A representation-adaptive kernel class produces, on a fixed sample, a union of reproducing-kernel Hilbert-space ellipsoids rather than one ellipsoid.
We introduce the minimum-trace common covariance that dominates the unrestricted empirical union generated by Brownian kernel ladders and develop its statistical, approximation-theoretic, and computational consequences.
The covariance value admits exact formulations through absolutely two-summing operators and covariance-dominated multipliers, and it yields a universal Gaussian-complexity bound.
A closed last-layer Dirac-trace reduction and a signed Brownian threshold representation convert the generic covariance problem into threshold, graph-coarea, and effective-resistance geometry.
These tools give deterministic depth laws, conditional Gaussian reverses, random-design and perturbation transfers, and an exact empirical Kolmogorov-width formula whose leading covariance eigenspaces approximate the complete adaptive ball simultaneously.
Finite contact, active semidefinite programs, verified separation, and a convex resistance-design relaxation provide complementary lower and upper certificates.
A finite covariance-indexed Brownian path on frozen representations illustrates the distinction between successful covariance certification and predictive selection: all reported path certificates succeed, whereas the locked predictive study misses one predeclared aggregate criterion.
The paper thereby identifies one finite-dimensional covariance object linking unrestricted kernel adaptation, Gaussian geometry, common subspaces, and certifiable computation.