Variation Brownian Kernel Ladders

Claims about the benefit of depth depend on the complexity assigned to a representation. We introduce the Variation Brownian Kernel Ladder (VBKL), a path-atomic function-space framework that separates nonlinear recursive dictionary construction from linear variation superposition. Starting from linear projections, each atom recursively composes unit-ball profiles from the Brownian reproducing kernel Hilbert space; the full VBKL space is then the signed-measure variation hull of the completed dictionary. We identify each recursive dictionary as a union of Brownian pullback RKHS balls and establish variation-controlled H"older regularity, compactness and attainment, and strict growth with depth under a local non-degeneracy condition whose trace lies in the support of the input measure. For associated finite lower-support architectures, we derive Rademacher and generalization bounds through Brownian quadratic chaos, signed threshold traces, and VC entropy. We also construct two-stage approximants by discretizing the outer measure and the selected outer Brownian profiles, obtaining an M-1/2+m-1/2 error bound, a sharp interpolation constant √A/2, and at most 2M active outer-profile basis contributions per evaluation. Controlled experiments illustrate the approximation mechanisms and indicate a favorable limited-data accuracy--complexity trade-off.