= Solution
Because the one-particle spectrum is bounded above, a <negative temperature> is possible. A population-inverted configuration with more particles in the $+\epsilon$ level than in the $-\epsilon$ level has
$$
\frac{N_+}{N_-}=e^{-2\beta\epsilon}>1,
$$
which requires $\beta<0$. Equivalently, above the maximum-entropy energy $E=0$, adding energy reduces the number of compatible microstates, so $\partial S/\partial E=1/T<0$. The limiting configuration with every particle at $+\epsilon$ has $E=N\epsilon$, zero entropy, and $T\to0^-$.
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