Polymer-compatible sparse attention

Integrate this into a local-attention or MoE-routing block between the token projection and the value aggregation. Let X in R^{L x d_model} be a sequence, and create a candidate set P containing connected intervals for text, connected patches for images, or connected subgraphs for graph data. Use lengths 1 through s_max; for each candidate S compute a pooled query/key representation q_S and k_S, a boundary estimate |partial S| equal to the number of neighboring positions outside S, and a score g_theta(S,X) = q_S^T k_S / sqrt(d_head). Set the polymer weight to w_tilde(S) = exp(-beta_theta |partial S| - h_theta |S| + g_theta(S,X)). Build a conflict matrix C_ab equal to 1 when d(S_a,S_b) < 2. First implement a greedy truncated expansion: sort candidates by log w_tilde, accept a candidate if it conflicts with no already accepted candidate, and stop after K polymers or when the cumulative weight falls below epsilon. A closer partition-function approximation can use K rounds of masked parallel top-k selection, with logits for candidate a set to negative infinity whenever C_ab = 1 for an accepted candidate b. Aggregate values inside each selected S using mean or learned intra-polymer attention, then scatter the resulting group outputs back to member tokens; add a residual path for uncovered tokens. beta_theta and h_theta may be fixed initially, for example beta=0.1 and h=log(1.5), and later learned with positivity enforced by softplus. The exact mathematics provides the connected-set compatibility product and exponential weight; candidate enumeration, boundary computation, truncation K, and the learned content score are engineering choices. Start on CIFAR-10 ViT-Tiny with 8x8 patch tokens and compare dense attention, windowed attention, and this router at equal hidden size. Measure attention FLOPs, peak activation memory, top-1 accuracy, and the fraction of tokens covered. The first success criterion is at least 2x lower attention memory or FLOPs with no more than 1 percentage point accuracy loss; a secondary signal is improved stability when h is increased to suppress oversized groups.