Spectral origin of the topological gap exponent d + eta: 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 about L d+η. We derive this analytically: the spectral integral I(α) = sum k not equal to 0 S conn(k),k α scales as L 2-α-η when IR-dominated, giving I(-2η) about L d+η. The decomposition I(-2η) = I 0 times I shape separates volume (I 0 proportional to N(1-m 2) about L d) from anomalous dimension (I shape about 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 the spectral integral is UV-dominated, explaining why density normalization is needed. An α-sweep for Potts q = 4 at L = 32 -- 256 finds α opt in [-0.75, -0.5], consistent with-2η Ising and inconsistent with-2η q=4 = -1; we flag this as tentative pending L at least 1024 confirmation. The m squared times I(-2η) hyperscaling product is dominated by the correlation r(m squared, I) about-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 Δ and I(-0.5) is primarily a magnetization correlation: R squared = 0.91 at L = 256 collapses to R squared about 0 once M is partialed out. Per-configuration evidence corroborates the I 0 Parseval identity but not the k-2η shape factor; the latter is established by ensemble L-scaling.