Spectrally admissible recurrent state

1. Exact integration point: create a recurrent or state-space layer whose transition matrix is a symmetric tridiagonal Jacobi matrix J. Parameterize its diagonal by unconstrained values b_n and its off-diagonal values by a_n = epsilon + softplus(alpha_n), with epsilon such as 1e-4. Use the transition h_(t+1) = tanh(J h_t + W x_t + u) or use J as the linear state matrix before a gated nonlinearity. Add the spectral penalty to the task loss. 2. Pseudocode: construct J from alpha and b; compute its eigendecomposition J = V diag(lambda_i) V transpose; for each c in {0.25, 0.5, 1.0}, compute M_c = sum_i V_(0,i)^2 exp(c absolute_value(lambda_i)); set L = L_task + lambda_spec times sum_c log(M_c) and backpropagate. For a two-sided model, compute separate left and right submatrices and sum both penalties. At every rollout, log M_c, the largest absolute eigenvalue, hidden-state norm, and gradient norm. 3. Computed from the paper: the Jacobi representation, positivity constraint, spectral-measure exponential moments, and the implication of spectral admissibility for the exact Toda flow. Estimated empirically: finite-dimensional moments, thresholds for the learned nonlinear recurrence, and the relationship between moment growth and exploding trajectories. 4. First cheap experiment: use a 32-dimensional recurrent layer on adding-task sequences of lengths 100, 500, and 1,000. Compare a vanilla tanh RNN, a spectral-normalized RNN, and the Jacobi layer with spectral regularization at matched parameter counts. Sweep the regularization coefficient and recurrent scale. The quantitative prediction is a sharp stability boundary: models maintaining M_1 below a threshold selected on short sequences should show a much lower frequency of hidden-state or gradient explosions at length 1,000, while removing positivity or allowing M_1 to increase by roughly one order of magnitude should sharply increase divergence frequency. Also test continuous dependence by perturbing initial parameters by norm delta and measuring trajectory distance; inside the admissible region, distance should grow smoothly with delta rather than displaying abrupt blow-up.