When Historical Energy Is Insufficient for Local Training Response

A training trajectory can be nearly one-dimensional in historical energy, yet discarding its weak second direction can lose an order of local prediction accuracy. For smooth fixed-loss gradient descent over fixed finite windows, we characterize the necessary and sufficient gradient and curvature capture rates for preserving the unrestricted linearized model's $O({\eta }^2)$ normalized parameter-error guarantee. With nonzero gradient and transverse curvature, even the best line through the anchor, fixed across the window and chosen using the complete future, has sharp $\Theta (\eta )$ error. Both the gradient-curvature plane and the exact noiseless historical top-two space attain $O({\eta }^2)$ error. History therefore contains a sufficient response space, but every fixed explained-energy threshold below 100% eventually discards its necessary second direction, whose energy fraction vanishes quadratically. Multi-scale tests in selected LoRA coordinates of eight LLM model-data blocks support the predicted orders and assess the sharp best-line coefficient, with model-dependent finite-scale deviations. Across four families, matched experiments locate over 96% of the one-direction predictor's squared residual in a direction carrying under 2% of future update energy (family medians). A varying-driver motion law separates driver and curvature contributions to historical compression; shuffled-AdamW diagnostics examine direction selection beyond fixed-loss GD.