Theorem: IX-A admission and compatibility boundary

For the fixed data of Section sec:fixed, there is an explicitly defined space ${\mathcal{P}}_{\Sigma }$ of actual compact-distribution polynomial functionals with a Hausdorff locally convex topology ${\tau }_{\Sigma }$ such that:

1. ${\mathcal{P}}_{\mathrm{mc}}\subset {\mathcal{P}}_{\Sigma }$ injectively, and the inclusion is continuous on every fixed-support, fixed-wavefront bulk stage carrying its normal topology;

2. every $B_f$ and every linear coordinate functional $\mathsf{J}(a)=\int a_{\mu }(Lh{)}^{\mu }$ belongs to ${\mathcal{P}}_{\Sigma }$;

3. ${\mathcal{P}}_{\Sigma }$ is closed under arbitrary finite pointwise products, smooth directional derivatives, and complex conjugation;

4. fixed-direction differentiation and complex conjugation are continuous, and pointwise multiplication is separately continuous;

5. $f\mapsto B_f$ and $a\mapsto \mathsf{J}(a)$ are continuous linear maps from their test-function spaces;

6. if $f\ne 0$ is real and $B_{f,\epsilon }$ is obtained by smoothing both original field arguments with a compact approximate identity, then

   $$B_{f,\epsilon }/⟶B_{f}in{\tau }_{\Sigma };$$

7. the underlying classical functional is independent of its admissible kernel presentation.

No quantum multiplication on ${\mathcal{P}}_{\Sigma }$ is included in the statement.