Zero-switch and flat controls

If the switch is specialized to zero in both the action and the coordinate prescription, then

and

The scalar interaction vertices also vanish, so ${\rho }_{1,0}={\rho }_{2,0}=0$.

This is a normalization specialization, not a gauge theorem for a nonempty plateau at $g=0$. The frozen gauge algebra belongs to the nonzero switch (eq:switch).

If $h=k=0$ with the original frozen switch, then likewise $w=v=0$. These checks fix the insertion and response signs. They do not cancel the nonzero pure-gauge witness.