Faithful Hypergraph Orthogonal Prototypes

Integrate this as a prototype table in a graph neural network or as a structured key/value table in attention. Let the hypergraph contain entities V and context triples E; each entity i owns a trainable vector u_i in R^3, and compute v_i=u_i/(||u_i||_2+epsilon) before every forward pass. At training time, add the context Gram penalty and anti-collapse penalty to the task loss. Pseudocode: initialize U in R^{|V|x3}; for each batch, set V=normalize_rows(U); compute the task output; for each triple e=(i,j,k), form G_e=V[e].T @ V[e] and accumulate ||G_e-I||_F^2; sample non-identical pairs (i,j), compute c=(V[i] dot V[j])^2, and accumulate exp((c-tau)/T); backpropagate the combined loss and update U. Here E is the context set, V_e is the matrix of context vectors, I_3 is the identity, tau is the tolerated squared ray overlap, and T is a temperature. The paper supplies the orthogonality equations and coordinate-plane propagation logic; lambda weights, tau, pair sampling, and optimizer are empirical choices. First test on Cora or synthetic 3-uniform hypergraphs with a small node-classification GNN. Compare unconstrained embeddings, context orthogonality alone, and the full model at equal parameter count. Record accuracy, minimum pairwise angular distance, context Gram residual, and collapse frequency across random seeds. Success means fewer collapsed prototypes and higher held-out-context accuracy without a major training slowdown.