Triple-point critical-window crossover

Establish the conjectured subcritical triple-point extension of the critical-line fluctuation theorem: when \(\varepsilon_N=1-\beta_N^2\to0\), \(N^{1/3-\\delta}\varepsilon_N\to\infty\) for some \(\delta>0\), and \(\gamma_N=1+w(N\varepsilon_N)^{-1/2}\), prove the stated convergence in distribution of the centered SKFI free energy to the quartic-integral law involving an independent standard Gaussian \(\zeta\).

Background

The critical-line theorem in the paper treats fixed $\beta<1$ with $\gamma_N=1+\theta N^{-1/2}$. The authors explain that as $\beta_N\uparrow1$, the relevant ferromagnetic crossover window changes scale to $(N\varepsilon_N)^{-1/2}$, eventually meeting the critical-temperature window near the triple point.

They explicitly conjecture an extension on the subcritical side, with a non-Gaussian quartic-integral limit. At and beyond the corner $\varepsilon_N\asymp N^{-1/3}$, they state that a sphere-to-cube comparison and spherical triple-point input would be needed.