Summed Resolvent Spectral Layer

Integrate this as a graph-nn or token-mixing layer between ordinary message-passing or attention blocks. Construct a sparse symmetric operator $\mathbf{L}$ from the batch graph Laplacian, normalized token-affinity Laplacian, or a fixed diffusion operator; add a small diagonal shift if needed so every $\mathbf{L}-p_j\mathbf{I}$ is nonsingular. Precompute $N$ poles $p_j$ and weights $w_j$ from a target spectral split, for example by fitting $r(\lambda)\approx 1$ on low eigenvalues and $r(\lambda)\approx 0$ on high eigenvalues over an estimated interval $[\lambda_{\min},\lambda_{\max}]$. Pseudocode: input $X$; for each pole $j$ in parallel, solve $(L-p_jI)Y_j=X$ using sparse factorization or fixed-iteration preconditioned CG/GMRES; accumulate $Z=\sum_jw_jY_j$; take the real part; apply a learned channel matrix $ZV$ and residual connection $X+\gamma ZV$. For complex poles, process conjugate pairs together. The mathematical computation is the additive resolvent evaluation; estimate $\lambda_{\max}$ by 10-step power iteration and measure solve residuals $\|(L-p_jI)Y_j-X\|/\|X\|$ empirically. First experiment: train a 4-layer GCN on Cora or a small 2D mesh node-classification problem with $N\in\{4,8,16,32\}$, identical poles, weights, optimizer, and FLOP budget. Compare the summed implementation with a recursive product implementation of the same rational filter and with a Chebyshev polynomial baseline. Track validation accuracy, gradient norm variance, filter error against a high-precision reference, solver residuals, and wall-clock time. Success means the summed version remains accurate as $N$ grows, avoids NaNs or exploding gradients seen in the product version, and improves accuracy or loss at equal solve cost; a secondary test should measure scaling when the independent solves are placed in separate GPU streams or batched sparse-solve kernels.