The saddle-point characterization of the Curie–Weiss critical curve

Let $f_\beta(u)$ be the function whose saddle-point equation is

$$f'_\beta(u)=\beta u-\beta\tanh(\beta u)=0,$$

and let $u^*(\beta)$ be the branch of solutions defined in the source, satisfying $u^*(1)=0$ for $0<\operatorname{Re}\beta\leq C$, with $C\approx1.3$. Let $\epsilon_0(R)$ be the critical-curve function from the preceding conjecture. Saddle-point characterization conjecture. The critical-curve relation holds with $\epsilon_0(R)$ defined by

$$\operatorname{Re} f_{1+\epsilon_0(R)+iR}\bigl(u^*(1+\epsilon_0(R)+iR)\bigr)=0.$$

This proposes that the phase-transition curve is determined by the vanishing of the real part of the relevant saddle-point action along the specified branch. The source gives no proof of this characterization, so its resolution is open.