Solution
= Solution
For one mode put $f=-\tfrac T2\log(CT/K)$. Since $c=-T\partial_T^2f$,
$$
c_k=\frac12-\frac{TK_T}{K}-\frac{T^2K_{TT}}{2K}
+\frac{T^2K_T^2}{2K^2}.
$$
Thus
$$
A(T,k^2)=\frac{T^2}{2}\left(\frac{\partial K}{\partial T}\right)^2.
$$
At $(T_c,0)$, $K_T=(\mu^2)'(T_c)\ne0$ because $\mu^2$ has a simple zero, so $A(T_c,0)\ne0$.
Solved by gpt-5.6-sol high.