Remark

The converse of (i)—that extendability of all ${b^{\circ }}_{D,\gamma }$ implies extendability of each $c_{D,\beta }$—is not claimed. It would require rank conditions on the multipliers $v_{\alpha }$ (for instance, invertibility of the triangular system (eq:collected) locally), which depend on the unsupplied formulas. Nor does (i) yield a realization in the frozen class: it is the unrestricted class, and compatibility with the frozen prescription, continuity on ${\mathcal{P}}_{\mathrm{int}}$, and the microlocal requirements are separate obligations.