Corollary: Complete ordinary-product convergence

On compact subsets of ${\mathcal{O}}_K$,

strongly in distributions, independently of the normalized compact mollifier and of a cutoff equal to one near the relative image.

The limit agrees with the inherited canonical value where that value is the ordinary contraction satisfying (eq:OP).

Proof. Apply Lemma lem:local-product to every pairing in Appendix app:inventory. Smooth weights, finite sums, and output derivatives preserve convergence. Each admissible mollifier has the same ordinary product as its limit.