Proposition: Admission does not remove the first-order switch gap

The domain and topology constructed here do not alter (eq:gaugeB). If a switched realization has zero forced response variation when $gK\xi =0$, a nonzero value of (eq:gaugeB) remains a first-order obstruction to that realization.

A modification beginning at coupling order ${\kappa }^2$ cannot remove this coefficientwise order-$\kappa$ gap.

Proof. Classical normalization retains the exact functional $B_f$. The topology does not change its value on any smooth configuration. Formal coefficient equality requires equality at each coupling order. Terms beginning at order ${\kappa }^2$ do not change the coefficient at order $\kappa$.