Spectral Geometry of Latent Representations: Calibrated LaplaceBeltrami Diagnostics for Abstention in Deep Architectures

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We develop a geometric and spectral framework for calibrated abstention in deep neural architectures. Hidden representations are modeled locally by samples from a smooth compact Riemannian manifold (M, g), and in-distribution coherence is characterized by low-frequency components of the LaplaceBeltrami operator. We introduce two complementary diagnostics a high-frequency spectral residual and a local heat-kernel deviation score. Unlike a universal geometric claim, the association between abnormal inputs and high spectral content is explicitly formulated as an empirical hypothesis to be calibrated on validation data. We derive the relevant heat-kernel asymptotics, discuss metric perturbations through the Lichnerowicz Laplacian, and provide a graph-based approximation suitable for nite activation clouds. The resulting rejection rule is a statistically calibrated decision procedure rather than a topologyonly guarantee. We specify consistency assumptions, limitations, and an evaluation protocol for out-of-distribution detection, adversarial detection, and selective prediction.