Solution
= Solution
Local thermal stability against energy exchange with a fixed-temperature reservoir requires $C_J>0$. Since $s\geq0$, this means
$$
2-2s-s^2>0
\quad\Longleftrightarrow\quad
s<\sqrt3-1.
$$
Equivalently,
$$
\boxed{\sqrt{2\sqrt3-3}<\frac{|a|}{M}<1}.
$$
The lower endpoint is the divergent-heat-capacity transition. The extremal endpoint $|a|=M$ has $T_H=0$ and $C_J=0$ and is obtained only as a limiting case.