Theorem: Continuity on the generated kernel stages

Fix common compact support reserves, a common plateau, and the positive width. Then:

1. the three kernel constructions are continuous in their smooth coefficient data on these reserves, into their stated compact normal microlocal stages;

2. contracting all quantum arguments except the scalar output in $M$ or $t_{\mathrm{self}}$ against fixed smooth fields gives an element of ${\mathcal{D}}_C$, continuously in those fields and data;

3. every two-slot smooth contraction of the symmetric matter cubic leaves a smooth test in ${\mathcal{D}}_C$;

4. a contraction leaving a graviton slot gives a smooth compact tensor test supported in $D_f$ in general, not necessarily in $C$;

5. the scalar tests in (ii)–(iii) pass continuously to $Q_m$.

The full mixed and cubic kernels belong to the polynomial carrier quotient $\mathcal{W}/\mathcal{J}$, not to the degree-one quotient $Q_m$.

Proof. The generated maps consist of finite smooth coefficient operations, fixed propagator distributions, the proper compositions of Lemma lem:sumzero, and fixed diagonal insertions. Their wavefront conditions and supports are already proved. The normal-topology operations therefore give continuity for these fixed-distribution, smooth-coefficient constructions BDH. No joint-continuity theorem for arbitrary varying singular kernels is asserted.

For smoothness of a one-slot remainder after partial contraction, its only possible wavefront covector would be $(0,\dots ,0,p)$ in $Z_n$. This forces $p=0$, so there is no wavefront. The support is the relevant projection of (eq:treesupport). Normal-topology pushforward with empty output cone is smooth convergence on compact sets. This proves the corresponding continuity into every smooth seminorm.

The quotient assertion follows from Lemma lem:closed.