= Solution
Combining the eigenvalue cases gives
$$
\boxed{\text{recurrent}\iff\mu\alpha\ge\lambda\beta,\qquad\text{transient}\iff\mu\alpha<\lambda\beta.}
$$
This agrees with the interior drift: the switching phase spends proportions $\beta/(\alpha+\beta)$ and $\alpha/(\alpha+\beta)$ in $C$ and $W$, giving level drift $(\lambda\beta-\mu\alpha)/(\alpha+\beta)$. The harmonic calculation establishes recurrence even at the zero-drift boundary, rather than inferring it from drift alone.
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