Common Covariance Geometry and Certification for Brownian Kernel Ladders

The paper introduces a new representation‑adaptive kernel class that, on a fixed sample, yields a union of reproducing‑kernel Hilbert‑space ellipsoids instead of a single ellipsoid. It defines a minimum‑trace common covariance dominating the empirical union generated by Brownian kernel ladders, and derives exact formulations, statistical and computational consequences, and a universal Gaussian‑complexity bound. The work further develops geometric reductions, deterministic depth laws, and exact empirical Kolmogorov‑width formulas, providing both lower and upper certificates for covariance certification and illustrating the distinction between successful covariance certification and predictive selection.