Connectivity-coded sparse masks

(1) Integration point: use the codebook as the mask generator for a sparse bipartite linear layer, a graph-neural-network message-passing block, or a token-to-expert routing matrix. Partition source and destination units into blocks of size d, construct a d by d K_{d,d} pattern in each block, and multiply the learned weight matrix elementwise by the selected binary mask M_a. The mask can remain fixed for a minibatch and change between minibatches or routing phases. (2) Pseudocode: choose d >= 4; sample or load a binary irreducible matrix T; compute powers P_i = T^i modulo 2; set ell[i,j] = P_i times e_j modulo 2; enumerate every nonzero a in F_2^d; compute mask[a,i,j] = dot(a, ell[i,j]) modulo 2; run BFS on every mask[a] and reject T unless each nonzero mask is connected and spanning. During training, maintain a state a and update it with a new nonzero increment u, using a_new = a XOR u. (3) The exact mathematics determines labels, masks, XOR transitions, and connectivity certificates. Empirical choices include transition frequency, block tiling, whether to learn a or sample it, and whether masks are applied to weights or activations. For larger layers, tile the certified block pattern and compare against random sparse masks at equal density. (4) First experiment: train a small sparse MLP or 16-expert MoE on CIFAR-10 or a small language-model corpus. Compare random masks, independently resampled masks, and connectivity-coded masks with identical FLOPs and parameter counts. Record final accuracy, loss spikes after mask changes, isolated-unit fraction, gradient norm variance, and MoE load imbalance. Success means lower instability or faster recovery after mask changes at equal compute, with no accuracy loss.