Assumption: IX-A-ST carrier and topology

The carrier is the already adapted $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$ of IX-A-ST IXAST, not a reconstruction of the original global Minkowski carrier. It has the following properties.

1. Its elements are actual finite polynomial functionals with compact symmetric distributional coefficient kernels. The admission condition is

   $$({\mathrm{D}}_{\mathbf{\eta }}F{)}_h^{\prime }(\Psi )\in {\mathcal{S}}_1={\mathcal{M}}_1+{\mathcal{Q}}_{\Sigma }$$

   for every smooth configuration and every finite list of smooth directions.

2. The sharp linear kernels and intrinsic trace are

   $$q_a=L^{\ast }a=C^{\ast }(\theta {G_{\mathrm{A}}}^{\mathrm{chart}}a),{\mathcal{Q}}_{\Sigma }=\{q_a:a\in \mathcal{D}(V;T^{\ast }U{)}_{\mathbb{C}}\},$$
   $$\mathfrak{b}(m+q_a)=({G_{\mathrm{A}}}^{\mathrm{chart}}a){|}_{\Sigma }.$$

   Here ${G_{\mathrm{A}}}^{\mathrm{chart}}$ is the supplied advanced chart comparison prescription restricted to the retained support. Elements of ${\mathcal{M}}_1$ have no nonzero nonspacelike wavefront covectors; compact smooth kernels belong to ${\mathcal{M}}_1$. The trace is well defined on actual linear kernels.

3. The sharp adjoint retains

   $$C^{\ast }(\theta b)=\theta C^{\ast }b+\delta (t)\mathcal{B}(b{|}_{\Sigma }).$$

   No surface term is dropped.

4. Classical pointwise multiplication, conjugation, and fixed smooth directional differentiation preserve ${\mathcal{P}}_{\Sigma }$. Classical multiplication is separately continuous. The Hausdorff topology ${\tau }_{\Sigma }$ records strong compact coefficient kernels and all intrinsic trace coordinates.

5. The generating maps $f\mapsto B_f$, $g\mapsto W_g$ are fixed-support continuous, and $B_f$, $W_g$, the selected mixed coefficient below, and the selected scalar-contracted self coefficient belong to ${\mathcal{P}}_{\Sigma }$. Their specified fixed-support families and trace coordinates are continuous.

6. For every nonzero real $f$, the unchanged regular approximants satisfy

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

No quantum product on all of ${\mathcal{P}}_{\Sigma }$, no joint classical continuity, and no quantum hypocontinuity is included.