The topological gap at criticality: scaling exponent d + η, universality, and scope

The topological gap $\mathrm{\Delta }=T{P}_{H_1}^{real}-T{P}_{H_1}^{shuf}$ – 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: $\mathrm{\Delta }(L,T)=A{L}^{d+\eta }{G}_{-}(L\mathrm{\mid }t\mathrm{/}{T}_c\mathrm{\mid })$, with $G_{-}(x)\sim (1+x\mathrm{/}{x}_0{)}^{-(1+\beta \mathrm{/}\nu )}$. For 2D Ising, $\alpha =2.249\pm 0.038$, matching $d+\eta =9\mathrm{/}4$ to $0.03\sigma$; the $G_{-}$ exponent $\gamma =1.089\pm 0.077$ is consistent with $1+\beta \mathrm{/}\nu =9\mathrm{/}8$ ($\mathrm{\Delta }{R}^2<{10}^{-5}$). 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 $1+\beta \mathrm{/}\nu =17\mathrm{/}15$. Scope boundaries: the law fails for 2D Potts $q=4$ ($\alpha =2.347\pm 0.017$, $9.3\sigma$ from $d+\eta =5\mathrm{/}2$) where logarithmic corrections prevent convergence, and for raw 3D Ising ($4\sigma$ from $d+\eta$), but density normalization $\mathrm{\Delta }\mathrm{/}\mathrm{\mid }M{\mathrm{\mid }}^{1\mathrm{/}2}$ 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 ($\omega \to 0$).