Theorem: Boundary carrier and nonconvergence

The space $({\mathcal{P}}_{\Sigma ,\min },{\tau }_{\mathrm{bd}})$ is Hausdorff. Fixed smooth directional differentiation and conjugation are continuous. Pointwise multiplication is separately continuous. Bulk inclusion is injective and continuous on fixed-support, fixed-wavefront normal-topology stages.

For every nonzero real $f$, the unchanged regular approximants

do not converge to ${\mathcal{B}}_f$ in ${\tau }_{\mathrm{bd}}$. No quantum product is included in this theorem.

Proof. Symmetric coefficient kernels determine an actual polynomial uniquely by polarization. Their continuous map into the Hausdorff distribution spaces therefore separates points.

Differentiation contracts coefficient kernels with fixed smooth directions, and sends each charge seminorm to another seminorm in the defining list. Conjugation has the same property.

On a fixed stage, multiplication by a fixed polynomial uses finite tensor products with fixed compact distributions and finite symmetrization. These are continuous on the relevant strong fixed-support spaces. Charges of every directional derivative of a product are finite Leibniz sums of evaluations and charges, which are continuous there. The inductive-limit universal property gives global separate continuity.

The normal topology includes the strong distribution seminorms. Bulk charges vanish, so fixed bulk-stage inclusions are continuous. They are injective because actual functionals are not quotiented further.

Use the witness configuration of Theorem thm:sharp. The charge of ${\mathcal{B}}_f$ is a nonzero smooth boundary tensor. Choose a compact boundary test detecting it. Every ${\mathcal{B}}_{f,ϵ}$ has a smooth kernel, hence zero charge at every configuration. The continuous charge pairing separates the proposed limit from all approximants.