Proposition: No cancellation by a second-order correction

Under Assumption ass:switch, a modification beginning at coupling order ${\kappa }^2$ cannot remove the order-$\kappa$ switch gap while the first-order prescription is retained.

Proof. The witness compares

If a corrected expression has formal variation

vanishing of the formal series requires the coefficient of $\kappa$ to vanish. It does not. Later coefficients cannot alter that coefficient.