Theorem: Convergence of the completed tests

For each fixed input satisfying the sheet, $d_{1,0},d_{2,0}\in {\mathcal{D}}_C(M)$, and for every $N$,

Thus $d_{j,\epsilon }\to d_{j,0}$ in ${\mathcal{D}}_C(M)$, and their free-field classes converge in $Q_m$.

Proof. Proposition prop:coordinate-convergence controls every coordinate derivative on a neighbourhood of $K$. The formulas (eq:bare) are finite smooth products and derivatives, multiplied by $f$; their differences are $O(\epsilon )$ in every compact-test seminorm. Lemma lem:reduction gives the same rate for the completed tests. Their supports lie in the fixed compact $C$, and the quotient map is continuous.