Tree Depth for Masked Diffusion: Exact Unmasking in Ferromagnetic Ising Models

Masked models predict every unrevealed coordinate in one evaluation, but independent parallel commitments can distort the joint law even when the one-coordinate conditionals are exact. Block size or marginal confidence alone does not determine which variables are conditionally independent given prior reveals, or the minimum number of rounds for exact sampling. To characterize both from the dependency graph, we study ferromagnetic Ising models, classical graphical models from statistical physics with local attractive interactions. We prove that independent parallel sampling is exact if and only if removing previously revealed vertices leaves no path between any two variables sampled together. Layers of a minimum-height elimination forest attain the minimum round count for exact fixed commit-once sampling, equal to tree depth, the minimum forest height. Below this round threshold, every fixed schedule incurs positive forward Kullback–Leibler divergence, which measures distributional mismatch; for learned conditionals, an exact decomposition separates this factorization cost from model error. With planner-aware training, our schedules improve conditional prediction quality at matched model-call budgets and better preserve dependencies in generated samples. In-distribution comparisons with dependency-guided baselines show lower prediction error at comparable or shorter end-to-end decoding times.