Nonlinear torsion positional encoding

Integrate this at the graph-data preprocessing or graph encoder input stage. For every graph, choose a nonempty boundary set B; if the task has explicit terminals, root nodes, source/sink nodes, or boundary labels, use them. Otherwise use a deterministic policy such as the lowest-degree nodes or k nodes farthest from a selected anchor. Construct symmetric edge weights w_ij, normally 1 or inverse geometric edge length. Solve for one or several torsion channels t^(p) with p in {1.5, 2, 3}. A practical solver is projected nonlinear gradient descent: initialize t=0 on nonboundary nodes, compute g_i=sum_j w_ij abs(t_i-t_j)^(p-2)(t_i-t_j)-1, update t_i <- max(0,t_i-lr*g_i), and reset t_b=0; use backtracking or Adam and stop when the normalized residual ||g||_2/sqrt(|V|) is below 1e-4. For larger graphs, use 50-200 iterations or warm-start from a nearby graph. Normalize each channel by (mean(t)+epsilon) or its standard deviation, then concatenate [x_i,t_i] to node features before the first message-passing layer. Optionally apply h_i <- h_i * sigmoid(a*t_i+b) after each layer. The paper supplies the source-driven p-Laplacian structure and variational objective; the graph discretization, boundary policy, and solver tolerance are engineering choices. First run a 3-layer GIN or GraphSAGE on synthetic bottlenecked trees, barbell graphs, and random geometric graphs, plus a small molecular or OGB node-classification subset. Compare raw features, degree, Laplacian eigenvectors, random-walk diffusion coordinates, and torsion channels at equal hidden size and preprocessing budget. Success is higher test accuracy or lower sample complexity on labels depending on global branch or boundary location, with no more than 10% preprocessing overhead and stable gains across p.