Selecting Spectral Memory from the Original Gradient

Selecting a spectral memory on the gradient used to fit it creates a dependence cost; independent evaluation avoids this dependence only by fitting candidates on noisier views. We keep every memory on the original observation and aggregate its polar action using Stein's unbiased risk estimate (SURE), conditional on the past. We establish the rank-boundary regularity and sequential risk comparison needed for this nonlinear setting. For strictly rectangular matrices, a predictable safety action makes the selected mixture continuously differentiable even at fitted rank loss, although individual corrected scores can have infinite variance. This justifies an exact, computable selection-response term and an expected-risk oracle against the unperturbed candidate rules under predictable feedback. The response is retained explicitly, not assumed small. Spectrum-saturated bounds connect the comparator risk to noise and memory bias without a lower bound on the true singular values. The leading-one oracle applies to the mixture and randomized isometric output; deterministic polar rounding has a factor-two guarantee. Across 272 fixed-plan synthetic streams, the rounded rule improves on one-way splitting in most cells, while symmetric splitting remains competitive and two zero/near-zero second-scale configurations incur substantial adaptation costs. The results identify when original-data fitting is useful and what dependence must still be charged.