Theorem: Regular quantum presentation

There is a canonical surjective homomorphism

sending linear symbols to causal generators, with kernel ${\mathcal{N}}_{\mathrm{reg}}[[\hslash ,\kappa ]]$.

Proof. Let $V=\mathcal{D}/{{\mathsf{K}}_{\mathrm{eq}}}^{\mathsf{T}}\mathcal{D}$, with the species grading. The regular classical quotient is $\mathrm{Sym}(V_{\mathrm{even}})\otimes \bigwedge (V_{\mathrm{odd}})$. The contractions descend to $V$ by the bisolution equations. Their graded antisymmetric part is $i\Delta$, so linear generators satisfy exactly (eq:CCR).

The graded symmetrization map from the causal presentation to this deformed symmetric algebra is bijective: on finite words its highest field-degree part is ordinary graded symmetrization, and all contraction corrections lower degree. The triangular map is invertible degree by degree. Every word uses a finite-dimensional coordinate span; no finite-dimensional assumption on $V$ is needed. Extending coefficientwise gives the formal isomorphism. Composing its inverse with the quotient map defines ${\pi }_{\mathrm{free},H}$ and proves the kernel claim.