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

The paper establishes a Loewner envelope for centered coefficient covariance under ordinary indexed fixed-size volume sampling followed by selected unweighted least squares, valid for every full-rank fixed pool, response, and budget d <= s <= m, with a globally sharp coefficient over the full-rank class. Building on prior work by Derezinski and Warmuth, the authors derive an exact fixed-design spectral phase via a feature-only margin, a response-aware change of measure, and nonvacuous lower certificates for cardinality decisions.