Proposition: Globalization with a common finite inventory

Let an open cover of $U$ carry verified expansions of the restricted actual family. Suppose all local expansions use one common finite power–logarithm inventory, allowing zero coefficients, and have compact-local bounds of the form (eq:remainder).

Then the local coefficients and finite parts agree on overlaps, define distributions on $U$, and give Assumption ass:expansion globally.

Proof. On an overlap, uniqueness in Theorem thm:decision, applied to compact tests there, identifies the coefficient restrictions. The distributions therefore glue.

For any compact $L$, select a finite subordinate localization near $L$. Lemma lem:assembly gives the local remainder bound for the glued expansion. The common inventory is already finite and independent of $L$.