Corollary: A higher coefficient cannot repair this defect

With fixed free input and the same first-order coefficient ${A_1}^{(g)}$, adding a regular coefficient ${\kappa }^2{B}_2$ cannot make the observable invariant under every gauge parameter of Proposition prop:witness.

Proof. In a formal series with nonnegative powers of $\kappa$, invariance requires the coefficient at each order to vanish. A nonzero coefficient at order $\kappa$ cannot be cancelled by a coefficient at order ${\kappa }^2$.