= Solution
Use the <physical rapidity strip> $0<\operatorname{Im}\vartheta<\pi$ for <poles> of the two-body <S-matrix>, and write $\vartheta$ for scattering <rapidity> to distinguish it from the theta angle in question 2. A denominator in the kink-antikink product vanishes at
$$
\vartheta=iu_k,\qquad u_k=\pi\left(1-\frac{k}{n}\right),\qquad k=1,\ldots,n-1.
$$
No numerator cancels these <poles>. For two equal-mass constituents with <rapidities> $\chi\pm iu_k/2$, their <four-momentum> vectors sum to
$$
M(\cosh(\chi+iu_k/2),\sinh(\chi+iu_k/2))
+M(\cosh(\chi-iu_k/2),\sinh(\chi-iu_k/2))
=2M\cos(u_k/2)(\cosh\chi,\sinh\chi).
$$
This gives the <relativistic bound-state mass from a rapidity pole>. The ordered <breather> spectrum is
$$
\boxed{m_k=2M\sin\frac{k\pi}{2n},\qquad k_{\max}=n-1.}
$$
It increases strictly with $k$. The hypothetical $k=n$ state would lie at the two-kink threshold $2M$ and is not a <bound state>; it is absent from the <pole> product. At these couplings $M=mn/\pi$ and $\beta^2=8\pi/(n+1)$. In particular $m_1\to m$ at weak coupling $n\to\infty$. This is the <Sine-Gordon breather spectrum at reflectionless couplings>. At $n=1$ there are no <breathers>; the subsequent processes involving a physical $\mathcal B_2$ require $n\geq3$.
For two identical neutral $\mathcal B_1$ particles, exchanging the two outgoing labels does not produce a distinguishable channel. In one spatial dimension the elastic final momenta are the incoming pair, up to interchange. Thus there is one scalar identical-particle amplitude, rather than separately observable transmission and reflection amplitudes.
Put $a=\pi/(2n)$, so the basic amplitude uses $\sin(2a)$. Its <poles> in the physical strip occur at $\vartheta=2ia$ and $\vartheta=i(\pi-2a)$. The first is the direct bound-state <pole>. Choosing constituent <rapidities> $\chi\pm ia$ gives real total energy-momentum
$$
p_1+p_2=2m_1\cos a\,(\cosh\chi,\sinh\chi).
$$
Since $m_1=2M\sin a$,
$$
\boxed{2m_1\cos a=2M\sin(2a)=m_2.}
$$
Both <energy> and <momentum> therefore match an <on shell> $\mathcal B_2$ with <rapidity> $\chi$. The complementary <pole> is its crossed-channel partner. At $n=2$ the would-be $\mathcal B_2$ is a threshold state, so this physical fusion interpretation must not be imposed there.
The <bound-state fusion of factorized S-matrices> treats a bound particle as its <on shell> constituents with analytically continued <rapidities>. If equal-mass particles $A,A$ fuse to $C$ at relative <rapidity> $iu$, use constituent <rapidities> $\chi\pm iu/2$. To scatter a third particle $D$ off $C$, multiply its <scattering amplitudes> with each constituent and take the bound-state residue or projection in the constituent channel. For a scalar amplitude this gives
$$
S_{C,D}(\chi-\chi_D)=
S_{A,D}(\chi-\chi_D+iu/2)S_{A,D}(\chi-\chi_D-iu/2).
$$
For particles with internal indices, the product is projected using the bound-state coupling tensors. The heuristic reason is <factorized scattering>: conserved higher charges prevent particle production and fix the <rapidity> data, so the third particle scatters through the constituents by successive two-body processes. Consistency of different orders is the <Yang-Baxter equation>. Without integrability, an independent three-body interaction would invalidate this simple bootstrap product.
For $\mathcal B_2=(\mathcal B_1\mathcal B_1)$, the constituent shifts are $\pm ia$, giving
$$
\boxed{S_{2,1}(\vartheta)=S_{1,1}(\vartheta+ia)S_{1,1}(\vartheta-ia).}
$$
It is useful to write this <Sine-Gordon breather fusion amplitude> in explicitly factorized form:
$$
\boxed{S_{2,1}(\vartheta)=
\frac{\sinh\vartheta+i\sin a}{\sinh\vartheta-i\sin a}
\frac{\sinh\vartheta+i\sin(3a)}{\sinh\vartheta-i\sin(3a)}.}
$$
To check the reduction, put $z=\sinh\vartheta$. Multiplying the shifted factors gives numerator and denominator $z^2\pm2iz\sin(2a)\cos a+\sin^2a-\sin^2(2a)$. Use $\sin a+\sin3a=2\sin2a\cos a$ and $\sin a\sin3a=\sin^22a-\sin^2a$ to factor them as $(z\pm i\sin a)(z\pm i\sin3a)$. The product tends to one at large positive real <rapidity>, fixing its overall phase in this bootstrap convention.
For $n\geq3$, the nearest <pole> to the real axis is $\vartheta=ia$. It is simple and comes from the first factor. In the crossed, or <t-channel>, the <momentum> carried between the external particles is their difference. With the $+,-$ metric its invariant is
$$
t=m_2^2+m_1^2-2m_2m_1\cosh\vartheta.
$$
At the <pole>, substitute the <breather> <masses>:
$$
\begin{aligned}
t&=4M^2\left[\sin^2(2a)+\sin^2a-2\sin(2a)\sin a\cos a\right]\\
&=4M^2\sin^2a=\boxed{m_1^2}.
\end{aligned}
$$
The trigonometric identity is applied with angles $2a$ and $a$. Thus the exchanged one-particle state is \b[the lightest <breather> $\mathcal B_1$], on its <mass shell>. This is <crossed-channel lightest-breather exchange>. The external momenta at a bound-state <pole> are analytically continued; <on shell> here means the invariant <mass> relation and conservation of the continued energy-momentum, not a <pole> at real physical <rapidity>. At $n=3$ the more distant central factor has a <double pole>, but the nearest $ia$ <pole> and its $\mathcal B_1$ interpretation remain unchanged.
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