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Non-autoregressive (NAR) generation reduces decoding latency by predicting many tokens in parallel, but iterative refinement often suffers from error accumulation and distribution shift under self-generated drafts. Masked diffusion language models (MDLMs) and their remasking samplers (e.g., ReMDM) can be viewed as modern NAR iterative refinement, where generation repeatedly revises a partially observed draft. In this work we show that \emph{training alone} can substantially improve the step-efficiency of MDLM/ReMDM sampling. We propose \textsc{DSL} (Discrete Stochastic Localization), which trains a single SNR-invariant denoiser across a continuum of corruption levels, bridging intermediate draft noise and mask-style endpoint corruption within one Diffusion Transformer. On OpenWebText, \textsc{DSL} fine-tuning yields large MAUVE gains at low step budgets, surpassing the MDLM+ReMDM baseline with \(\sim\)4$\times$ fewer denoiser evaluations, and matches autoregressive quality at high budgets. Analyses show improved self-correction and uncertainty calibration, making remasking markedly more compute-efficient.