The question and the exact change of admission

A compact distributional kernel need not belong to a chosen quantum product domain. Admitting the kernel to a larger vector space does not prove that its contractions close there. Finally, choosing a finite part for a numerical coefficient does not prove that the finite parts arise from one associative product.

This manuscript addresses the first of these steps and its topology. It is the IX-A module of a larger coherent-product problem.

The sharp insertion is kept fixed:

No field equation is imposed on $h$ or $\varphi$. The observation test $f$ is smooth, compactly supported, and strictly to the future of the initial surface.

The required change from normal-topology admission is explicit. The new topology records the Cauchy trace responsible for the sharp surface term in metric derivatives. Regular kernels have zero such trace. The sharp insertion need not. Therefore the exact sharp insertion is not declared to be the continuous limit of its unchanged regular approximants.