A Geometric Phase Boundary for Volume-Sampled Linear Readouts

Global sharpness of a sampling bound does not determine whether the bound is attainable on a particular fixed design. We study ordinary fixed-size volume sampling followed by selected unweighted least squares, with the feature pool and response fixed; subset selection is the only randomness. We first prove a globally sharp all-budget Loewner envelope for centered, full-Gram-whitened coefficient covariance. We then characterize the fixed-design question left open by global sharpness. On the positive-loss, no-coloop strict-interior domain, a feature-only geometric margin is positive if and only if every compatible residual has strict covariance slack at every strict-interior budget, whereas zero margin holds if and only if one compatible residual reaches the Loewner ceiling in at least one coefficient direction at every strict-interior budget. For real whitened designs without coloops, the phase sign is equivalently determined by a pairwise Naimark-complement minor test, which also yields an explicit lower slack certificate. Residual augmentation exposes the response-aware contraction, and a critical equal-leverage specialization identifies an explicit geometric boundary. Any verified positive lower bound on the margin therefore yields a conservative certificate for strictness and for the subset-refit variance term in fixed-query squared loss. Together, these results give a design-specific phase characterization for this finite-pool randomized linear-readout primitive.