Fourier-Mode Stability Shaping

Implement a cyclic state-space or recurrent layer with \(N\) channels and \(m\) forward shifts. For the phase version, update \(\theta_i^{t+1}=\theta_i^t+h[\omega+\kappa\sum_{\ell=1}^{m}\sin(\theta_{i+\ell}^t-\theta_i^t)]\). For a real-valued SSM, use the corresponding circulant shift matrix and either a sinusoidal interaction or its Jacobian approximation. At initialization and after optimizer steps, compute the analytic factors \(\mu_k(q)\). If the shift weights are learnable, replace each unit coefficient by its learned value \(a_\ell\), giving \(\mu_k=\sum_\ell a_\ell\cos(2\pi q\ell/N)[1-\cos(2\pi k\ell/N)]\). Add a barrier \(\lambda\sum_{k\in K}\operatorname{softplus}(\epsilon-\mu_k)^2\) for modes that must contract, or target an interval \([\mu_{\min},\mu_{\max}]\) to balance stability and memory. The formula is computed exactly from the layer parameters; nonlinear Jacobians and finite-step corrections should be estimated empirically. First test copy-memory and delayed sine prediction with \(N=32\), \(m\in\{1,2,4,8\}\), comparing unconstrained and Fourier-shaped circulant layers. Inject a small perturbation independently in each Fourier mode and fit its log-amplitude slope. The quantitative prediction is slope \(-\kappa\mu_k\) for small timestep and perturbation. The observed stability boundary should be within 20 percent of the analytic zero crossing \(\mu_k=0\). Negative factors must produce exponential growth, while all-positive factors must produce decay except for the neutral global phase mode.