Model Error Resonance: The Geometric Nature of Error Dynamics

This paper introduces a geometric theory of model error, treating true and model dynamics as geodesic flows generated by distinct affine connections on a smooth manifold. When these connections differ, the resulting trajectory discrepancy--termed the Latent Error Dynamic Response (LEDR)--acquires an intrinsic dynamical structure governed by curvature. We show that the LEDR satisfies a Jacobi-type equation, where curvature mismatch acts as an explicit forcing term. In the important case of a flat model connection, the LEDR reduces to a classical Jacobi field on the true manifold, causing Model Error Resonance (MER) to emerge under positive sectional curvature. The theory is extended to a discrete-time analogue, establishing that this geometric structure and its resonant behavior persist in sampled systems. A closed-form analysis of a sphere--plane example demonstrates that curvature can be inferred directly from the LEDR evolution. This framework provides a unified geometric interpretation of structured error dynamics and offers foundational tools for curvature-informed model validation.