= Three-particle bound state from equal-mass fusion
{title2=$m_3=m\sin(3u/2)/\sin(u/2)$}
For the <unit-modulus hyperbolic scattering block> with elementary <mass> $m$, a pair bound at <rapidity> difference $iu$ has <mass> $m_2=2m\cos(u/2)$. <Bootstrap fusion> with another elementary particle gives a further direct <pole> at $3iu/2$. It is inside the <physical rapidity strip> with positive imaginary <residue> only for $0<u<2\pi/3$. In that range the new <bound state> has rest-frame constituent <rapidities> $iu,0,-iu$, so its energy is $m(1+2\cos u)=m\sin(3u/2)/\sin(u/2)$. Charges add at the bound-state vertex. At $u=2\pi/3$ the apparent pole cancels; beyond that point it lies outside the physical strip. The formula is conditional on this particular scattering block and its direct-pole interpretation, not a universal three-body binding law.
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