Lemma: Affine second-metric slot

There is a symmetric bilinear test map $\mathcal{Q}[h_1,h_2]$ such that

For fixed smooth backgrounds,

Their dependence on the backgrounds near the retained causal reserve and on fixed-support observation tests is continuous in the smooth topologies.

Proof. The $k$-linear parts of $A,B,L_2$ are respectively $a_1[k],b_1[k],L_1[k]$. Equation (eq:coordinates) and uniqueness give

The $k$-part of $c_2$ is therefore $c_1[k]$. Subtracting $L_1[k]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$ proves the affine assertion; polarize the remaining quadratic expression.

The bare tests are supported in $K$. Each term with an $L_j$ is supported in $\mathrm{supp}g$, and advanced support restricts it to $J_{\eta }^{-}(K)$. Finite-time smooth Cauchy estimates, Green continuity, and finite differentiation prove continuity.