= Solution
For attractive nearest-neighbour interactions, $J>0$, the transformed model is a ferromagnetic <Ising model>. In three dimensions and at $T<T_c$, its infinite-volume <spontaneous magnetization> has two branches. As the effective field crosses zero, the equilibrium mean spin changes discontinuously from a negative value to a positive value. Since $(\partial h/\partial\mu)_T=1/2$, varying the physical <chemical potential> crosses this same <first-order phase transition>.
The particle-hole-symmetric coexistence line is
$$
\boxed{h=0\quad\Longleftrightarrow\quad \mu_{\rm coex}(T)=-2qJ-k_BT\ln\frac b{\lambda^3}.}
$$
The mean Ising spin is a rescaled occupation density. Writing $n=\langle N\rangle/V$ for the physical <number density>,
$$
\boxed{m_{\rm I}=\frac1M\sum_\alpha\langle\sigma_\alpha\rangle
=2\frac{\langle N\rangle}{M}-1=2bn-1.}
$$
The negative-spin phase is a dilute gas and the positive-spin phase is a dense, liquid-like phase. If the spontaneous spin magnitude at coexistence is $m_0(T)>0$, their number densities are
$$
n_{\rm gas}=\frac{1-m_0(T)}{2b},\qquad n_{\rm liquid}=\frac{1+m_0(T)}{2b},\qquad
n_{\rm liquid}-n_{\rm gas}=\frac{m_0(T)}b.
$$
Thus the physical <order parameter> is a density or occupation imbalance, not an actual magnetic <moment> of these <particles>. In a closed system with an intermediate mean density, the same transition appears as <phase coexistence> between the two densities. At $T=T_c$ the density contrast vanishes continuously; above $T_c$, sweeping the <chemical potential> gives no coexistence jump. A true discontinuity requires the <thermodynamic limit>; a finite sum over finitely many cells remains analytic.
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