Theorem: Unfiltered first-order derivation obstruction

On the unfiltered centrally extended algebra above, no operation of the form

can be a ${\star }_W$-derivation for every compact $\xi$, while retaining the stated central second-jet relations and canonical kernel $E$.

Proof. The relation $[M_a,Z_g{]}_{{\star }_W}=0$ would imply

At leading order,

The resulting first-order commutator is $2i\hslash E({A_{\xi }}^{†}a,g)\mathrm{id}$.

Terms from $\hslash R_{\xi }$ have no first-order commutator contribution because ordinary multiplication at order zero is commutative. Lemma lem:witness makes the displayed coefficient nonzero, a contradiction.