Gaussian-resummed sparse interaction layer

(1) Integration point: insert the block after token or graph embeddings and before the next residual layer. Given X in R^{n x d}, compute q_i=W_q x_i and define J_ij=(beta/n)tanh(q_i^T q_j/sqrt(d)) for i!=j and J_ii=0; symmetrize J as (J+J^T)/2. Enforce lambda_max(J)<=1-kappa with 3-5 power-iteration steps and the rescaling J <- J*min(1,(1-kappa)/(lambda_max(J)+1e-6)). (2) Pseudocode: compute Q=I-J; obtain base=-0.5*logdet(Q) using Cholesky for n<=256 or stochastic Lanczos for larger n. For every subset S with 1<=|S|<=R, evaluate c_S using Gauss-Hermite quadrature over z_S under the Gaussian covariance Q^{-1}; for each scalar residual potential r_i(z_i), subtract its degree-0, degree-1, and degree-2 Taylor terms before multiplying residual factors, ensuring the correction starts at degree three. Sum base and all c_S, then use the result as a global residual bias or interaction feature. (3) Use the paper's tail expression to choose the smallest R satisfying (C*R/n)^(R/2)<epsilon/2. Estimate C from the maximum normalized degree-three residual on a calibration batch; estimate the e^{-cn} contribution by comparing outputs at R and R+1. For large n, replace exact subset enumeration by uniform or magnitude-weighted Monte Carlo subset sampling with unbiased reweighting. (4) First experiment: train a small transformer with sequence lengths 32, 64, and 128 on a synthetic task whose target is a non-Gaussian Gaussian-coupled partition function, where brute-force quadrature is available for n<=12. Compare dense softmax attention, linear attention, and this block at matched hidden size and wall-clock budget. Measure partition-function relative error, training loss, gradient variance, peak memory, and throughput. A successful result is below 1% approximation error at n=32, lower gradient variance near the coupling boundary, and faster loss reduction or lower memory than dense attention at comparable accuracy.