Corollary: No higher-order repair of the fixed residual

With ${A_1}^{(g)}$, the free input, and the larger-support identification fixed, no regular addition beginning at ${\kappa }^2$ makes the observable invariant under all those compact generators. Terms of positive $\hslash$-degree cannot cancel the nonzero $\kappa {\hslash }^0$ residual either.

Proof. Equality of formal series with nonnegative powers is coefficientwise. Extract the coefficient of $\kappa {\hslash }^0$ and use (eq:negativegap).