Proposition: F9a on the completed degree-two blocks

The coefficient families in (eq:degree2maps) are jointly continuous on fixed-support generating test spaces, with values in $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$. This does not assert continuity of ${q_L}^{-1}$ or ${q_R}^{-1}$ in the inherited subspace topology.

Proof. Multiplication of each fixed graph distribution by a varying smooth joint coefficient is continuous in its wavefront-controlled distribution space. The proper pushforwards and fixed transpositions in Proposition prop:existence then give strong coefficient continuity BDH.

For trace coordinates use (eq:treeadmission). The one-vertex pushforwards are continuous into smooth functions by the verified empty-wavefront pushforward bound. Applying the fixed smooth advanced trace map to $a_h$ is continuous. Replacing $h$ by any fixed direction proves the same for all nonzero directional trace coordinates. For the scalar parent, a fixed distribution has a finite order on every compact set, which bounds product-test pairings by finitely many test seminorms. No inverse presentation map appears in these estimates.