Twisted Z2 classifier on graph cycles

Integrate this as a classification head on top of a graph neural network, point-cloud network, or transformer whose tokens have known adjacency and periodic identifications. First construct a sparse graph G=(V,E) using k-nearest neighbors or known grid edges, retain edge weights w_ij, and compute a cycle basis B from a spanning tree plus one cycle for each non-tree edge. The backbone produces scalar logits z_i=h_theta(x_i). Add one trainable scalar a_ij=a_ji per undirected edge and define a differentiable transport s_ij=tanh(alpha*a_ij), or use sign(a_ij) with a straight-through estimator. For every labeled node, apply ordinary cross-entropy to sigmoid(z_i). Add the covariant smoothness term w_ij*(z_j-s_ij*z_i)^2. For each basis cycle, compute m_gamma as the product of its edge transports and add the cycle penalty (1-m_gamma)^2. To avoid the trivial solution s_ij approximately zero, increase alpha during training or add lambda_s*(1-s_ij^2)^2. The graph transport is the learned discrete connection; cycle products are computed exactly from current edge parameters, while the cycle basis is computed once. First test on a 2D torus grid with XOR labels at 4, 16, and 100 labeled points, comparing an MLP, ordinary GNN, and this head. Success means lower test error at fixed labels and stable learned cycle parity, especially when the ordinary MLP violates periodic identification.