The Primitive and Diamond surfaces locally minimize the variance of Gauss curvature

What the math gives to ML

The paper reduces a geometric variance objective to a finite configuration of branch points on the sphere and to log-partition integrals of exponentials of a Green-function potential. The transferable asset is a differentiable, globally coupled energy for spherical point sets, together with normalized Gibbs measures that expose which regions dominate the objective. A practical neural-network use is to initialize or regularize learnable spherical prototypes, attention keys, or MoE router centroids so that they avoid clustering while retaining a controllable potential-based notion of balance.