A Curvature–Frequency Routing Law for Local Optimization Dynamics

Mini-batch ordering changes the temporal structure of gradient noise. To understand how this temporal structure interacts with local loss geometry, we analyze optimizer dynamics along individual positive-curvature modes. SGD, EMA momentum, and Adam with a frozen second-moment preconditioner reduce to a common positive-curvature mode. This common form reveals a frequency–curvature routing law: the infinite-horizon harmonic gain is maximized by flatter modes at low frequencies and sharper modes near the optimizer-specific frozen stability edge at high frequencies. In edge-normalized coordinates, memory preserves the maximizing mode while narrowing the half-power band. Controlled 32-step nonlinear probes are consistent with the predicted columnwise relative ridge and qualitative selectivity while separating steady-state theory from finite-window measurement.