The topological gap Δ= TP_(H₁)ʳᵉᵃˡ - TP_(H₁)ˢʰᵘᶠ – the excess H₁ total persistence of the majority-spin alpha complex over a density-matched null – encodes critical correlations in spin models. We establish finite-size scaling: Δ(L,T) = A L^(d+η) G₋(L|t/T_c|), with G₋(x) ∼ (1+x/x₀)^(-(1+β/ν)). For 2D Ising, α= 2.249 ± 0.038, matching d+η= 9/4 to 0.03σ; the G₋ exponent γ= 1.089 ± 0.077 is consistent with 1+β/ν= 9/8 (ΔR² < 10⁻⁵). For 2D Potts q=3 with L up to 1024, α= 2.272 ± 0.024 (0.2σ from d+η= 2.267), with two-term corrections to scaling (R² = 0.9999). The G₋ exponent γ= 1.114 (68% CI [1.053, 1.173]) matches 1+β/ν= 17/15. Scope boundaries: the law fails for 2D Potts q=4 (α= 2.347 ± 0.017, 9.3σ from d+η= 5/2) where logarithmic corrections prevent convergence, and for raw 3D Ising (4σ from d+η), but density normalization Δ/|M|^(1/2) recovers α= 3.06 ± 0.04 (0.6σ). The framework fails for first-order, BKT, and percolation. The criterion: α= d+η holds when corrections to scaling are algebraic (ω> 0) but fails when logarithmic (ω→ 0).