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

arXiv: https://arxiv.org/abs/2604.01484 · HTML: https://mlsanity.com/paper/2604.01484/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 3 references that are in the map.
- First posted on arXiv 1 Apr 2026.

## Builds on (most cited first, 1 of 1)

- [How much of persistent homology is topology? A quantitative decomposition for spin model phase transitions](https://mlsanity.com/paper/2603.29072/) (2026)

## Built on by (most cited first, 1 of 1)

- [Spectral origin of the topological gap exponent d + η: mechanism, kernel, decomposition, and scope](https://mlsanity.com/paper/2609.09159/) (2026)

## Abstract (arXiv)

The topological gap Delta = TPH1real - TPH1shuf -- 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: Delta(L,T) = A Ld+η G_-(L|t/Tc|), with G_-(x) ~ (1+x/x0)-(1+β/nu). 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/nu = 9/8 (Delta R2 < 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/nu = 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 Delta/|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).