One-step lowest-variance selection in a Gaussian random-field model motivated by masked diffusion: Total correlation and a square root collision threshold

Discrete diffusion, including remasking and uniform-state samplers, generate a sequence by writing multiple token positions per step, drawing each from a per-position distribution and choosing which positions to write from those same distributions. For domains of general interest (pixels, phonemes, or words) there are inherent dependencies between tokens. We show that a step matches the training distribution only when the positions it writes are conditionally independent given the tokens already fixed, that no product of per-position distributions can match a dependent group, and that per-position distributions do not determine whether a group is dependent: two joint distributions can have identical per-position marginals while differing in which combinations of values occur. On ScanAndAdd, a synthetic task whose joint distribution is available in closed form, we verify that every group of two or more undetermined positions a confidence ranking writes is dependent, and measure the generated distribution to be $29\times$ the sampling-noise floor total variation while per-sample metrics are $1.0$.

We study masking diffusion for discrete sampling and introduce a path-resolved measure of data geometry called the \emph{unmasking growth complexity} ({\textsf{UGC}\xspace}). Its local increments directly control Kullback--Leibler (KL) discretization error, yielding a unified analysis of Bernoulli-subset and fixed-cardinality unmasking schemes. In log-reveal-odds coordinates, this structure yields optimized single-block and multi-block schedules, and quantifies the gains from adapting computational effort to data geometry. Crucially, we show how {\textsf{UGC}\xspace} increments can be estimated from samples via KL increments along coupled reveal trajectories. This leads to \emph{certified-optimal} samplers that achieve a prescribed KL error with high probability and iteration complexity within a constant factor of the corresponding oracle procedure. Collapsing the \ugc path yields the aggregate {\textsf{UGC}\xspace} mass, which connects to classical multivariate dependence measures and complexity measures from previous analyses of discrete diffusion. In the fine-partition limit, the squared integral of the square-root {\textsf{UGC}\xspace} density determines the sharp leading-order optimal Euler discretization error. Examples exhibit substantial dimension-dependent gains over coarse schedules, including $\widetildeΩ(\sqrt{d})$ improvements achievable with a constant number of adaptively placed blocks.

Masked diffusion language models (MDMs) uniquely support any-order generation, with confidence-based decoding currently serving as the de facto standard inference policy. To optimize for this, recent training schemes attempt to align training mask patterns directly with those observed during generation. However, we argue that confidence-based decoding is inherently misaligned with the logical-flow trajectories required for complex reasoning, and that confidence-aligned training actively entrenches this misalignment. We make this concrete using multi-digit addition, where the decoding strategy prematurely predicts locally easy digits before resolving their long-range dependencies, producing high-confidence errors on challenging inputs. While traditional random masking keeps the failure rate low on this challenging tail, confidence-aligned training amplifies the error rate by an order of magnitude. Across five distinct reasoning tasks, this same pattern emerges with task-dependent severity: confidence-based decoding induces failures on highly complex inputs, and confidence-aligned training exacerbates them. In contrast, random masking -- despite its perceived inefficiency -- robustly preserves the reasoning-trajectory conditionals essential for solving the challenging tail.