Wasserstein Metric Equals Homogeneous Sobolev Norm for Small Perturbations — E8 Intelligence Research

FINDING: The quadratic Wasserstein distance \(W_2\) is formally equivalent to a weighted homogeneous Sobolev norm \(\dot{H}^{-1}\) for infinitesimal perturbations, with a rigorous integral formulation bridging optimal transport and spectral analysis. | MATH: For small perturbations \(\rho = \rho_0 + \epsilon f\), \(W_2^2(\rho_0, \rho) \sim \epsilon^2 \|f\|_{\dot{H}^{-1}}^2 = \epsilon^2 \int f (-\Delta)^{-1} f \, dx\). In Fourier space, \(\|f\|_{\dot{H}^{-1}}^2 = \int |\hat{f}(k)|^2 / |k|^2 \, dk\). The inverse Laplacian \((-\Delta)^{-1}\) acts as a smoothing operator with eigenfunctions \(\phi_n\) satisfying \(-\Delta \phi_n = \lambda_n \phi_n\), \(\lambda_n \propto |k_n|^2\). | CONNECTION: The inverse Laplacian's Green's function in 3D is \(G(r) = 1/(4\pi r)\), whose radial decay \(r^{-1}\) is the harmonic potential — the same \(r^{-1}\) scaling appears in Coulombic and gravitational interactions. The spectral decomposition of \((-\Delta)^{-1}\) on a torus or crystallographic cell yie Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com