Ising auxiliary mean-field free energy ($A_{\mathrm{aux}}/N=qJ{m}^2/2-k_{B}T\log [2\cosh ({\beta }_T(h+qJm))]$)

The approximate partition function obtained by decoupling each Ising bond has this free energy. The quadratic constant compensates double counting. Its stationary condition is the self-consistency equation; away from stationarity $m$ parametrizes the assumed effective field and need not equal that distribution's mean Ising spin. It agrees at stationary points with the Bragg-Williams free energy of the Ising model, although their off-equilibrium expansions differ.