Mixed-Subgroup Conservation Router

Integrate the layer immediately before MoE expert selection or multimodal or relational feature fusion. Assume l input branch tensors x_i with shape [batch, hidden]. Choose a small finite abelian group, initially G=Z_4 or Z_8, and define subgroup masks H_i. For Z_m, use the subgroup of size d_i dividing m, represented by {0,m/d_i,2m/d_i,...,(d_i-1)m/d_i}. Each branch has a linear classifier producing logits a_i over H_i, followed by p_i=softmax(a_i). Enumerate valid tuples A once at initialization. For each batch item, compute log weights s(g_1,...,g_l)=sum_i log p_i(g_i), set s=-infinity outside A, and obtain q=softmax(s over A). Associate every valid tuple with either an expert index or a learned code e_tuple. Compute y=sum_tuple q_tuple e_tuple, then use y to gate experts or add it to the fused hidden representation. Pseudocode: logits_i=Linear_i(x_i); p_i=softmax(logits_i); scores=stacked_sum(log(p_i) indexed by A); q=softmax(scores); y=q @ E; output=MoE(x, gate=y) or LayerNorm(concat(x_i)+Proj(y)). For large l, avoid full Cartesian enumeration by dynamically programming the modular sum: repeatedly convolve branch distributions on G, then recover tuple samples with a constrained beam or compute exact marginal probabilities using the probability that the remaining branches cancel the current label. The paper supplies the exact support constraint; all neural logits, code vectors, and expert weights are learned. First test on a synthetic compositional classification task where three branches contain Z_4 labels and the target is their zero-sum class, comparing an unconstrained Cartesian router, a masked router with H_i=G, and mixed subgroups such as sizes (4,2,2). Then test a small relational GNN on modular addition or colored-edge composition. Measure validation accuracy at fixed samples, number of active joint states, routing entropy, and wall-clock cost. Success means faster learning and higher accuracy when the conservation rule is correct, with no degradation on a control task using randomized labels.