Assumption: Already adapted carrier

The pair $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$ is the IX-A-ST actual-functional carrier and topology, with the following inherited properties IXAST,CheungIXA.

1. Its elements are finite polynomial functionals with compact symmetric distributional coefficient kernels. Their admission criterion is

   $$({\mathrm{D}}_{\mathbf{\eta }}F{)}_h^{\prime }(\Psi )\in {\mathcal{S}}_1\text{for every smooth }\Psi \text{ and finite smooth direction list }\mathbf{\eta }.$$

   Here

   $${\mathcal{S}}_1={\mathcal{M}}_1+{\mathcal{Q}}_{\Sigma },{\mathcal{Q}}_{\Sigma }=\{L^{\ast }a:a\in \mathcal{D}(V;T^{\ast }U)\},$$

   and ${\mathcal{M}}_1$ consists of compact linear metric kernels with no nonzero nonspacelike wavefront covectors. The trace of a represented linear derivative is the intrinsic advanced-Cauchy trace defined by that carrier.

2. The topology records strong compact-distribution coefficients and all those trace coordinates. It is Hausdorff. Fixed-direction differentiation and conjugation are continuous. Classical pointwise products stay in ${\mathcal{P}}_{\Sigma }$ and are separately continuous. Joint continuity and hypocontinuity are not assumed.

3. The bulk inclusion is faithful. The functionals $B_f,W_g$ and the selected one-contraction coefficient with $W_g$ belong to ${\mathcal{P}}_{\Sigma }$, with all directional descendants required by admission.

4. For every nonzero real $f$, the unchanged compact-kernel approximants satisfy

   $$B_{f,\epsilon }/⟶B_f\text{in }{\tau }_{\Sigma }.$$

   A normalized continuous boundary separator detects this nonconvergence.

No quantum multiplication on all of ${\mathcal{P}}_{\Sigma }$ is part of this assumption.