The sharp curl-Sobolev inequality

What the math gives to ML

The paper supplies an explicit sharp coercive inequality for curl potentials on the round sphere, together with the exact equality family generated by Killing forms and conformal transformations. This can become a mathematically calibrated regularizer for neural vector-field models: it prevents the learned curl from becoming disproportionately concentrated relative to its helicity, while providing an exact lower-bound target rather than an arbitrary penalty coefficient. The most practical transfer is to neural PDE or physics models on three-dimensional spherical domains, using a divergence-free gauge and a differentiable discrete curl operator.