Corollary: Class, ordering correction, enlarged sector

$Q_H{{\Phi }^{(2)}}_{\mathrm{D}}(f,g)\in {\mathcal{E}}_H$ has even parity and ghost number zero. For an admissible smooth change $d=H^{\prime }-H$,

with $T$ the species-symmetric kernel of ${{\Phi }^{(2)}}_{\mathrm{D}}$ and ${\ell }_d$ compact and smooth. The coefficientwise $(\hslash ,\kappa )$-adic completion of the algebra generated by XII-D's enlarged named sector and all $Q_H{{\Phi }^{(2)}}_{\mathrm{D}}(f,g)$ is mapped by ${\bar{\alpha }}_d$ onto its ${\star }_{H^{\prime }}$ counterpart generated by the unchanged formulas.

Proof. XII-A's free-shell descent applies to any element of ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$. The ordering computation is XII-D's Proposition on the regular ordering correction, whose proof uses only that the cubic kernel is compact with $\mathrm{WF}\subset Z_3$: a nonzero wavefront point of ${\ell }_d$ would need $(0,0,p)$ in the cubic wavefront set, forcing $p=0$. Sector stability follows as in XII-D: each new generator is transported to itself plus a regular linear field, which already belongs to the old sector; the inverse map gives the reverse inclusion; formal order is preserved.