Geometry-Aware Distributional Change-Point Inference

A distributional change in independent manifold-valued observations can preserve location summaries and remain hard to detect even after a representation makes it visible. We study offline testing and localization through the contrast retained by geometric features and the variation assigned to that contrast. Our scan combines multiscale diffusion features with geodesic distance critics and controls finite-sample type-I error by jointly permuting all channels under the iid null. For fixed channels with stable standardizers, we derive local feature-drift limits and consistent localization whenever a retained contrast is nonzero. The leading trace-standardized signal is the fraction of pooled feature variance explained by the change. This characterization motivates projecting distance features off a finite diffusion span: when the diffusion contrast vanishes, projection preserves the distance contrast and yields an explicit variance-reduction factor. Circle, sphere and torus experiments demonstrate sensitivity to matched-moment changes and distinguish detection from localization. A frequency-band comparison recovers the main residual gain at lower cost, showing that variance allocation, as well as signal retention, determines the value of a geometric representation.