Microlocal agreement with its convergence hypotheses

Let $V\subset U$ be an open domain. A sufficient localized agreement hypothesis is the following: after the needed localization, each pair consisting of an internal component and a pulled-back crossed vertex component has closed wavefront bounds ${\Gamma }_K,{\Gamma }_C$ satisfying

and the smoothed crossed components converge to the unsmoothed ones in the normal topology with that bound ${\Gamma }_C$.

Under these hypotheses, the fixed-factor product converges to the ordinary product by hypocontinuity BDH (Theorem 6.1). Smooth weights, output derivatives, and the finite sums preserve strong distributional convergence. Hence any existing characteristic limit agrees on $V$ with the ordinary vertex contraction.

The supplied Em01-B-v3 excerpt reports such localized ordinary-product convergence on its verified domain Em01B. That report is used only with its stated localization and topology hypotheses; the mere existence of an ordinary product is not treated as convergence in an unspecified stronger topology.

If two characteristic limits have the same reference on $V$, their difference is supported in $U\setminus V$. Support in $N$ requires a common reference throughout $U\setminus N$. No equality

is asserted.

The supported-sector reference of XIV-J concerns $K_{\mathrm{supp}}=K_6+K_8+K_9$, not the full retained sum XIVJ. It does not identify the latter agreement domain.