Ordinary-product agreement

A sufficient localized reference package has closed bounds ${\Gamma }_K,{\Gamma }_C$ satisfying

and normal-topology convergence of the pulled-back smoothed crossed components in the fixed cone ${\Gamma }_C$. Then fixed-factor multiplication converges by hypocontinuity BDH (Theorem 6.1). The actual pullbacks and supports must already be verified.

Smooth weights, derivatives, and finite sums preserve the resulting strong distributional convergence. This identifies an existing limit with the ordinary contraction on that proved domain. Merely having an ordinary product does not supply convergence in an unspecified stronger topology.