Product-Matched Spectral Trust Region

1. Exact integration point: replace the recurrent transition matrix in a small RNN or linear state-space model by a continuous-time cyclic generator Q. For hidden state x in R^N, define Q_{i+1,i}=f_i, Q_{i-1,i}=b_i, and Q_{i,i}=-(f_i+b_i), with periodic indices; use x_{t+1}=expm(hQ)x_t+Bu_t. Parameterize f_i=softplus(theta_i)+epsilon and b_i=softplus(phi_i)+epsilon so rates remain positive. 2. Pseudocode: compute logf=log(f), logb=log(b); fbar=exp(mean(logf)); bbar=exp(mean(logb)); for m=1,...,floor(N/2), set q=2*pi*m/N and omega_bound[m]=abs((fbar-bbar)*sin(q)); add L_spec=alpha*mean(relu(abs(imag(eigvals(Q)))-omega_bound[mode(eigenvalue)])^2), or estimate the largest observed oscillation with Arnoldi and compare it with the relevant bound. Add a damping margin requiring Re(lambda)<=-delta for modes used by the layer. 3. Computed from the paper: affinity, geometric means, comparison eigenvalues, and the equality prediction that uniform rates saturate the envelope. Estimated empirically: the winding-sector assignment of a finite learned generator's eigenvalues and the actual eigenspectrum, using dense eigendecomposition for N<=128 or Arnoldi for larger N. Since the theorem is for a Markov cycle, do not claim validity for arbitrary dense recurrent matrices; first test the exact cyclic parameterization. 4. First cheap experiment: train a 32-unit cyclic continuous-time RNN on permuted-MNIST or a synthetic delayed sinusoid-copy task, comparing unconstrained positive rates, a standard spectral-radius penalty, and this product-matched penalty. Log maximum imaginary eigenvalue by winding sector, the geometric-rate nonuniformity coefficient, long-horizon hidden-state norm, and task loss. The falsifiable signature is that measured mode frequencies do not exceed the comparison envelope, that the envelope gap grows with rate nonuniformity, and that a uniform-rate control approaches equality. For step h, discrete-time instability should occur when h times the largest positive real part crosses zero; the constrained model should maintain |z_m|<1 whenever Re(lambda_m)<0.