Lemma: The undeformed source is not a quantum presentation

There is no parity-preserving unital algebra homomorphism from ${{\mathcal{A}}^{\mathrm{cl}}}_{\mathrm{off}}[[\hslash ]]$ to ${\mathcal{A}}_0(M)$ sending every linear smearing to its corresponding causal generator.

Proof. The image of a graded-commutative source is graded commutative. But the scalar commutator is $-i\hslash ⟨u,E_{m}v⟩1$, nonzero for suitable tests.