Conditional permutations in a revealed switching environment

Fix half the labels in a Thorp shuffle on $2^d$ positions and reveal their complete trajectories. We prove that, after an explicit absolute number of coordinate sweeps, the conditional permutation of the remaining labels approaches uniform in expected total variation as $d\to \infty$, uniformly in the initial layout. The resulting constructions give full-deck mixing in a constant multiple of d physical shuffles.