Lemma: One-order loss

There is a well-defined coefficient shift

Numerator data through order ${\hslash }^3$ determine a normalized quotient only through order ${\hslash }^2$.

Proof. The coefficient expansion gives the shift uniquely. In ${\mathcal{P}}_{\Sigma }{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4)$, multiplication by $\hslash$ annihilates every ${\hslash }^{3}C$. Hence that final coefficient of a putative quotient is undetermined. Equivalently, $\hslash$ is not invertible in $\mathbb{C}[\hslash ]/({\hslash }^4)$.