The Clarke tangent and normal cones to decomposable sets in Lebesgue spaces

What the math gives to ML

The paper establishes an exact decomposability principle: for finite-p Lebesgue-space selection constraints, the regular tangent cone of the global feasible set equals the set of measurable directions that belong to the pointwise tangent cone almost everywhere. This can support constrained neural operators, dynamics models, and prediction heads whose outputs must satisfy hard, possibly nonconvex, per-input constraints. A practical adaptation is to project each output-space descent direction into its local Clarke tangent cone, then fit those feasible directions with a parameter update. The theorem guarantees first-order function-space feasibility, but not finite-step feasibility or good parameter-space conditioning, so experiments should measure both constraint violation and optimization speed.