Unit-Weight Posterior Coresets: Sharp KL Geometry and Certification Limits

We characterize posterior fidelity for literal evidence subsets, with no reweighting and an exact cardinality. Under additive logits, omitted-evidence range equals the maximum posterior log-odds distortion. We derive the exact minimum and maximum forward Kullback-Leibler (KL) divergence at a prescribed range for a fixed full posterior; the minimum depends only on its two smallest masses. The lower envelope turns a range-solver lower bound into a KL optimality certificate. With smallest posterior mass $\beta$, exact range initialization followed by KL-decreasing refinement achieves the sharp factor $(1-\beta )/\beta$. A matching binary construction establishes sharpness even with a unique full winner and any fixed exchange radius. For nonadditive consumers, aggregate signed transfer error connects posterior control to decision certification. In 1,800 controlled instances, refinement reduces mean true range by 10% relative to KL-greedy, with mean true KL .3089 versus .3166. RACE, DREAM, and SciFact evaluations with two neural targets identify 172–232 winner-preserving omissions per operating point, while all four calibrated constructions issue zero nontrivial joint winner certificates. Together, the exact envelopes, optimization diagnostics, and source-aware neural evaluations characterize when posterior control supports informative certification.