Uniformly Mixing Coulomb Particle Bank

Integrate the method at a prototype-bank or latent-particle tensor. Let Z in R^{N x 2} contain N learnable or sampled two-dimensional coordinates. These coordinates can be used directly for routing, or passed through a learned map e_i = g_theta(z_i) to produce high-dimensional prototypes. At initialization or during periodic refreshes, run K unadjusted Langevin steps. For each step, compute pairwise differences D_ij = z_i - z_j and squared distances r2_ij = ||D_ij||^2. Set R_i = (1/N) sum_{j != i} D_ij / (r2_ij + epsilon), then update z_i <- z_i - eta (z_i - R_i) + sqrt(2 eta / beta) xi_i, where xi_i is a standard two-dimensional Gaussian. The confinement term z_i comes from the quadratic potential, the factor 1/N is the paper's mean-field scaling, beta controls noise, and epsilon prevents numerical blow-up at collisions. For a routing experiment, use logits l_i(h) = -||W h - z_i||^2 / tau and train W and the downstream model while either freezing Z after initialization or refreshing Z periodically. The paper supplies the N-independent LSI claim for the exact singular target; estimate the effective mixing rate of the discretized sampler empirically using autocorrelation of mean radius, pairwise distance, and assignment histograms. Start with CIFAR-10, a small MLP or ViT, and N in {16, 64, 256}. Compare Gaussian prototypes, repulsion-only gradient descent, and Coulomb Langevin at matched initialization cost. Record nearest-neighbor distance, dead-prototype fraction, assignment entropy, validation accuracy, and autocorrelation time. Success means stable minimum distances, fewer dead routes, and roughly unchanged mixing statistics as N increases, together with equal-or-better accuracy at fixed compute.