The topological gap at criticality: scaling exponent d + {\eta}, universality, and scope

Abstract

The topological gap $\Delta = TP_{H_1}^{real} - TP_{H_1}^{shuf}$ -- the excess $H_1$ total persistence of the majority-spin alpha complex over a density-matched null -- encodes critical correlations in spin models. We establish finite-size scaling: $\Delta(L,T) = A L^{d+\eta} G_-(L|t/T_c|)$, with $G_-(x) \sim (1+x/x_0)^{-(1+\beta/\nu)}$. For 2D Ising, $\alpha = 2.249 \pm 0.038$, matching $d+\eta = 9/4$ to $0.03\sigma$; the $G_-$ exponent $\gamma = 1.089 \pm 0.077$ is consistent with $1+\beta/\nu = 9/8$ ($\Delta R^2 < 10^{-5}$). For 2D Potts $q=3$ with $L$ up to 1024, $\alpha = 2.272 \pm 0.024$ ($0.2\sigma$ from $d+\eta = 2.267$), with two-term corrections to scaling ($R^2 = 0.9999$). The $G_-$ exponent $\gamma = 1.114$ (68% CI $[1.053, 1.173]$) matches $1+\beta/\nu = 17/15$. Scope boundaries: the law fails for 2D Potts $q=4$ ($\alpha = 2.347 \pm 0.017$, $9.3\sigma$ from $d+\eta = 5/2$) where logarithmic corrections prevent convergence, and for raw 3D Ising ($4\sigma$ from $d+\eta$), but density normalization $\Delta/|M|^{1/2}$ recovers $\alpha = 3.06 \pm 0.04$ ($0.6\sigma$). The framework fails for first-order, BKT, and percolation. The criterion: $\alpha = d+\eta$ holds when corrections to scaling are algebraic ($\omega > 0$) but fails when logarithmic ($\omega \to 0$).