Theorem: Failure of the naive restriction ideal

For every nonempty $W$, the space ${\mathcal{N}}_W[[\hslash ,\kappa ]]$ is not a two-sided ideal of the inherited product. Its image

is not a two-sided ideal in ${\mathcal{E}}_H$ either. These assertions hold for every admissible ordering.

Proof. Choose $q,h$ from Lemma lem:compact-pair. If the formal vanishing space were a two-sided ideal, both $L_q{\star }_H{G}_h$ and $G_h{\star }_H{L}_q$ would belong to it. Their difference would then belong to it. But (eq:commutator) is the formal scalar $i\hslash C(q,h)1$.

The zero configuration belongs to ${\mathcal{C}}_W$, so a nonzero constant cannot vanish on ${\mathcal{C}}_W$. Its $\hslash$ coefficient is therefore outside ${\mathcal{N}}_W$. This is a contradiction. The same commutator descends through the full-shell ideal, and constants survive that quotient, proving the on-shell assertion. The causal scalar is ordering-independent.