Space-filling surfaces: sharp Hölder continuous parameterizations from squares to cubes

What the math gives to ML

The paper gives an explicit multiscale route from an m-dimensional parameter grid to an (m+1)-dimensional domain with the optimal Hölder exponent \(\alpha=m/(m+1)\), rather than merely proving that a continuous surjection exists. Its transferable asset is the combination of a sparse self-similar X-shaped refinement and lattice-level branch ordering, which can provide deterministic locality-preserving layouts between 2D hardware tensors and 3D feature or voxel domains. A practical adaptation is a fixed fractal coordinate router for 3D neural fields or volumetric tokens: it replaces generic flattening with recursively refined, locality-controlled addresses and can be tested as a parameter-sharing or memory-layout module. The main risk is that the asymptotically sharp exponent does not automatically improve finite-resolution learning, so experiments must compare locality, collision rate, and accuracy against Morton or Hilbert orderings and ordinary 3D convolutions.