Poisson Bridges: Spectral Geometry of Discrete Optimal Transport on Grids — E8 Intelligence Research

Abstract

FINDING: Discrete optimal transport on grids reduces to solving a Poisson equation whose eigen-structure encodes the geometry of the underlying lattice — a bridge between combinatorial optimization and spectral graph theory. | MATH: Optimal transport dual potentials \(u_i, v_j\) satisfy \(u_i + v_j \le c_{ij}\) (cost matrix); on a grid, the discrete Poisson equation \(\Delta_h u = f\) (with \(\Delta_h\) the 5-point stencil) yields eigenvalues \(\lambda_{k,l} = \frac{4}{h^2}\left(\sin^2\frac{k\pi}{2N} + \sin^2\frac{l\pi}{2M}\right)\) for an \(N\times M\) grid. The network simplex solves the dual via successive shortest augmenting paths — equivalent to a discrete gradient flow on the potential lattice. | CONNECTION: The grid eigenvalues are sums of two squared sines — the same structure as root system \(A_1 \times A_1\) (square lattice) weights. The ratio of the two smallest non-zero eigenvalues for a square grid is exactly 1:1 (degenerate), but for a rectangular \(N\times M\) grid with Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com