Finite-Step Adaptation Decisions: Loss Responses and Retained Directions

A restricted update can match a full update's loss while omitting directions that affect another execution or target. We study when scalar observations certify those omitted effects. For a radial family containing ordinary and normalized gradient descent, we derive a response law at every finite derivative order. Its leading matrix depends affinely on one gain slope. This transfers two explicit containment designs across the slope interval $[0,1]$. The normalized endpoint also admits sharp, realizable cancellation examples. A second observation uses a normal-gradient correction at a saved state. Its loss decrease gives a positive energy measurement. We characterize four-prefix designs by an exact rank condition and derive a finite transfer bound for unobserved steps. A finite residual certificate then allows changing batches, refreshed spaces, and separate validation losses. Reference derivative actions yield a tighter centered interval at additional cost. On stored RoBERTa feature-head trajectories, that interval certifies a nontrivial tolerance decision. The results identify how response structure, observations, and execution cost determine retained-direction adequacy.