The IX-A carrier and the output boundary

For self-contained use of the carrier name, we recall its admission criterion, not its proof. Let ${\mathcal{P}}_{\mathrm{dist}}$ denote actual finite polynomial functionals on smooth configurations, with compact symmetric distributional coefficient kernels. Let ${\mathcal{M}}_1$ be the space of compact linear metric kernels with no nonzero nonspacelike wavefront covectors, and put

Here ${G_{\mathrm{A}}}^0$ is the massless advanced inverse. IX-A defines

The empty direction list is allowed IXA (def:Padm).

The topology ${\tau }_{\Sigma }$ is the initial locally convex topology for the strong compact-distribution coefficient inclusion and all boundary-jet maps

IX-A proves that this trace is intrinsic, that ${\tau }_{\Sigma }$ is Hausdorff, and that fixed-direction differentiation is continuous IXA (lem:charge, def:topology, prop:top-basic). Its multiplication theorem concerns separately continuous classical pointwise multiplication only.

IX-A admits $B_f$, $W_g$, and the selected two-factor coefficients. This does not supply a quantum multiplication on all of ${\mathcal{P}}_{\Sigma }$, nor an assignment of either sharp triple association. A future compact polynomial output must satisfy (eq:admission); a numerical finite part alone is not a substitute for that test.