A quantitative Schur comparison theorem for curves in CAT(k) spaces

What the math gives to ML

The paper turns Schur comparison into a quantitative sensitivity formula: changing a curve's curvature measure changes endpoint distance by the integral of a signed moment arm. This is more useful for neural networks than the qualitative theorem itself because it supplies a spatially weighted curvature penalty, distinguishing bends that obstruct endpoint separation from bends that do not. The most direct transfer is to hidden-state trajectories in residual networks, neural ODE discretizations, or state-space models: penalize curvature at points with large positive moment arm relative to the trajectory's endpoints, while monitoring whether endpoint separation and signal propagation improve. The guarantee is mathematically exact only for suitable curves in model or CAT(k) spaces, so the first implementation should use Euclidean latent trajectories and treat the formula as a geometry-inspired surrogate.