Lemma: Normalized information at the available truncation

Suppose a formal numerator in ${{\mathcal{V}}_{\Sigma }}^{[3]}$ has zero classical term:

Its normalized quotient by $\hslash$ is determined only modulo ${\hslash }^3$:

It is not uniquely determined as an element of ${{\mathcal{V}}_{\Sigma }}^{[3]}$.

Proof. Multiplication by $\hslash$ in ${{\mathcal{V}}_{\Sigma }}^{[3]}$ annihilates ${\hslash }^{3}F$ for every $F\in {\mathcal{P}}_{\Sigma }$. Thus two quotients differing by such a term have the same numerator modulo ${\hslash }^4$.

Conversely, comparison of coefficients shows that the constant, linear, and quadratic quotient coefficients are exactly $N_1,N_2,N_3$. These are therefore uniquely determined in ${\mathcal{P}}_{\Sigma }{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^3)$.