Contour-resolvent state-space layer

Integrate this at the state-transition operation of a linear SSM, recurrent block, or continuous-depth residual block. Let the hidden width be $n$, choose a descriptor matrix $E$ and a trainable dynamics matrix $A(\theta)$, and replace Euler or bilinear propagation with the contour update. Choose $q$ conjugate-symmetric nodes, for example $s_j=\sigma+\rho e^{i\phi_j}$ on a parabolic or Talbot-like contour, together with quadrature weights $w_j$. For each node, solve $(s_jE-A)z_j=Eh_k$ and set $h_{k+1}=\operatorname{Re}\sum_jw_je^{s_j\Delta t}z_j+b_k$. If $A$ and $E$ are fixed across sequence positions, cache one LU factorization per node; if $A$ changes slowly, reuse preconditioners or use batched Krylov solves. With $E=I$, this is a drop-in stiff SSM; with singular $E$, it becomes a descriptor SSM. The paper supplies the resolvent propagation rule, while contour shape, node count, and truncation tolerance should be selected by comparing against a finer quadrature and monitoring the residual of the propagated system. First test a width-128 SSM on a stiff synthetic oscillator and on a small Long Range Arena task. Compare Euler, bilinear/Tustin, and exact matrix-exponential baselines at equal FLOPs. Measure loss descent, maximum stable $\Delta t$, gradient-norm variance, wall-clock time, and final sequence accuracy. A positive result is faster loss descent or a substantially larger stable step size at matched accuracy; a practical target is roughly $2\times$ larger stable steps or fewer propagation evaluations without perplexity degradation.