Normal convergence versus boundary convergence

The kernel approximants converge strongly as compact distributions. They also converge in the indicated normal stage. Neither statement forces their boundary charges to converge.

The new topology deliberately retains information that is not continuous in that normal stage. The nonzero witness proves this rather than assuming it.

This is a topology for a subsequent admission problem, not a renormalization prescription. Any proposed product must still prove its own continuity properties in this topology or specify a further change.