Weighted Tree Dirichlet-to-Neumann Layer

Integrate this as a sparse multiscale interaction layer after token or patch embeddings and before a Transformer block. Given N boundary feature vectors X in R^{Nxd}, construct a balanced b-ary tree by recursively grouping nearby tokens, with depth K chosen so b^K is approximately N. Here b replaces p, while n is a user-chosen effective ultrametric dimension and s controls scale selectivity. For every edge from level k-1 to k, set conductance w_k=b^{k(s-n)}. Initialize each internal-node feature as the weighted average of its children. Hold boundary features fixed and perform T Jacobi updates on internal nodes: u_v <- (sum_{z~v} w_vz u_z)/(sum_{z~v}w_vz). This is the discrete weighted-harmonic equation. Compute the boundary flux q_x=sum_{z~x}w_xz(u_x-u_z), then add a learned projection of q to X through a residual connection. For small trees, compute the exact Dirichlet-to-Neumann matrix L_BB-L_BI L_II^{-1}L_IB; for large trees, use the finite relaxation approximation and backpropagate through its T steps. Add lambda times E_T(u)=1/2 sum_edges w_e||u_v-u_z||^2 to the task loss if an explicit regularizer is desired. The paper supplies the conductance law, harmonic update, flux interpretation, and energy identity; estimate the tree from token locality or clustering and tune s, lambda, and T empirically. First test on CIFAR-10 with 256 image patches or a small language-model task with 256 tokens, comparing against dense attention at matched feature width. Measure validation accuracy or perplexity, wall-clock time, peak memory, and scaling with N. Success is equal or better accuracy at lower cost, with energy decreasing across relaxation steps and stable performance across tree depths.