Squarefree Cycle Positional Encoding

1. Integration point: modify the input embedding of a graph neural network. For each graph with n nodes and weighted adjacency A, append the vector z_i=(z_{i,3},...,z_{i,K}) to the original feature h_i before the first message-passing or graph-transformer layer; optionally append graph-level totals C_k(1) after pooling. 2. Pseudocode: construct A; set p[i,k]=0 and total[k]=0; for each k in 3..K, enumerate simple cycles with DFS using the smallest vertex as a canonical start and one canonical orientation; compute q=product of the edge weights around the cycle; add q to p[i,k] for every node i in that cycle and to total[k]. Normalize with z[i,k]=log(1+p[i,k]/(epsilon+total[k]/n)), concatenate z to h_i, and run the unchanged GNN. For small dense graphs, an alternative is symbolic propagation of tr((XA)^k), retaining monomials represented by distinct vertex bitmasks. 3. The paper supplies the 2k normalization, squarefree criterion, and trace/log-determinant identities. The engineer computes cycle lists exactly; no theorem constants are estimated. For K<=5, use exact DFS on small graphs; for larger K use capped DFS or randomized simple-cycle sampling. 4. First experiment: train a 5-layer GIN and a graph transformer on MUTAG, PROTEINS, and a synthetic degree-matched dataset whose classes differ only in triangle or 4-cycle counts. Compare the baseline, Laplacian eigenvector features, and these features at equal parameter count. Success is higher held-out accuracy, especially on the degree-matched task, with feature computation below 10% of training time for graphs of at most 100 nodes.