Parallel samplers, including masked diffusion models, generate several coordinates per round from conditional marginals, and their accuracy depends on which coordinates are revealed together. A schedule can be improved in two natural ways: adaptively, by choosing later batches from the values already generated, or by randomization, by drawing the schedule at random. We show that randomization is strictly more powerful. For every $r \geq 1$ we construct a full-support distribution on $9r$ binary coordinates such that every deterministic three-round sampler, even with value-dependent batches and arbitrary product kernels, has forward Kullback-Leibler divergence at least $γr - \log 4$ with $γ\approx 0.0966$, whereas drawing one of four fixed schedules with two fair bits drives the divergence below $18 r β^r$ with $β\approx 0.966$. Both samplers query conditionals of the same target. The separation holds for every fixed number of rounds $T \geq 2$, a strict advantage survives perturbations of the target and errors in the conditional predictor, and exact rational arithmetic certifies the gap at nine coordinates, where trained neural predictors preserve it. The mechanism is a hidden mixture: each schedule preserves the dependence pattern of one component, while no deterministic first batch can serve them all.

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