= Unit-modulus hyperbolic scattering block
{title2=$S_u(\theta)=\sinh[(\theta+iu)/2]/\sinh[(\theta-iu)/2]$}
For real $u$, this scalar <S-matrix> block satisfies $S_u(\theta)S_u(-\theta)=1$ and <Hermitian analyticity of a two-particle S-matrix>. It therefore has modulus one for real <rapidity> differences. For $0<u<\pi$ its <pole> at $\theta=iu$ has <residue> $2i\sin u$ and admits a direct-channel <bound-state pole> interpretation. Its crossed block is $\cosh[(\theta-iu)/2]/\cosh[(\theta+iu)/2]$. The endpoints $u=0$ and $u=\pi$ are removable degeneracies giving constant amplitudes, so a particle cannot be inferred there from the uncancelled denominator alone. This block specifies an amplitude; a complete theory must also obey the remaining <crossing symmetry> and <bootstrap fusion> constraints.
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