Spectral origin of the topological gap exponent d + {eta}: mechanism, kernel, decomposition, and scope

arXiv:2609.09159v1 Announce Type: new Abstract: The topological gap Delta -- the excess H_1 total persistence of a critical point cloud over a density-matched null -- scales as Delta sim L^{d+eta}. We derive this analytically: the spectral integral I(alpha) = sum_{kneq 0} S_{conn}(k),|k|^{alpha} scales as L^{2-alpha-eta} when IR-dominated, giving I(-2eta) sim L^{d+eta}. The decomposition I(-2eta) = I_0 dot I_{shape} separates volume (I_0 propto N(1-m^2) sim L^d) from anomalous dimension (I_{shape} sim L^{eta}); the volume factor accounts for the magnetization-driven per-configuration variance of Delta. We prove the mechanism requires d < 2 + eta (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 alpha-sweep for Potts q = 4 at L = 32--256 finds alpha_{opt} in [-0.75, -0.5], consistent with -2eta_{Ising} and inconsistent with -2eta_{q=4} = -1; we flag this as tentative pending L geq 1024 confirmation. The langle m^2 dot I(-2eta)rangle hyperscaling product is dominated by the correlation r(m^2, I) approx -0.98 via the shared I_0 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: R^2 = 0.91 at L = 256 collapses to R^2 approx 0 once |M| is partialed out. Per-configuration evidence corroborates the I_0 Parseval identity but not the |k|^{-2eta} shape factor; the latter is established by ensemble L-scaling.

Source: arXiv cs.LG | 2026-09-10

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