Cross-ratio dynamics and the dimer cluster integrable system

The paper proves that the AFIT Poisson structure, Casimirs, and Hamiltonians for cross-ratio dynamics coincide with the Goncharov–Kenyon dimer integrable system on Γ_n, via a Poisson map π_{α} and identification of the dimer spectral polynomial as det(zI ± Π(w)) (Theorems 1.1, 1.2, 6.7, 7.1). It computes Poisson compatibility in u-variables and matches Hamiltonians through Π(w) and the AFIT monodromy trace identity, including the odd/even n cases. The candidate constructs an explicit birational Poisson map Φ through triple-crossing diagrams and local intersection computations in c-variables, also matching Casimirs and spectral polynomials. The core identifications and integrals agree with the paper; the routes differ (paper: u/Π(w); model: local cycle intersections/c). See Theorem 1.1 and the bracket/casimir formulæ for c-variables and zig-zags, and the Π(w) construction in the paper .