Stability-Constrained Approximation in Spline KANs: Exact Layer Balancing and Budget-Compatible Saturation

Deep spline superposition networks face a tension between approximation order and stability across depth. We study approximation under a hard layerwise Lipschitz budget, and organise it around two quantities: the factorisation stability complexity of a given deep factorisation, and the budget-compatible approximation complexity of a discretisation operator. First, we solve exactly the finite-depth diagonal balancing problem for a fixed chain of nonnegative envelope matrices: the optimal uniform layer budget equals |ML-1cdots M0|∞→∞1/L, attained by an explicit one-pass minimiser, for rectangular layers, with a complete treatment of degeneracies and non-attainment. The optimum can be arbitrarily larger than the Lipschitz constant of the network itself, because passing to envelopes destroys sign cancellation. Second, we give a constructive spline discretisation theorem preserving the budget up to a controlled slack, with an explicit grid threshold. Conversely, for linear spline-valued operators that preserve the budget exactly, we prove budget-compatible minimax lower bounds on classes constrained simultaneously in the first and third derivative norms -- a constraint pair that is forced by the problem and that rules out the usual scaling escapes. Finally, we show that the corresponding layer errors need not cancel under composition: for every operator of the class there is a stable depth-L tower realising a constant fraction of the accumulated error, so the linear-in-depth accumulation of the upper bound is not a proof artefact.