When Is the Sharp Covariance Envelope Tight? Feature-Only Geometry for Volume-Sampled Least Squares

Abstract

Prior analyses by Derezinski and Warmuth established all-size sampling identities, selected-OLS unbiasedness, and inverse moments for ordinary volume sampling, while their exact arbitrary-fixed-response loss and prediction-covariance formulas are at the rank-size endpoint s=d. We establish a Loewner envelope for centered coefficient covariance for every full-rank fixed pool, response, and legal budget d<= s<= m under ordinary indexed fixed-size volume sampling followed by selected unweighted least squares; its coefficient is globally sharp over the full-rank class. Global sharpness does not determine attainability on the pool in hand. Under positive loss, strict-interior budgets, and no coloops, a feature-only margin nu_A gives the exact fixed-design spectral phase: nu_A>0 if and only if the normalized spectral envelope is strict for every compatible residual, whereas nu_A = 0 if and only if some compatible residual is spectrally tight; the same zero-margin residual is tight at every strict-interior budget. A residual-augmented change of measure supplies the response-aware mechanism and a one-sided quantitative slack bound, while support saturation proves the attainment direction. Critical equal-leverage geometry interprets the boundary, and sound lower certificates yield conservative same-primitive cardinality decisions. Frozen-feature examples show that the certificate is nonvacuous and measure the fixed-pool cost of its authorized reduction. The claims concern conditional centered, full-Gram-whitened coefficient covariance, not population generalization.