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 $d = 3$ the spectral integral is UV-dominated, explaining why density normalization is needed.
An $α$-sweep for Potts $q = 4$ at $L = 32$--$256$ finds $α_{\mathrm{opt}}$ in $[-0.75, -0.5]$, consistent with $-2η_{\mathrm{Ising}}$ and inconsistent with $-2η_{q=4} = -1$; we flag this as tentative pending $L \geq 1024$ confirmation.
The $\langle m^2 \cdot I(-2η)\rangle$ 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 $Δ$ 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 $|k|^{-2η}$ shape factor; the latter is established by ensemble $L$-scaling.