Lemma: Positive quantum orders cannot repair a classical residual

In a vector-valued formal series, adding $\hslash C$ cannot change the coefficient of ${\hslash }^0$. In particular, it cannot cancel a nonzero classical residual such as (eq:cutoffbreaking).

Proof. Formal equality is coefficientwise equality. The added series has no ${\hslash }^0$ coefficient.