E3: admission and named continuity

For every coefficient, the proof must establish existence of its distributional pairings, compact configuration kernels, the metric derivative condition (eq:admission), all required descendants, and annihilation of actual-presentation kernels.

Existence on a test family does not establish admission. In particular, an expression omitting the boundary contribution in (eq:sharp) cannot be justified by referring to a different presentation of the same functional.

For definiteness, a possible weak fixed-support target is the following. Give ${\mathcal{P}}_{\mathrm{adm}}$ the locally convex topology generated by

for all configurations and finite direction lists under consideration. For every named test-function presentation with supports in fixed compact sets, require each $T_j$ to be continuous from the product of the corresponding ${C^{\infty }}_K$ spaces into this topology.

This is one explicit continuity strength; it is not asserted to follow from the supplied F9a statement. It neither claims normal-topology continuity on the full carrier nor hypocontinuity of multiplication. A different topology may be chosen, but it must be named and proved, rather than inferred from the absence of the joint VII-C assumptions.