Cρ-stable recurrent transition

(1) Integration point: replace the recurrent transition h_{t+1}=Th_t+Bx_t, or the SSM state matrix discretization, with a trainable complex or real matrix T. Apply the certificate as a training-time regularizer and optionally as a projection step after each optimizer update. Use ρ=1 for a strict contraction baseline, ρ=2 for a numerical-radius-constrained model, and test intermediate values such as 1.25 and 1.5. For real T, evaluate the complex phase condition using complex arithmetic or represent T as a real 2d×2d block matrix. (2) Pseudocode: initialize raw T; at each update construct T; for j=0,...,M-1 set z=exp(i*2πj/M), Q_j=I-2*(1-1/ρ)*Re(conj(z)*T)-(2/ρ-1)*abs(z)^2*T.H*T; compute v_j=-lambda_min(Q_j); add Vρ=(1/M)Σ softplus(τv_j) to the task loss; backpropagate and update T. Every K steps, optionally subtract η times the gradient of the largest violation. (3) Compute the smallest eigenvalues with 5–10 Lanczos iterations rather than a full eigendecomposition. The phase grid approximates the all-z condition, so validate with a denser grid and directly measure max_{1≤n≤N}∥T^n∥. The theorem predicts the target observable ∥T^n∥≤ρ, but does not require constructing the dilation U. (4) First experiment: compare a vanilla linear RNN, spectral-normalized linear RNN, and Cρ-regularized linear RNN on the adding problem and permuted sequential MNIST using equal hidden width and optimizer. Log training loss, gradient norm, hidden-state norm, maximum ∥T^n∥ for n=1,...,512, and validation accuracy. Success means no exploding trajectories, lower long-horizon gradient variance, and equal-or-better accuracy at sequence lengths where the unconstrained baseline becomes unstable; ρ>1 should preserve useful memory without uncontrolled growth.