When Does Posterior Geometry Matter for In-Context Demonstration Selection?

Posterior-based demonstration selection uses label-distribution compatibility to refine semantic retrieval. Understanding these selections requires identifying which posterior components can change the selected examples. We analyze candidate posterior tails, comprising probabilities assigned to labels other than the observed candidate label, at fixed query posteriors and observed-label probabilities. For Jensen–Shannon alignment, we derive exact attainable score intervals and establish a necessary and sufficient condition for selection invariance. Every non-invariant observed set admits a constructive counterexample involving at most two candidate tails. This characterization determines when posterior variation can reverse the ordering at the selection boundary. A five-task audit certifies 32.71% of selected sets under equal task and seed weighting, with rates ranging from 0.05% to 89.64%. In controlled nine-task comparisons, full posterior alignment improves macro accuracy over a query-label control by 0.83 percentage points on Mistral-7B, while the two selectors achieve similar means on Qwen2.5-7B. The certificate permits recovery of the full selector's demonstration set without candidate-tail access on certified queries. Code is available at https://anonymous.4open.science/r/posterior-geometry-5376.