Spectral origin of the topological gap exponent d + η: mechanism, kernel, decomposition, and scope
The topological gap $Δ$ -- the excess $H_1$ total persistence of a critical point cloud over a density-matched null -- scales as $Δ\sim L^{d+η}$. We derive this analytically: the spectral integral $I(α) = \sum_{k\neq 0} S_{\mathrm{conn}}(k)\,|k|^α$ scales as $L^{2-α-η}$ when IR-dominated, giving $I(-2η) \sim L^{d+η}$. The decomposition $I(-2η) = I_0 \cdot I_{\mathrm{shape}}$ separates volume ($I_0 \propto N(1-m^2) \sim L^d$) from anomalous dimension ($I_{\mathrm{shape}} \sim L^η$); the volume factor accounts for the magnetization-driven per-configuration variance of $Δ$. We prove the mechanism requires $d < 2 + η$ (IR dominance), confining it to $d = 2$ for physical systems; in $d = 3$ th...
| Authors | Matthew Loftus |
|---|---|
| Primary category | cs.LG |
| Published | 25 May 2026 |
| Last updated at source | 25 May 2026 |
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