What a Point Set Knows About a Riemannian Minimizer: A Scaling Law at the Cut Locus

Quasi-Monte Carlo (QMC) point sets come with integration certificates: a discrepancy bounds the error of every integrand in a function class. Riemannian optimization asks where the minimizer of the sampled loss lies. One object connects the two. The point set minus the target law is a signed measure, the discrepancy charge; through the loss it generates a potential whose gradient, the field, is the error of the sampled gradient. The minimizer error is the field at the minimizer times an inverse Hessian, and reusing a point set, refreshing it, or solving once reads the field in three norms, which gives their sample complexities. What a certificate says about the field passes through one integrand, the gradient of the loss, and on a manifold this integrand is singular at the cut locus. For positive rules, dilation predicts that a singularity of order $m$ on a set of codimension $c$ leaves the exponent $(m+c)/(c+\beta )$ of the discrepancy, where $\beta$ is the dilation exponent of the certificate norm; we prove this scaling law, and that it cannot be improved, for point singularities on spheres in Sobolev norms and for point and polygonal singularities in the square in Hardy–Krause variation. On spheres the Fr\'echet problem is electrostatics: the cap discrepancy is the energy of the charge, the value error its potential at the antipode, the gradient error its field there. This gives the sharp exponent $2d/(2d+1)$ for the cap discrepancy on ${\mathbb{S}}^d$; a floor of order $1/N$ on the worst-case gradient error, which spherical designs meet up to $N^{ϵ}$; and a chart that removes the singularity when it blows up the cut point. Experiments on positive definite matrices, Grassmannians and spheres test the main predictions.