Lemma: Balanced one-line composition

Let $F,G$ have compact homogeneous cubic kernels with wavefront sets contained in $Z_3$. A contraction of one derivative slot from each kernel through a balanced retarded Green line exists. The remaining degree-four kernel is compact and has wavefront set contained in $Z_4$.

Proof. Denote the contracted variables by $y,x$ and the four uncontracted variables by $z_1,\dots ,z_4$. The derivative kernels have covectors

whose sums vanish; the line has covectors $(k,-k)$ at $(x,y)$.

A forbidden cancellation in the full distribution product would have zero covectors at every uncontracted variable. The two sum-zero relations then force $q_y=q_x=0$. The line covector must also vanish, contradicting a nonzero cancellation. The product therefore exists by the distribution product criterion.

Both derivative kernels are supported in fixed compact sets. Their product with the line has compact support, so projection over $x,y$ is proper. The wavefront pushforward condition sets the contracted output covectors to zero. Summing all covectors leaves ${{\sum }_{j=1}}^4{p}_j=0$. Thus the output bound is $Z_4$. Fiber sums and normalized symmetrization preserve these conclusions.