Conditional coordinate sweeps and analytic transfer

For coordinate sweeps with near-uniform line laws and line sizes among powers of two in a suitable fixed interval, we prove a Schatten-moment bound after conditioning on any feasible collection of prescribed card trajectories. An analytic transfer to binary sweeps then shows that a fixed number of sweeps mixes the Thorp shuffle on $2^d$ cards in $O(d)$ physical steps, matching the order of the support lower bound.