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This paper extends the work of Derezinski and Warmuth by establishing a Loewner envelope for centered coefficient covariance in the context of ordinary indexed fixed-size volume sampling followed by selected unweighted least squares. The authors demonstrate that the envelope's coefficient is globally sharp across the full-rank class, while also revealing that global sharpness does not guarantee attainability for a given pool. They introduce a feature-only margin that characterizes the spectral phase of the residuals, providing insights into the conditions under which the normalized spectral envelope is strict or tight.
The sharpness of the covariance envelope reveals critical insights into the conditions that dictate the attainability of optimal sampling strategies in least squares regression.
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.