Theorem: Hausdorffness and separate continuity

The carrier is Hausdorff and ${\mathcal{N}}_{\mathrm{sol}}$ is closed. The formal shell quotient is Hausdorff. Quantum multiplication is globally separately continuous. The maps $s_0$, ${\alpha }_d$, $Q_H$, and their descended maps are continuous; ${\bar{\alpha }}_d$ is a homeomorphism.

Proof. On a fixed support stage the map to ${\mathcal{E}}^{\prime }(M^n{)}_{\beta }$ is continuous: for bounded $B\subset C^{\infty }(M^n)$ choose a compact cutoff $\chi =1$ near the support, and use the bounded test family $\chi B\subset \mathcal{D}(M^n)$. The universal property gives a continuous injection of each inductive limit into the Hausdorff distribution space. The direct sum is consequently Hausdorff.

Every Grassmann coefficient of evaluation at a fixed smooth shell configuration is continuous on every stage, by the same cutoff argument, hence on the direct sum. Their common kernel is ${\mathcal{N}}_{\mathrm{sol}}$, which is closed. The coefficientwise quotient map is the product of open linear quotient maps; its kernel is ${\mathcal{N}}_{\mathrm{sol}}[[\hslash ,\kappa ]]$. This identifies the formal quotient topology with the product topology on ${{\mathcal{F}}^{\mathrm{sol}}}_{\mathrm{mc},\mathrm{pol}}[[\hslash ,\kappa ]]$.

Fix a polynomial $G$. On each input stage, multiplication by $G$ is a finite sum of continuous linear contractions into target stages, by Theorem thm:stagecontinuity. The inductive-limit and direct-sum universal properties give global continuity of that linear map. For a fixed formal $G$, any output coefficient depends on finitely many input coefficients and finite contraction orders. The same argument proves formal separate continuity.

The differential acts stagewise by differential operators. Each coefficient of ${\alpha }_d$ acts by smooth multiplication and proper pushforward. These maps are globally continuous by the same universal properties. They preserve the null ideal, so continuity descends through the quotient. The inverse ordering map has the same construction.