# Spectral origin of the topological gap exponent d + η: mechanism, kernel, decomposition, and scope

arXiv: https://arxiv.org/abs/2609.09159 · HTML: https://mlsanity.com/paper/2609.09159/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 13 references that are in the map.
- First posted on arXiv 25 May 2026.

## Builds on (most cited first, 4 of 4)

- [On the choice of weight functions for linear representations of persistence diagrams](https://mlsanity.com/paper/1807.03678/) (2018)
- [How much of persistent homology is topology? A quantitative decomposition for spin model phase transitions](https://mlsanity.com/paper/2603.29072/) (2026)
- [Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology](https://mlsanity.com/paper/2109.10960/) (2021)
- [The topological gap at criticality: scaling exponent d + η, universality, and scope](https://mlsanity.com/paper/2604.01484/) (2026)

## Abstract (arXiv)

The topological gap Delta -- the excess H1 total persistence of a critical point cloud over a density-matched null -- scales as Delta ~ Ld+η. We derive this analytically: the spectral integral I(α) = sumkneq 0 Sconn(k),|k|α scales as L2-α-η when IR-dominated, giving I(-2η) ~ Ld+η. The decomposition I(-2η) = I0 · Ishape separates volume (I0 propto N(1-m2) ~ Ld) from anomalous dimension (Ishape ~ Lη); the volume factor accounts for the magnetization-driven per-configuration variance of Delta. We prove the mechanism requires d < 2 + η (IR dominance), confining it to d = 2 for physical systems; in d = 3 the spectral integral is UV-dominated, explaining why density normalization is needed. An α-sweep for Potts q = 4 at L = 32--256 finds αopt in [-0.75, -0.5], consistent with -2ηIsing and inconsistent with -2ηq=4 = -1; we flag this as tentative pending L ≥ 1024 confirmation. The langle m2 · I(-2η)rangle hyperscaling product is dominated by the correlation r(m2, I) ≈ -0.98 via the shared I0 amplitude, so we report it as a covariance-correction analysis. Under a heuristic argument extending Divol--Polonik to inhomogeneous Poisson intensities, the bare PH kernel is flat; the effective kernel acquires k-dependence only at criticality. The per-configuration agreement between Delta and I(-0.5) is primarily a magnetization correlation: R2 = 0.91 at L = 256 collapses to R2 ≈ 0 once |M| is partialed out. Per-configuration evidence corroborates the I0 Parseval identity but not the |k|-2η shape factor; the latter is established by ensemble L-scaling.