= Full S-matrix phase from a classical soliton delay
{title2=$d\delta/dE=\Delta T/\hbar,\quad S=e^{i\delta}$}
A narrow outgoing energy packet has phase $-ET/\hbar+\delta(E)$, so <stationary phase> gives the arrival shift $\Delta T=\hbar\delta'(E)$. For two equal-mass <solitons> with relative <rapidity> $\theta$, $E=2\mathcal M\cosh(\theta/2)$. The convention $S=e^{2i\delta_{\rm pw}}$ instead gives $2\hbar\delta_{\rm pw}'=\Delta T$. Classical time delays determine only the energy-dependent phase difference, not its constant branch.
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