Corollary: No second-order cancellation of the witness

No modification beginning at order ${\kappa }^2$ makes the retained functional gauge invariant on ${\mathfrak{g}}_U$ while preserving its first-order coefficient.

Proof. Formal invariance requires equality of each coefficient separately. Theorem thm:preflightstop gives a nonzero coefficient of $\kappa$. A coefficient beginning at ${\kappa }^2$ cannot change it.