Too Sharp, Too Sure: When Calibration Follows Curvature

Modern neural networks can achieve high accuracy while remaining poorly calibrated, producing confidence estimates that do not match empirical correctness. Yet calibration is often treated as a post-hoc attribute. We take a different perspective: we study calibration as a \emph{training-time} phenomenon on small vision tasks, and ask how it co-evolves with loss-landscape geometry. We identify a tight coupling between calibration, curvature, and margins during training of deep networks under multiple gradient-based methods. Empirically, Expected Calibration Error (ECE) closely tracks curvature-based sharpness throughout optimization. Mathematically, we show that both ECE and Gauss--Newton curvature are controlled, up to problem-specific constants, by the same margin-dependent exponential tail functional along the trajectory. Causal tests via interventions that target sharpness versus directional curvature confirm the mechanism, with directional interventions yielding more reliable in-sample calibration gains.