Theorem: Free-shell descent

The space ${\mathcal{N}}_{\mathrm{sol}}[[\hslash ,\kappa ]]$ is a two-sided quantum ideal. Hence

is a surjective algebra homomorphism. There is an injective homomorphism $j_H:{\mathcal{A}}_0(M)[[\kappa ]]\to {\mathcal{E}}_H$ making

commute.

Proof. Fix $F\in {\mathcal{N}}_{\mathrm{sol}}$, $G\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, and $\Phi \in {\mathcal{C}}_{\mathrm{sol}}$. Graded polarization of the identically vanishing restriction gives that $w=F^{(k)}(\Phi )$ annihilates every solution tuple. By Corollary cor:fullseparation, $w={\sum }_i{{{\mathsf{K}}_{\mathrm{eq}}}_{\mathsf{i}}}^{\mathsf{T}}{r}_i$.

By Lemma lem:smoothing, $S=H^{\otimes k}{G}^{(k)}(\Phi )$ is a smooth multisolution. Consequently

The primitives need not be microcausal: they are paired with a smooth multisolution, not inserted into a distributional product.

The $k=0$ term is $F(\Phi )G(\Phi )=0$. Every higher contraction vanishes by the preceding argument. For $G{\star }_{H}F$, use the transposed orientation from Lemma lem:smoothing; no symmetry of the noncommutative product is assumed. Kernel evaluation and the displayed derivative contraction are the same proper composition, so no approximation step is needed. This proves the ideal property and formal descent.

Finally the restriction of $Q_H$ to ${{\mathcal{A}}^{\mathrm{reg}}}_{\mathrm{off},H}$ has kernel ${\mathcal{N}}_{\mathrm{reg}}[[\hslash ,\kappa ]]$ by Proposition prop:regularideal. The induced map $j_H$ is therefore injective, and the diagram commutes.