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Split the solitary-wave energy into longitudinal, transverse and interaction contributions,Consider the longitudinal dilation ${\psi }_{\lambda }(x,y,z)=\psi (x,y,\lambda z)$ for $\lambda >0$. Changing variables to $Z=\lambda z$ givesThe renormalized momentum of a condensate is unchanged: its single $z$ derivative contributes $\lambda$, cancelling the measure factor ${\lambda }^{-1}$. Since the wave is a critical point of $E-Up$, differentiating this admissible scaling variation at $\lambda =1$ givesTherefore the longitudinal scaling identity for a condensate solitary wave yieldsThis directional Pohozaev identity assumes the finite energy and sufficient decay to justify the dilation variation; it can equivalently be obtained with cutoffs followed by a limit. It is an identity for solitary-wave critical points, not for arbitrary localized fields.