A channel-level check of the two-state control

For the supplied dynamical control, $c$ is the collected coherent-state overlap $c_R(t)$, and the residual channel is locally isometric to ${\Delta }_{c_R(t)}$. The preceding proof checks its channel formulas, not the Hamiltonian-to-overlap calculation. No noncommuting recovery window, scrambling estimate, or out-of-time-order-correlator bound follows.