Hodge-Metric Sector Coordinates

1. Integration point: add a continuous sector coordinate to a graph encoder, neural operator, or surrogate model for twisted Hamiltonians. The input is a graph or sample x together with a character coordinate theta. Encode theta using Fourier features after whitening by the Hodge matrix G, then inject the embedding into the first MLP block or into graph-message biases. 2. Pseudocode: estimate a positive-definite matrix G; compute a Cholesky factor G=L L^T; for every sector theta compute the wrapped coordinate delta=wrap(theta) and q=L^T delta; form e_theta=[sin(Bq), cos(Bq)] for a fixed or learned frequency matrix B; predict f(x,theta)=Net(x,e_theta). During training, sample sector pairs theta_a and theta_b and add the kernel-weighted consistency loss from the displayed formula to the task loss. 3. In the paper, G is the Hodge inner-product matrix, proportional to the bottom-band Hessian. In a generic implementation it can be estimated by finite differences of a scalar response E_0(theta): G_ij approximately equals [E_0(h e_i+h e_j)-E_0(h e_i-h e_j)]/(4h^2), followed by symmetrization and eigenvalue clipping. The loss weight, bandwidth sigma, and Fourier bandwidth B are empirical choices. 4. First cheap experiment: use a small twisted-graph Hamiltonian generator with genus g=2, sample many theta sectors, and train a three-layer MLP or GNN to predict the lowest eigenvalue and selected eigenvector observables. Compare raw-coordinate Fourier features, ordinary Euclidean smoothness, and Hodge-whitened coordinates at equal sample count. Measure interpolation error and extrapolation error on held-out sectors, especially along anisotropic directions. Success is lower error per training sector while preserving sharp nonconstant spectral variation.