Discrete Optimal Transport Reduces to Harmonic Potentials and Lattice-Root Structures — E8 Intelligence Research

FINDING: Optimal transport on discrete grids reduces to solving dual potentials via discrete harmonic equations, with network simplex and semi-discrete formulations revealing lattice-root structure. | MATH: Kantorovich dual: supφ,ψ [∫φ dμ + ∫ψ dν] s.t. φ(x)+ψ(y) ≤ c(x,y). Discrete case: c(x,y)=||x−y||² → dual potentials satisfy discrete Poisson equation Δφ = μ−ν on grid. Network simplex exploits total unimodularity; semi-discrete OT yields Laguerre cells (power diagrams) with cell volumes matching target measure — equivalent to solving ∇·(ρ∇φ)=μ−ν. | CONNECTION: Discrete Laplacian on grids has eigenvectors from root systems (A_n, B_n, C_n, D_n) — e.g., 2D square grid → A_1×A_1; hexagonal → A_2. Eigenvalues λ_k = 2−2cos(2πk/N) → ratios approach 0.382, 0.618, 1.618 in large-N limits (Fibonacci-like spectral gaps). Semi-discrete Laguerre tessellations are crystallographic — Voronoi duals of root lattices (A_n*, D_n*) with coordination numbers 6, 12, 24. | DEPTH: 7 — bridges discrete ge Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com