Coverage-Constrained Neural Exponential Embedding: Exact Test-Time Projection for Frozen Generators

Test-time control of a generator is often implemented by prompting, scalar guidance, or reweighting a candidate bank. These mechanisms can improve a preferred attribute while silently concentrating probability on very few outputs. We instead ask for the closest distribution to a frozen generator that matches target statistics to a declared tolerance and retains a minimum effective sample size. The resulting finite-bank problem is a convex, coverage-constrained information projection. We prove existence and uniqueness, derive its low-dimensional dual, and show that an active coverage constraint changes the density ratio from an exponential tilt to a Lambert-W deformation. Thus classical empirical tilting is recovered only when coverage is inactive; rescaling a fitted tilt along a ray is generally not the constrained optimum. A reference-only bootstrap calibrates moment tolerance without evaluation leakage. The fitted density ratio can be used directly for bank resampling or as the terminal potential of learned twisted sequential Monte Carlo. Across frozen image and language generators, the exact projection improves held-out distribution matching over ray temper- ing, remains competitive with density ratio and stable-balancing baselines, and produces substantially better diffusion targets at matched fit-bank coverage. The method separates target specification, optimal projection, and sequential realization, making both its guarantee and its support limitations explicit.