{
 "artifacts": null,
 "category": "architecture",
 "description": "Use a cyclic forward-neighbor recurrent or state-space layer and regularize its coupling so selected discrete Fourier modes are contracting while task-critical modes remain weakly damped. The paper's exact mode factors make instability falsifiable: a mode becomes unstable when its scalar factor changes sign, producing a measurable transition rather than a vague smoothness prior.",
 "formulas_latex": [
  "$$\\dot\\theta_i=\\omega+\\kappa\\sum_{\\ell=1}^{m}\\sin(\\theta_{i+\\ell}-\\theta_i),\\qquad \\phi_i=\\frac{2\\pi q i}{N}$$",
  "$$u_i^{(k)}=\\exp\\left(\\frac{2\\pi\\mathrm{i}ki}{N}\\right),\\qquad \\mu_k(q)=\\sum_{\\ell=1}^{m}\\cos\\left(\\frac{2\\pi q\\ell}{N}\\right)\\left[1-\\cos\\left(\\frac{2\\pi k\\ell}{N}\\right)\\right]$$",
  "$$\\delta_k(t)\\approx\\delta_k(0)\\exp[-\\kappa\\mu_k(q)t]\\exp(\\mathrm{i}\\nu_k t)$$",
  "$$\\mu_{k_*}(q)\u003c0\\ \\Longrightarrow\\ |\\delta_{k_*}(t)|\\propto\\exp[\\kappa|\\mu_{k_*}(q)|t]$$"
 ],
 "id": 2842,
 "implementation": "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.",
 "math_summary": "Consider \\(N\\) cyclic units with forward-neighbor dynamics \\(\\dot\\theta_i=\\omega+\\kappa\\sum_{\\ell=1}^{m}\\sin(\\theta_{i+\\ell}-\\theta_i)\\), with indices modulo \\(N\\). A \\(q\\)-twisted profile is \\(\\phi_i=2\\pi qi/N\\). Linearization around this profile is diagonalized by Fourier vectors \\(u_i^{(k)}=\\exp(2\\pi\\mathrm{i}ki/N)\\). Define \\(\\mu_k(q)=\\sum_{\\ell=1}^{m}\\cos(2\\pi q\\ell/N)[1-\\cos(2\\pi k\\ell/N)]\\). The global phase mode has \\(\\mu_0=0\\). Apart from an imaginary advection term, mode \\(k\\) decays at rate \\(-\\kappa\\mu_k(q)\\). Stability modulo global phase requires \\(\\mu_k(q)\u003e0\\) for all \\(k=1,\\ldots,N-1\\); any negative factor gives an exponentially growing perturbation.",
 "math_tags": [
  "dynamical-systems",
  "spectral-theory",
  "linear-algebra",
  "bifurcations"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "memory",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.29630",
  "arxiv_url": "https://arxiv.org/abs/2608.29630",
  "summary_what_math_gives_to_ml": "The paper provides a constructive theory of nonzero-frequency relative equilibria in directed Kuramoto networks, including existence conditions, orbital stability modulo the global phase direction, invariant neighborhoods, and exact discrete-Fourier stability tests for forward-neighbor graphs. The most transferable mechanism is a phase-based recurrent or state-space layer whose desired rotating solution is designed through an equitable partition or twisted profile, while common phase drift is treated as a neutral gauge direction rather than an instability. A second mechanism is mode-selective spectral shaping: explicit Fourier-mode factors predict whether a cyclic recurrent layer contracts or amplifies particular modes. These ideas are most credible for oscillatory RNNs, cyclic state-space models, and latent dynamical systems.",
  "title": "Existence and Stability of Dancing Equilibria in Asymmetric Kuramoto Networks",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 8,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "memory",
  "accuracy"
 ],
 "title": "Fourier-Mode Stability Shaping",
 "url": "https://synthcore.org/idea/2842/fourier-mode-stability-shaping",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}