Second-order Fusion Asymptotics for Sine\b{eta} Correlation Functions

What the math gives to ML

The paper gives an explicit local expansion for an unordered point-process density when m points fuse: the leading term is Vandermonde repulsion, and the first finite-scale correction is a universal quadratic penalty in pairwise separations. This is transferable to neural samplers or score models that generate point configurations, where it can provide an analytic short-range prior instead of forcing the network to learn repulsion from limited data. The useful asset is the combination of a singular repulsive score and a computable second-order correction, together with the validity condition m beta > 1. A practical test is to add this local score target to a point-set diffusion model and measure pair-correlation accuracy and collision frequency at fixed model size.