Elastic channel momentum magnitude and direction freedom

Consider a two-product elastic channel in a finite isolated collision: the complete incoming and outgoing lists each have exactly two legs, their recorded rest masses agree pairwise, the total four-momentum is timelike, and the balance postulate is adopted. In the center-of-momentum frame, let the incoming momenta be $k{\mathbf{e}}_{\mathrm{i}\mathrm{n}}$ and $-k{\mathbf{e}}_{\mathrm{i}\mathrm{n}}$ in the spatial rest space $n_Q^{\perp }$, where $k\ge 0$ is fixed by Complete two-body rest-frame kinematics. Every admissible outgoing pair has the same momentum magnitude $k$. If $k>0$, any unit vector ${\mathbf{e}}_{\mathrm{o}\mathrm{u}\mathrm{t}}\in n_Q^{\perp }$ gives an on-shell outgoing pair $k{\mathbf{e}}_{\mathrm{o}\mathrm{u}\mathrm{t}},-k{\mathbf{e}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ with the same total four-momentum; thus kinematics alone leaves the full direction sphere available. If $k=0$, both products are at rest and the direction is undefined. No isotropic or uniform distribution of directions follows.