One of the biggest promises of Diffusion LLMs is parallel generation: predicting multiple tokens at once to bypass the sequential bottleneck of autoregressive models. However, parallel generation comes with a price. For example: Should the sentence “He is from [MASK] [MASK]” be filled with [New] [York] or [San] [Diego]? If a diffusion model predicts both at the exact same time, it assumes independence and may produce... [San] [York]. 🤦‍♂️ We argue this arises from a structural misspecification: models are restricted to fully factorized outputs because parameterizing the full joint distribution would require a prohibitively massive output head. This is the Factorization Barrier crippling parallel generation. Here is how we broke it with CoDD.