Coefficients and unordered placements

This proposition does not evaluate a smooth Hessian directly on singular fields without justification: the finite jet formula specifies the distributional expression, and Lemma lem:sign establishes every required punctured product.

The numerical kernels are translation invariant. Their coefficients generally depend on $y,w$ separately. The source jet depends on $y$; the residual field and interaction switch depend on $w$.