Proposition: Equality of the complete tested kernels

Formula (eq:fullregroup), with (eq:B9root), gives the same attachment distribution $L_G$ as the sum ${{\sum }_{j=4}}^9⟨K_j,G⟩$.

For every such $G$, this kernel is compact and satisfies

This is an intermediate attachment-kernel bound, not admission of the final quantum coefficient.

Proof. The exact smooth-slot identity proves equality on tensor-product tests $h(a)\psi (b)$. Finite sums of these tests determine a distributional kernel, giving the asserted equality.

Each inherited $K_j$ is a compact tree kernel with the covector sum-zero bound in $(a,b,z)$. Multiplication by the smooth residual test and pushforward over its compactly supported variable set the residual covector to zero. The resulting covectors therefore satisfy $\xi +\eta =0$. Smooth testing and the finite sum preserve compactness Hormander (Section 8.2).