Diffusion language models (DLMs) accelerate generation by decoding multiple tokens in parallel, but pushing them to fast decoding often sacrifices accuracy. We attribute this to a static-to-dynamic mismatch: training optimizes local denoising under fixed schedules, while fast inference skips steps and encounters irregular mask patterns. Existing accelerators largely address only one side of this problem. Training-free inference methods speed up decoding without changing the model, leaving the mismatch unresolved; training-side distillation methods adapt the model, but often tie the student to a specific decoding trajectory, limiting composability and still suffering accuracy loss. We propose CD⁴LM, a composable framework for accurate DLM acceleration. Its core training component, Discrete-Space Consistency Distillation (DSCD) trains a trajectory-invariant student on diverse masked states, reducing the mismatch without fixing a decoding policy. We pair DSCD with a lightweight Confidence-Adaptive Decoder, while keeping the trained student compatible with external inference-side accelerators. On GSM8K, CD⁴LM preserves accuracy while achieving a 5.18× speedup; stacking CD⁴LM with per-step cache accelerators further pushes speedup to 9.73× at teacher-level accuracy, suggesting a scalable path for combining training-side robustness with future decoding advances.