Sharp Accuracy Thresholds for Parallel Masked Sampling of Noisy Pairs

How accurately must conditional heads be learned to preserve two-round generation of independent noisy pairs? We study masked samplers that select a batch, draw independently from the original heads, and permanently commit the values. For $n$ pairs, the critical total population excess-loss budget at fixed depth $K\ge 2$ is ${\Theta }_{K,\epsilon }(n^{-1/[2(K-1)]})$ for total variation (TV) accuracy $\epsilon \in (0,1)$. The heads are coherent: they are conditionals of one model, and the budget equals joint KL. Each pair has a fair representative and a mismatch bit; the model allows every positive exchangeable mismatch law independent of the representatives. A deterministic geometric schedule attains the threshold uniformly. A single model gives the converse against fully informed, randomized, value-adaptive policies. With logarithmically vanishing KL and uniformly vanishing head error, some models require $\Omega (\log n/\log \log n)$ expected rounds; a growing-depth upper matches this order for budgets $(\log n{)}^{-\zeta }$ with $\zeta >11/3$. The lower obstruction also survives heterogeneous representative biases. Within the fair exchangeable class, target-aware selection of sampled candidates instead yields vanishing TV in three rounds whenever the KL budget vanishes. abstract