Corollary: Coefficientwise order obstruction

Suppose the first coefficient is fixed by (eq:candidate). No choice of the coefficient ${{\mathcal{A}}_{f,g}}^{(2)}$ makes

invariant under (eq:free-identification) on the full stated domain. Nor can terms of positive $\hslash$-degree cancel the fixed nonzero $\kappa {\hslash }^0$ gap in a coefficientwise formal identity.

Proof. Extract the coefficient of $\kappa$, and then, if applicable, of ${\hslash }^0$. The value $\Delta \ne 0$ is unaffected by terms starting at higher orders.