Selector-Driven Hierarchical Permutation Mixer

Integrate the module before the MLP in a Transformer, or use it as a message-passing layer on a rooted tree. Represent every token by a path of symbols in a fixed branching tree and store its local support and level. Choose a small support Y, such as three sibling positions, and precompute sparse permutation indices for 3-cycles and their inverses. Use two base sparse permutations P_b1 and P_b2. At level n, select the word s_n using n modulo 4, compose its permutation, apply a local cycle P_sigma, and conjugate the cycle: P_out=P_s P_sigma P_s^{-1}. Replicate the same local permutation over all descendant supports with the same parent type, exactly as in the diagonal embedding. The forward pass is: for n in levels: sigma=local_cycle[n]; s=selector[n%4]; P_out=compose(P_s, sigma, inverse(P_s)); H=H+alpha[n]*gather(P_out,H)@W[n]. Keep P fixed initially, then optionally learn logits over a small set of valid local cycles using a straight-through categorical estimator. The paper determines the replication rule and selector schedule; estimate alpha_n and the number of active levels empirically. Begin with CIFAR-100 using a four-stage vision Transformer and synthetic balanced-tree sequence classification. Compare dense attention, window attention, and this mixer at equal hidden size and parameter count. Measure validation accuracy, peak memory, tokens per second, and effective receptive field. A successful result is at least 1.5x throughput or 30 percent lower activation memory with no more than one percentage point accuracy loss.