Conclusion and the boundary of IX-A

The IX-A construction closes the following chain:

The carrier consists of actual functionals. Its bulk inclusion is faithful, its classical presentations agree, and its directional derivatives and arbitrary finite pointwise words remain admitted. Smooth test variation is continuous.

The change from normal-topology admission is explicit and proved: the unchanged regular approximants have zero boundary trace and cannot converge to a sharp insertion with nonzero trace. This is not an altered value for their old Wick products.

The completed topological multiplication statement is separate continuity of classical pointwise multiplication. Nothing in the proof establishes quantum hypocontinuity, a renormalized associative product, or a continuous extension of the failed mixed-Wick coefficient.