Proposition: Fixed-support first-stage continuity

On fixed compact observation supports, $(f,g)\mapsto Q_{f,g}$ and its reverse are jointly continuous into the strong coefficient topology. Their metric-derivative trace coordinates vanish, also after all required directional insertions. The first-stage generating families are continuous into $({{\mathcal{P}}_{\Sigma }}^{[3]},{\tau }_{\Sigma })$.

Proof. The coefficient of $Q$ is obtained by multiplying the fixed distribution $K$ by $f\otimes g$ and taking one fixed scalar-slot derivative. On common supports, finite-order distribution estimates bound this operation by finitely many smooth seminorms of $f,g$, uniformly on bounded smooth test sets.

Contracting its scalar slot with a smooth configuration leaves a compact smooth metric kernel, by the selected output property in the carrier input. Replacing the scalar configuration by a fixed scalar direction has the same property. A metric insertion removes the only metric slot. All higher derivatives vanish. Thus every relevant trace coordinate is zero. The classical first-stage term has the inherited generating test-family continuity. Combining its coordinates gives the last assertion.