Boundary-State Renormalization GNN

1. Integration point: use this as a hierarchical graph encoder for graphs supplied with a recursive cell decomposition, such as synthetic gaskets, tree-like meshes, hierarchical molecules, or multiscale circuit graphs. At the leaves, compute a state vector from local node and edge embeddings; at each parent cell, replace ordinary cross-cell message passing by the boundary-state update. Each state sigma encodes the pairing or ordering of active boundary terminals and whether each terminal is unused, following the admissibility rule. A small MLP produces leaf log-scores u_0,sigma. 2. Pseudocode: enumerate the finite state set and admissible tuple table offline from the first-level incidence graph. Initialize leaf scores. For each level m, each parent, and each output state tau, compute terms = log_c[tau, tuple] + sum(u_child[i, sigma_i]) over admissible tuples and set u_parent[tau] = logsumexp(terms). Concatenate parent state scores and pass them to a graph-level task head. Use temperature T through T times logsumexp(terms divided by T); T equal to 1 is the partition-function version and small T approaches max-plus dynamic programming. 3. Compute admissible tuples and coefficients exactly from topology; learn only leaf encoders, optional state biases, temperature, and task-head parameters. Subtract the maximum term before exponentiation for numerical stability. 4. First experiment: generate level-4 through level-7 triangular or polygonal gasket graphs with random node features and labels determined by whether two boundary vertices admit a simple path with a prescribed terminal pattern. Compare against a width-matched message-passing GNN. Measure accuracy versus graph size, wall-clock cost, number of propagation layers, and gradient norms. Success means accuracy is retained as graph depth grows, computation uses substantially fewer propagation steps, and gradients remain stable where the flat GNN oversmooths or violates simple-path constraints.