Remark: The necessary identity fails

The first-order identity required for coordinate covariance is

Its actual value is (eq:preflight). There is no claim that $L_{g,\epsilon }$ globally factors as an operator on $Ch$: the switch-derivative term in (eq:CHswitch) prevents that simplification. On the pure-gauge test, however, the failure is exactly the failure of the regulated inverse to return $\xi$.