Theorem: Ordering-stable enlarged named sector

The inherited ordering isomorphism maps ${{\mathcal{S}}_{\mathcal{H}}}_{\mathrm{D},H}$ onto the corresponding ${\star }_{H^{\prime }}$-sector generated by the unchanged new formulas:

Proof. XII-A transports the old sector onto its counterpart. By Proposition prop:ordering, each new generator is transported to the unchanged new generator plus a regular linear field. Regular linear fields already belong to the old sector. The inverse ordering map gives the reverse inclusion. Formal-order preservation gives equality of the coefficientwise completions.