When Should Measurements Speak? Directional Guidance for Diffusion Inverse Problems

Diffusion inverse solvers combine learned priors with measurement guidance, yet most rules use one scalar per reverse step or fixed measurement geometry. We introduce trajectory-conditioned observability (TCO), the spectrum of a noise-whitened measurement Jacobian restricted to the diffusion prior's current uncertainty subspace. In the exact local linear model, TCO yields the unique linear minimum-MSE correction and, under Gaussianity, the posterior mean and shrinkage. We prove an exact lower bound: whenever two active TCO singular values differ, every scalar measurement-gradient rule incurs strictly positive excess Bayes risk. For nonlinear operators and estimated geometry, a perturbation theorem separates curvature, subspace, residual, and trust-region errors. These results motivate DynaObs, a training-free plug-in that refreshes local geometry and applies mode-wise guidance. Across linear and nonlinear inverse problems, trajectory spectra predict failures better than endpoint diagnostics, and DynaObs improves matched-compute reconstruction while geometry-mismatched controls remove the gain.