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

ArXi:2604.01484v1 Announce Type: cross 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.