Boundary-Only Cell-Complex Network

1. Integration point: use this as the input parameterization and message-passing interface of a graph neural network on an oriented 2D cell complex. Store encoder-produced plaquette features H2 in R^{n2 x d}; do not independently store link features. Construct the sparse plaquette-to-link incidence matrix B2, with one row per link and entries identifying its incident plaquettes. For binary Z2 data, compute XOR exactly. For continuous features, use signed plaquette states S2 = tanh(H2) and reconstruct each link feature by elementwise multiplication of its incident plaquette states; concatenate this with fixed link attributes before link processing. 2. Pseudocode: H2 = encoder(plaquette_input); S2 = tanh(H2); Sl[e] = product(S2[p] for p in incident_plaquettes[e]); H1 = link_mlp(concat(Sl, link_attributes)); H2_next = plaquette_update(H2, aggregate_incident(H1)); output = readout(H2,H1). For discrete sampling, use f = Bernoulli(sigmoid(H2)) and b = (B2 @ f) % 2. 3. Compute B2 and B1 directly from the mesh; no learning or estimation is required. Verify the hard constraint by measuring max(abs((B1 @ b) % 2)), which must be zero. Compare against a baseline that independently predicts link and plaquette features and adds a Gauss-law penalty. 4. First experiment: train a small cell-complex GNN on synthetic 16x16-grid and triangulated-mesh tasks such as boundary-loop prediction from plaquette fillings. Report parameter count, activation memory, test accuracy, mesh-size transfer, and exact constraint violations. The method succeeds if it has zero violations without penalty, lower memory at equal hidden width, and equal or better accuracy on unseen mesh topologies. A useful stress test is random plaquette masking followed by boundary-loop reconstruction.