On the sharp constants in curl-Sobolev inequalities on $\mathbb{S}^n$

What the math gives to ML

The paper gives a nonstandard Sobolev quotient for middle-degree differential forms whose numerator measures an L^{2n/(n+1)} norm of curl while its denominator is the quadratic helicity pairing with curl. The transferable asset is not the sharp constant itself, but the combination of a non-elliptic differential operator, a large exact-form gauge kernel, and a stability result selecting the helicity quotient J1 over the seemingly similar gauge-invariant quotient J2. This suggests a spherical or volumetric neural feature regularizer that removes exact/gauge components and rewards stable curl-coherent representations, rather than penalizing raw feature magnitudes. The most direct first test is a 3D vector-feature network using discrete exterior calculus, comparing the J1-inspired regularizer with ordinary gradient, divergence, and J2-inspired penalties.