Lemma: Causal support of every field component

For every structural proposal in ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ and every smooth input,

Every component vanishes on an open time band preceding $S_{\xi }$.

Proof. The parameter-local terms are supported in $S_{\xi }$. The retarded output is supported in its causal future. Differentiation and multiplication by smooth coefficients do not enlarge distributional support. Since $S_{\xi }$ is compact inside $\{t>\epsilon \}$, it has a minimum time strictly above $\epsilon$.