Information retention and local attribution after covariance-spectrum compression

Replacing a data matrix by sample-covariance eigenvalues discards entries,
eigenvectors, and directional labels. We ask which local departures from Gaussianity
remain detectable and whether retained evidence can distinguish covariance shape from
non-Gaussianity. We study sample-covariance experiments around real and proper-complex
Gaussian/Wishart roots. In the real model, we compute the complete finite-dimensional
Gaussian-root Fisher metric for standardized iid entry-law perturbations: odd Hermite
directions vanish, even-cumulant information follows an exact hierarchy, and the standardized
geometry collapses to the fourth-cumulant direction as dimensions grow. Along an explicit
positive standardized fourth-cumulant path, the score uses only the first two spectral traces,
admits a dimension-uniform likelihood expansion, yields a one-spectrum testing ceiling, and
is locally asymptotically sufficient under replication. In the proper-complex model, we derive
the singular trace-preserving covariance-shape score and its exact joint Fisher geometry
with a canonical fourth-order tangent. For 𝑝 > 1, these two specified local mechanisms
are identifiable exactly when 𝑛 > 1; at 𝑛 = 1 their scores are pointwise collinear. We
establish one-sided differentiability in quadratic mean, half-plane local asymptotic normality,
boundary inference, and physical-source maps. For iid rank-one sources, source excess
kurtosis determines the fourth-order coordinate; for dependent blocks, exact score means
depend on normalized block-energy variance