Assumption: Carrier, charge, and classical operations

The carrier ${\mathcal{P}}_{\Sigma }$ consists of actual finite polynomial functionals with compact symmetric distributional kernels, subject to

for every smooth configuration $\psi$ and every finite list of smooth configuration directions $\mathbf{\eta }$.

The linear sharp kernels and intrinsic charge are

The charge is well defined on actual linear kernels and vanishes on ${\mathcal{M}}_1$. Every element of ${\mathcal{M}}_1$ has no nonzero nonspacelike wavefront covector; compact smooth metric kernels belong to ${\mathcal{M}}_1$.

The sharp adjoint retains its surface term:

Classical multiplication, smooth directional differentiation, and conjugation preserve ${\mathcal{P}}_{\Sigma }$. Classical multiplication is separately continuous in the locally convex topology ${\tau }_{\Sigma }$, which records strong compact-distribution coefficients and all intrinsic trace coordinates.

The generating test maps are injective and have fixed-support test-family continuity. The inherited nonconvergence $B_{f,\epsilon }\nrightarrow B_f$ in ${\tau }_{\Sigma }$ for nonzero real $f$ is retained, but is not used as a charge witness.