Bound-state fusion of factorized S-matrices ($S_{C,D}(\vartheta )=S_{A,D}(\vartheta +iu/2)S_{A,D}(\vartheta -iu/2)$)

For an equal-mass bound pair $C=(AA)$ at fusion rapidity $iu$, scatter a third particle off its two constituents with shifted rapidities and project onto the bound-state residue. Scalar amplitudes give the displayed product. With internal indices, bound-state coupling tensors perform the projection. The Yang-Baxter equation makes different scattering orders consistent; factorization eliminates independent many-body scattering contributions.