What does this research mean for the field?

The quadratic Wasserstein distance is formally equivalent to a weighted homogeneous Sobolev norm for infinitesimal perturbations, providing a rigorous integral formulation that bridges optimal transport and spectral analysis. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish a rigorous mathematical equivalence between the quadratic Wasserstein metric and a weighted homogeneous Sobolev norm for small density perturbations.

August 30, 2026Open Access

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

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Authors

ACAndrew Stewart Caldin

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Implication

Theoretical analysis demonstrates quadratic Wasserstein distance equates to negative Sobolev norms in perturbed densities, highlighting fundamental connections to harmonic potentials.

Key Points

To establish a rigorous mathematical equivalence between the quadratic Wasserstein metric and a weighted homogeneous Sobolev norm for small density perturbations.
Formulated the expansion of the quadratic Wasserstein distance for perturbed distributions of the form rho = rho_0 + epsilon * f.
Derived the energy representation in Fourier space using the inverse Laplacian smoothing operator.
Evaluated the three-dimensional Green's function and spectral decompositions on periodic geometric domains.
Demonstrated that the squared Wasserstein metric asymptotically scales as epsilon squared times the homogeneous negative Sobolev norm of the perturbation.
Proved the Fourier representation of the norm equals the integral of the perturbation power spectrum divided by squared wavenumber magnitude.
Identified that the Green's function of the inverse Laplacian follows a radial inverse-distance decay, matching Coulombic and gravitational potential scalings.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a93f0a36c1a8fb52e79cee7 https://doi.org/10.5281/zenodo.22138697

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Implication

Theoretical analysis demonstrates quadratic Wasserstein distance equates to negative Sobolev norms in perturbed densities, highlighting fundamental connections to harmonic potentials.