Proposition: Smooth first-derivative kernels

Every first derivative of each off-shell smeared coefficient, at a fixed smooth configuration, has a smooth compactly supported kernel with support in $D_f$. In particular, no initial-surface delta occurs in its metric derivatives.

Proof. Equations (eq:firstderivatives) and Lemma lem:kernel handle all but (eq:O2h). Put $b={\mathsf{E}}_{\mathrm{A}}a$. Transposing (eq:dv) gives

The first line is multiplication of $H$ by an explicit smooth tensor coefficient. The source $j$, extended by zero to negative times, is smooth and flat at $t=0$, and is compactly supported below the terminal surface. The extended transposition statement and Lemma lem:kernel apply again to $⟨j,LH⟩$. Thus (eq:O2h) is represented by a smooth compact tensor.

All constructions use nested advanced pasts of the original test, so their supports lie in $D_f$. Flatness of $\sigma$ eliminates initial boundary distributions.