Hyperuniform Delone Realizations and Rigidity

What the math gives to ML

The paper provides a realization theorem showing that arbitrary ergodic measurable R^d dynamics can be encoded by a Delone point process with arbitrarily strong finite-order suppression of low-frequency fluctuations, while retaining the original measurable dynamics. The transferable mechanism is hyperuniform sampling: enforce a structure factor that vanishes as the wave number approaches zero, thereby reducing long-wavelength sampling noise and count fluctuations. A practical neural-network adaptation is to replace iid collocation or minibatch locations by bounded-displacement, hyperuniform point sets and test whether smooth-loss integration and gradient estimates exhibit the predicted variance reduction. The theorem is not an explicit neural sampler, so implementation should use lattice-jitter, blue-noise, or Fourier-relaxation approximations and treat the low-frequency scaling law as the falsifiable target.