Subcritical Ancestral Attention

Integrate this into the attention-pattern construction before computing the usual QK^T logits. For each query token i, initialize one active history particle S={i} at layer L and run it backward for T routing steps. At each step and for every active particle, sample a branch event with probability alpha. If branching occurs, replace the particle by B candidate ancestors, using B=4 or 8 for hardware efficiency; otherwise retain one candidate with probability p_survive and kill the history otherwise. Candidate ancestors may be selected from a local sequence window, graph neighbors, or approximate retrieval. Deduplicate ancestors and cap the set at K_max; compute attention logits only between the query and the resulting set, then renormalize softmax over that set. Pseudocode: initialize S={i}, J=0; for t=1,...,T, create new_S; for j in S, sample branch~Bernoulli(alpha); if branch, add B candidates and increment J; otherwise add one candidate with probability p_survive; set S=unique(new_S); if |S|>K_max, retain the largest-logit candidates; use S as the sparse mask. Numerically integrate the ODE for u(s,theta,t) and choose the largest alpha whose predicted moment u remains below a memory target M. The paper-derived computation is the branching moment and tail estimate; candidate locations and truncation are empirical choices. First test a 6-layer Transformer on WikiText-2 against dense and fixed-window attention at equal wall-clock FLOPs. Record perplexity, average and 99th-percentile |S|, J, activation memory, and gradient variance. Success means at least 2x lower attention memory or FLOPs with no more than 1% perplexity degradation and a lighter tail of history sizes.