Swarmalator Token Organizer

Integrate the module immediately before a transformer attention block or graph message-passing layer. Give each token i two trainable state variables x_i and θ_i, initialized uniformly on the circle, and retain its ordinary embedding h_i. At every recurrent substep, compute pairwise wrapped differences, evaluate the two coupling sums, and update x_i and θ_i with Euler or RK2. Use cos(x_i), sin(x_i), cos(θ_i), and sin(θ_i) as additional embedding channels, or use the coordinates to bias attention logits. Pseudocode: compute dx_ij=x_j−x_i and dθ_ij=θ_j−θ_i; set fx_i=v_i+(J/N)Σ_j sin(dx_ij)cos(dθ_ij); set fθ_i=ω_i+(K/N)Σ_j sin(dθ_ij)cos(dx_ij); update x_i=wrap(x_i+h fx_i) and θ_i=wrap(θ_i+h fθ_i); then apply attention to the augmented embeddings. Compute R_x and R_θ after every substep. The paper supplies the flow and qualitative four-state/cusp organization; estimate the finite-N Jacobian spectrum using automatic differentiation and estimate transition curves by parameter sweeps. The first cheap experiment is a small transformer on CIFAR-10 patch sequences or Penn Treebank, comparing vanilla attention with one or three swarmalator substeps. Sweep J and K and plot R_x, R_θ, attention entropy, and loss. The predicted signature is a regime map with four regions, cusp-like convergence of transition curves, non-monotonic R_θ as K increases at fixed J, and Euler instability when the measured spectral radius of I+hD_zF crosses one.