Proposition: The switch witness is a separate gate

Retain the copied first-order witness, on its original hypotheses,

In a coupling expansion, a correction beginning at ${\kappa }^2$ cannot cancel this nonzero order-$\kappa$ variation.

Proof. The order-$\kappa$ coefficient of a correction divisible by ${\kappa }^2$ is zero. The copied nonzero coefficient therefore remains unchanged.