= Paramagnetic auxiliary-field curvature
{title2=$\Omega''(0)/N=U^{-1}-\int\nu(\epsilon)[-n_F'(\epsilon)]d\epsilon$}
For a uniform exchange field with cost $NM^2/(2U)$ and spin-band energies $\epsilon_k-\sigma M$, the <grand potential> is even in M. Its curvature at zero field is $N[U^{-1}-\chi_0]$, with $\chi_0=\int\nu(-n_F')d\epsilon$ and spin-summed <density of states> per site. Negative curvature proves local paramagnetic instability. It does not by itself exclude a first-order transition before that instability, nor determine the stable finite-field amplitude.
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