How Much Jointness Must a Denoiser Predict? Harmonic Elicitation Limits for Parallel Diffusion Language Models

Parallel masked diffusion can update many tokens simultaneously, but its usable information is determined jointly by the masking schedule and the conditional distribution exposed by the denoiser head. We model this mask--head pair as a statistical experiment. At a product reference law, a complete (r)-joint head has exact order-(d) eigenvalue (g_{d,r}(\nu)=\EE_{T\sim\nu}\PP(1\leq\Bin(d,T)\leq r\mid T)). We prove the conservation law (\sum_{d\geq1}g_{d,r}(\nu)/d=(1-\nu({0}))Hr) and the constant-factor minimax theorem (\sup\nu\min_{d\leq D}g_{d,r}(\nu)=\Theta(Hr/HD)), attained by an explicit multiscale schedule. An exact nonlinear witness shows that the same coefficient controls global report contraction and the minimax error of every report-based decoder, including arbitrary remasking and revision. It is also the exact Fisher-information fraction and effective-sample fraction in the induced training experiment. Finally, any regular report universally sufficient through order (r) on (k) (q)-ary tokens needs at least (\sum_{m\leq r}\binom{k}{m}(q-1)^m) coordinates. Thus masks allocate interaction information, whereas heads determine how much exists to allocate.