The mean-field exponent is universal because changes in the character of solutions of analytic equations are always described by catastrophes in the Taylor series, which is a polynomial equation. By symmetry, the equation for H must only have odd powers of H on the right hand side. Changing β should only smoothly change the coefficients. The transition happens when the coefficient of H on the right hand side is 1. Near the transition: H = ∂ ∂ ( β β F ) ∂ ∂ h = ( 1 + A ε ε ) H + B H 3 + ⋯ ⋯