Signed-triangle Nyström lookahead

Integrate this into the landmark-selection stage of a Nyström attention, kernelized transformer, or inducing-point layer; do not change the forward attention computation initially. Given an SPD kernel matrix K over n tokens or inducing candidates, maintain a selected set S of size k and its Cholesky factor C of K_{S,S}. At each selection round, compute the current residual diagonal r_a=K_{aa}-K_{a,S}K_{S,S}^{-1}K_{S,a} for every unselected candidate a, using triangular solves with C. Form a shortlist Q containing the q candidates with largest r_a, where q is typically 8, 16, or 32. For each unordered pair (i,j) in Q, temporarily factor the 2-by-2 Schur update conditioned on S, compute E(S union {i,j}) from the trace formula E(S)=tr(K_{T,T}-K_{T,S}K_{S,S}^{-1}K_{S,T}), and choose the pair with the largest measured reduction. Add either both columns or the better member if only one budget slot remains, then update the Cholesky factor by block Cholesky. The paper supplies the exact Nyström and SDD/SDDM structure; the implementation estimates gains from the observed kernel matrix and does not assume submodularity. A diagnostic experiment should use a 2-layer transformer on CIFAR-10 or a small language-model attention block, compress a 256-token attention kernel to 16 or 32 landmarks, and compare ordinary greedy selection against pairwise lookahead at equal landmark count. Track approximation Frobenius error, attention-output error, validation accuracy or perplexity, and selection wall time. The expected signal is lower kernel/output error at the same memory budget, especially on synthetic SPD kernels containing a signed three-variable interaction; on benign kernels, accuracy should match greedy while adding only the controlled O(q^2) shortlist overhead.