Beta-fragmented hierarchical attention

(1) Integrate this at the attention-mask and routing level of a transformer. Given n token states, recursively partition every active token block until its size is at most a leaf capacity B. Add one learned summary vector to each internal node. Run ordinary multi-head self-attention within each leaf, then aggregate child summaries upward from leaves to the root, and finally broadcast parent summaries downward to update token states. The local attention cost is O(nB d), while tree communication costs roughly O(n d log n), instead of O(n^2 d), where d is hidden width. (2) Pseudocode: initialize a root block A containing all token indices; while size(A) > B, sample a split fraction s, set left_size = clip(round(s size(A)), 1, size(A)-1), create left and right child blocks, and attach summary u_A. Apply local attention to each leaf. For each level from leaves to root, compute u_A = Attention(u_A, concatenate(u_left,u_right)). For each level from root to leaves, update child token states using cross-attention to u_A. (3) Compute phi_beta(1) with adaptive quadrature and set a target maximum depth near 2 log(n) / phi_beta(1). Before training, sample many trees and measure the empirical 95th-percentile depth; reject, rebalance, or resample trees exceeding that limit. The mathematical quantities are phi_beta(1) and the logarithmic depth prediction; block size, summary dimension, and whether summaries use one or several vectors are empirical design choices. (4) First experiment: use a 6-layer transformer on ListOps or CIFAR-10 with 196 to 256 tokens. Compare full attention, a fixed balanced binary tree, uniform random recursive splits, and beta values -1, 0, 1, and 2. Match parameter counts and report validation accuracy, tokens per second, peak memory, attention FLOPs, maximum depth, and accuracy variance across sampled trees. A useful success criterion is at least 1.5x lower attention FLOPs and lower peak memory than full attention with no more than one percentage point accuracy loss; additionally test whether observed maximum depth scales linearly with log(n) and whether beta-controlled trees reduce depth variance.