What Should a Diffusion U-Net Preserve? Predictive Sufficiency at Multiscale Interfaces

Downsampling in diffusion U-Nets is typically treated as a geometric compression operation. We instead view each resolution transition as a target-specific predictive information interface: a representation is useful only insofar as it preserves the information needed for its assigned denoising target. We formalize this idea through a predictive sufficiency gap and derive exact conditions for linear Gaussian interfaces. For rank-matched Gaussian measurements, sufficient geometry generically varies with diffusion SNR, yielding a conditioned-versus-static separation. A tractable Haar model then shows that a constrained local three-tap pre-projection transform can exactly realize this SNR-dependent geometry at fixed retained dimension and yields a strict pre-versus-post separation. In a controlled CIFAR-10 diffusion U-Net interface, correct time conditioning and access to fine-space directions improve held-out prediction, and the learned retained measurement geometry varies with diffusion time. A covariance-defined Gaussian-linear diagnostic further shows that, over the first 15 of 20 evaluation timesteps, the learned Time Pre measurement is closer than Fixed to the surrogate sufficient geometry under both surrogate PSG and subspace distance; the ordering reverses in the five lowest-SNR timesteps as that surrogate geometry approaches the fixed Haar subspace. In an end-to-end diffusion U-Net, matched fine-space access produces a small denoising advantage concentrated in the lowest-noise regime, while a DDPM generation evaluation does not preserve the matched ordering in FID. These results support predictive sufficiency as a principled criterion for analyzing and constraining multiscale diffusion-denoiser interfaces, while showing how interface-level information preservation contributes within the broader dynamics of the generative process.