Support requires a proved reference domain

Support in $N$ follows only if the reference has been proved throughout $U\setminus N$, or if a direct localization proof establishes that support. The supported-sector reference is not automatically a reference for the complete kernel.

For ordinary-product agreement, one sufficient microlocal package consists of localized nonopposing closed wavefront bounds for each actual internal/crossed pairing and normal-topology convergence of the smoothed crossed factors in those bounds. Fixed-factor product convergence then gives the ordinary contraction on that verified domain BDH.

A failed sufficient product criterion is not, by itself, a nonexistence theorem.