Remark: The plan's trichotomy in this module

For every tested null presentation $a\in H$ the outcome is never “defect on an open set away from the puncture.” Either the defect vanishes on all of $U$ (when $a\in H_T$) or it is a nonzero distribution carried by $Z\times \{0\}$. The second case is the genuine content of “vanishes only on the canonical domain”: it is produced not by the insertion structure but by tangential annihilator relations $\beta T^{\circ }=0$ that fail for $T$.