Filtered Stable Rank and Tail-Local Limits on Oversampling in Randomized SVD

Randomized SVD is often run with a fixed oversampling parameter, even though approximation quality is driven by the spectrum and by the power-iteration count. We test whether the stable rank of a matrix can govern the oversampling needed to reach a target relative error, and whether this yields an a-priori rule that is independent of ambient size and target rank. The answer is conditional. In finite Gaussian range finding, global stable rank alone does not remove target-local tail dependence: for power-law spectra with exponent greater than one half, stable rank can remain bounded while a normalized tail quantity grows with the target rank. We define a power-filtered rank that contracts under subspace iteration, derive a conditional oversampling threshold under an explicit intrinsic-tail premise, and validate the algebra with seven symbolic checks. Executed experiments on 1,728 synthetic randomized SVD runs and 648 real-matrix runs show mixed empirical support. Normalizing by filtered rank improves collapse only slightly on average, with mean relative spread improvement 0.019 over global stable rank. Conditional tail fits are mostly decreasing in oversampling per filtered rank, but a finite two-parameter rule has only 80\% training and held-out coverage and no feasible cell in the tested calibration grid. On SuiteSparse SNAP/ca-GrQc, MovieLens 100K, and a Fashion-MNIST train slice, matched-accuracy transfer rows show zero positive sketch-column savings. The supported conclusion is therefore not a universal stable-rank-only law. Effective rank is useful when paired with power filtering and tail-local diagnostics, and the resulting rule should be treated as a scoped calibration rather than a general replacement for existing oversampling heuristics.