Proposition: Order loss under coefficient shifting

The linear coefficient-shift map

is well defined. Order-three numerator data do not determine an order-three normalized coefficient in ${\mathcal{P}}_{\mathrm{adm}}{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$.

Proof. The coefficient expansion in ${\mathcal{P}}_{\mathrm{adm}}{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$ is unique, so $L$ is well defined. Within ${\mathcal{P}}_{\mathrm{adm}}{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$, multiplication by $\hslash$ annihilates every ${\hslash }^{3}C$. Hence, for every $C\in {\mathcal{P}}_{\mathrm{adm}}$,

The coefficient $C$ is undetermined. Equivalently, $\hslash$ has no inverse in $\mathbb{C}[\hslash ]/({\hslash }^4)$.