Lemma: Finite localization assembly

Let $L⋐U$, and let a fixed finite smooth partition ${{\sum }_{\ell =1}}^N{\zeta }_{\ell }=1$ hold near $L$. Suppose each exact localized complete family $t_{\delta }({\zeta }_{\ell }\Phi )$ has an expansion with a compact-uniform vanishing finite-order remainder.

Their sum, with equal factors collected, gives such an expansion on tests supported in $L$. Its inventory is finite on this support.

Proof. Sum the exact localized identities. Multiplication by the fixed cutoffs is bounded in every required test seminorm. Take the maximum derivative order and sum the finitely many remainder envelopes. The resulting envelope tends to zero.