Conditioning Without Guidance: When the Observation Is Already a Diffusion Marginal

Conditioning a diffusion model usually approximates an unavailable likelihood score at inference. We identify a restricted case where this is unnecessary: when an observation has the law of a reachable forward marginal, conditioning reduces to restarting the reverse process there. With an exact score and exact marginal match, restart samples the Bayes posterior. We instantiate this idea with scale-time diffusion, which destroys Fourier modes in scale order. A two-segment schedule removes the unobserved band and then matches sensor noise. Noiseless observations remain fixed by the reverse dynamics; noisy ones are denoised before they freeze. Across two chaotic dynamical systems involving turbulent flows, the method substantially reduces reconstruction error relative to learned baselines in the first three missing octaves while retaining fine-scale power through repeated assimilation. A progressive clock uses a finer unseen observation with an exactness–utility tradeoff. The theorem covers matched projection sensors. The production endpoint has a climatology-averaged joint-law total-variation bound of $9.5\times 10^{-4}$; sparse sensing and useful off-design entry remain outside exact matching.