Mean-field onset of two-state ferromagnetism ($m\sim \sqrt{3(1-T/T_C)}$)

For $m=\tanh (m/t)$ and positive temperature, the positive solution exists uniquely for $0<t<1$: the slope at zero is $1/t>1$, while the hyperbolic tangent is concave and below one. For $t\ge 1$ only $m=0$ remains. Expanding to cubic order near $t=1$ gives $m^2\simeq 3t^2(1-t)$ and $m\simeq t\sqrt{3(1-t)}$, with mean-field critical exponent $1/2$. The prefactor $t$ follows from the truncated equation; it is not controlled beyond leading critical order.