Pole-Certified SSM Initialization

(1) Integration point: add a small diagonal or 2-by-2 real-block state-space branch to an existing sequence model, such as an S4-style layer, gated residual block, or recurrent residual stream. Collect a scalar or low-dimensional probe sequence y_0,...,y_{T-1} from a frozen teacher model, sensor stream, or intermediate-token feature over time. Initialize the SSM state matrix A from certified bases lambda_j instead of random or logarithmic initialization. (2) Pseudocode: choose order N and shifts S. For every shift s, build H0[i,j]=y[s+i+j] and H1[i,j]=y[s+i+j+1]. Solve H1 v=lambda H0 v using SVD truncation of H0. Retain eigenvalues with modulus below 1+epsilon and acceptable pencil condition number. Cluster eigenvalues across shifts, fit q(lambda)=product(lambda-lambda_j), and map each lambda_j to a real scalar mode or a 2-by-2 real block for complex-conjugate pairs. Set A to these blocks and fit input/output vectors B,C by least squares on y_t approximately C A^t B. (3) Compute the Hankel pencil, generalized eigenvalues, annihilating polynomial, and contour winding count directly. Estimate background and noise from the residual after fitting retained modes. For each candidate annulus contour in p-space, evaluate Q on 128-512 boundary points, compute discrete phase winding, and record m_Gamma; reject candidates whose count changes across shifts or whose margin is below 5-10 times the residual scale. (4) First experiment: use a 1-2 layer GRU or diagonal SSM on synthetic sums of damped sinusoids plus white noise, then test a character-level Penn Treebank task or long-range-copy benchmark. Compare random HiPPO/log-spaced initialization, learned initialization, and pole-certified initialization at equal parameter count. Success means faster early loss descent, fewer exploding or vanishing gradients, and improved extrapolation at sequence lengths 2-4 times the training length; target at least 10% lower validation loss or 2x longer stable rollout at equal FLOPs.