Proposition: Existence and internal order

All seven kernels exist. Their separated-patch order bounds are

The complete kernel has local order at most four.

Proof. The two-cone seeds are tensor-product pullbacks under the invertible maps

The response seed is a tensor product in $(a,b-a,z-a,y)$. All seeds have order zero.

The first three rows have at most four derivatives, and $K_6$ has two. The root rows have at most two source derivatives and one outer derivative, or no outer derivative in the mass row.

For $K_8$, commute the inner $b$-derivative through (eq:response) and use the separated $B$-derivative identity. It becomes a smooth multiple of an $R$-seed with two source derivatives and one outer derivative. This proves its order-three bound.