The sharp discrete Hardy inequality on $\Z^3$

What the math gives to ML

The paper proves a sharp coercivity relation between nearest-neighbor differences and inverse-square-weighted energy on a three-dimensional lattice. The transferable asset is that a feature field cannot concentrate near a designated lattice singularity without paying a quantitatively controlled discrete-gradient cost. This can be used as a Hardy barrier for 3D voxel CNNs or graph neural networks, either as a concentration regularizer or as a diagnostic constraint for learned feature fields around an object center, sensor location, or defect site. The payoff is most plausible for improving robustness against localized activation spikes, although the construction is specialized and should first be tested on small 3D models.