of ${\mathcal{D}}_f$.

In particular, their functional support in the metric variables is contained in ${\mathcal{D}}_f$. Equality with all of ${\mathcal{D}}_f$ is not asserted.

Proof. First-order scalar and coordinate solutions at $y$ use only their sources in the truncated past of $y$. All coefficient operators are local differential operators.

A nested second-order term has an intermediate point $z$ in the past of $y$. Every further point affecting that term is in the past of $z$, hence in the past of $y$. The same reasoning applies to $v$, including ${{\mathcal{Q}}_h}^{(0)}w$. Taking the union over $y\in \mathrm{supp}f$ proves the statement. Agreement on a neighborhood includes the derivatives occurring in these formulas.