One-Step Curvature Probes Miss the Fitting Operator: Retained Capacity and Terminal Null-Space Correction for Continual Learning

A one-step curvature probe evaluates an initial direction, whereas continual learners are judged after reaching comparable new-task fit. These quantities need not agree. In an overparameterized linearization, projected gradient descent converges to ${\Delta }_P=P{J}^{\top }(JP{J}^{\top }{)}^{-1}r$, and its squared-displacement inflation is exactly the reciprocal of the direction-specific retained fitting capacity $c_P(r)=\Vert J^{+}r{\Vert }^2/\Vert P{J}^{\top }(JP{J}^{\top }{)}^{-1}r{\Vert }^2$. More generally, the terminal old-task quadratic ratio factorizes as $1/[c_P(r)G_{\mathrm{end}}(P,r)]$, where $G_{\mathrm{end}}$ compares curvature along the normalized endpoint directions. In the rank-one case, $G_{\mathrm{end}}$ equals the one-step probe gain; for multiple outputs, the two gains need not agree. The terminal quadratic further decomposes into a common curvature-optimal fitting floor and an algorithm-dependent null-space excess. This geometry identifies the variable omitted by a matched-norm probe and yields terminal null-space correction, a post-hoc operation that preserves linearized new-task outputs on its Jacobian batch. In separate controlled checks, the local quadratic has slope $1.0012$ ($R^2=0.9997$); projection reduces the matched-norm quadratic by $4.03\times$ while terminal forgetting changes by only $5.34\times 10^{-4}$. Across five common-threshold configurations, projection produces the larger signed old-task loss change in $73/99$ matched pairs. In the eligible-subset probe study, median gain ranges from about $3\times$ to $>1000\times$ (eight pairs at the upper end). The common-threshold rank sweep shows median retained fitting capacity falling from $0.88$ to $0.19$. A fixed-rank comparison separates terminal forgetting when probe gain is approximately matched, whereas the capacity-matched contrast remains compatible with effects of either sign. On Permuted MNIST and Split CIFAR-100, terminal correction decreases signed old-task loss in $166/180$ method–dataset–seed pairs under the all-seed intention-to-correct analysis. Because acceptance and outcome reporting use the same held-out split, this result is descriptive and test-conditioned. Together, the results replace gain-only interpretation with a matched-fit account: in the fixed-Jacobian quadratic model, terminal cost depends jointly on retained fitting capacity, endpoint-direction curvature, and the null-space component selected by optimization.