Lemma: Affine slot and compact scalar support

There is a symmetric bilinear test map $\mathcal{Q}$ such that

For fixed smooth data, $d_1[l],d_2[h,k]\in {\mathcal{D}}_C$, continuously in the smooth data near the retained causal reserve.

Proof. The $k$-linear part of the second coordinate equation is the first coordinate equation with input $k$. Uniqueness gives $v[h,k]=v[h,0]+w[k]$. The $k$-part of $c_2$ is therefore $c_1[k]$; the $k$-part of $L_2$ is $L_1[k]$. This proves the affine identity, with polarization of the remaining quadratic expression.

The bare representatives are supported in $K_f$. Every $L_j$-term is supported in $\mathrm{supp}g$, and advanced propagation restricts it to $J_{\eta }^{-}(K_f)$. Smooth Cauchy estimates and Green continuity prove the stated continuity.