Blackwell-Greatest Rank-Constrained Linear Observation for Gaussian Covariance Rays

This work studies optimal linear observation of a one-parameter Gaussian covariance family under a hard rank constraint. The leading generalized covariance modes are shown to Blackwell-dominate every competing linear observer of rank at most m through an explicit parameter-independent Gaussian attenuation-and-noise kernel. A strict-gap converse characterizes the equality cases and identifies the optimal observation subspace. The framework also yields an exact Bhattacharyya consistency criterion, describes the transition when the observation rank is allowed to grow, establishes a one-sided local asymptotic normality result, and derives a fourth-root separation law in the weak-amplitude regime. Particular emphasis is placed on making the comparison mechanism explicit: the optimal experiment can be transformed into any competing rank-constrained experiment using a Gaussian Markov kernel that does not depend on the unknown covariance parameter. This provides a direct statistical interpretation of the generalized covariance modes and their optimality. Supplementary material provides additional derivations, boundary cases, and numerical checks. The accompanying reproducibility package contains source files, deterministic verification code, audit outputs, and machine-readable numerical results.