Magnetization scaling-function matching ($m=|t{|}^{2-\alpha -\Delta }{g}_{m,\pm }(h/|t{|}^{\Delta })$)

Differentiate $f_s=|t{|}^{2-\alpha }{g}_{f,\pm }(h/|t{|}^{\Delta })$ with respect to its conjugate field. A finite nonzero one-sided ordered-branch amplitude gives $\beta =2-\alpha -\Delta$. To approach a finite critical isotherm as $t\to 0$ at fixed small $h$, the large-argument function must behave as $g_m(x)\sim C\mathrm{sgn}(x)|x{|}^{1/\delta }$. Cancelling the remaining thermal power yields $\beta =\Delta /\delta$. An analytic background, branch selection and thermodynamic limit must be distinguished from the singular free energy being differentiated.