{
 "artifacts": [
  {
   "name": "dw_experiment.py",
   "url": "https://synthcore.org/code/1036/dw_experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1036/report.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1036/results.json"
  }
 ],
 "category": "dynamics",
 "description": "Build a recurrent or state-space network from heterogeneous dynamical modules and characterize each module through sampled frequency-response passivity and Davis–Wielandt shell bounds. Constrain inter-module coupling so that the composed frequency response retains a positive passivity margin, providing a model-based alternative to blindly shrinking all recurrent weights.",
 "download_zip": "https://synthcore.org/download/1036",
 "formulas_latex": [
  "$$\\mathcal{DW}(G)=\\{(z,q)\\in\\mathbb{C}\\times\\mathbb{R}_{+}:z=x^{\\ast}Gx,\\ q=x^{\\ast}G^{\\ast}Gx=\\|Gx\\|_{2}^{2},\\ \\|x\\|_{2}=1\\}.$$",
  "$$p(G)=\\inf_{\\omega\\in[0,\\omega_{\\max}]}\\lambda_{\\min}\\!\\left(\\frac{G(j\\omega)+G(j\\omega)^{\\ast}}{2}\\right),\\qquad g(G)=\\sup_{\\omega\\in[0,\\omega_{\\max}]}\\|G(j\\omega)\\|_{2}.$$",
  "$$p_{\\mathrm{comp}}\\ge p_{\\mathrm{blocks}}-\\|K\\|_{2}g_{\\mathrm{blocks}}^{2}\u003e0.$$",
  "$$G_r(e^{j\\omega})=C_r(e^{j\\omega}I-A_r)^{-1}B_r+D_r,$$"
 ],
 "id": 2682,
 "implementation": "1. Integration point: use a heterogeneous recurrent or state-space model with modules indexed by r. Each module has local dynamics x⁽ʳ⁾_{t+1}=A_r x⁽ʳ⁾_t+B_r u⁽ʳ⁾_t and output y⁽ʳ⁾_t=C_r x⁽ʳ⁾_t+D_r u⁽ʳ⁾_t; couple outputs through u=-Ky+B_u v_t. Apply the constraint during optimizer steps or as a periodic projection of K. 2. Pseudocode: every M updates, freeze parameters and collect representative hidden states; linearize each nonlinear module around those states; evaluate G_r(e^{jω}) on a logarithmic grid using the displayed transfer formula; compute p_r=minω λ_min((G_r+G_r*)/2) and g_r=maxω σ_max(G_r); set p_blocks=min_r p_r and g_blocks=max_r g_r; calculate p_bound=p_blocks-||K||₂g_blocks². If p_bound\u003cε, scale K by α=min(1,(p_blocks-ε)/(||K||₂g_blocks²)) or reject the update. 3. The paper-derived quantities are shell coordinates, passivity margins, gain estimates, and the compositional inequality. The minibatch linearization, finite frequency grid, and frequency cutoff are empirical approximations. 4. First cheap experiment: compare a 2–4 block gated SSM or GRU on sequential MNIST and copy-memory against unconstrained coupling and spectral normalization. Sweep coupling scale α and record p_bound, hidden-state norm, Jacobian amplification, and long-horizon error. The quantitative prediction is a degradation boundary near p_bound=0: positive-margin runs should have bounded rollout amplification, while negative-margin runs should exhibit rapidly increasing hidden-state norms or prediction error. The measured critical coupling should be within 20% of ||K||₂=p_blocks/g_blocks².",
 "math_summary": "For an operator or frequency-response matrix G(jω), its Davis–Wielandt shell is the set of pairs (z,q) generated by unit vectors x, where z=x*Gx is the numerical-range response and q=||Gx||² is squared amplification. Define the local passivity index p(G) as the minimum Hermitian-part eigenvalue over frequencies and the gain index g(G) as the maximum singular value over frequencies. For feedback coupling u=-Ky, use the conservative compositional certificate p_comp ≥ p_blocks - ||K||₂ g_blocks²; enforce p_comp\u003e0. Here G is a module transfer matrix, ω is angular frequency, x is a unit complex vector, K is the inter-module coupling matrix, and ||·||₂ is the spectral norm.",
 "math_tags": [
  "control-theory",
  "dynamical-systems",
  "linear-algebra",
  "spectral-theory",
  "numerical-analysis"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "architecture",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.21984",
  "arxiv_url": "https://arxiv.org/abs/2608.21984",
  "summary_what_math_gives_to_ml": "The paper provides a compositional stability mechanism for heterogeneous dynamical components: characterize each device by a Davis–Wielandt shell together with local passivity and imaginary-axis indices, then quantify how interconnection changes the global shell without requiring detailed internal models. This is transferable to recurrent and state-space neural networks whose heterogeneous blocks interact through feedback or residual coupling. The most promising implementation is a shell-aware stability monitor and coupling constraint that estimates frequency-domain response of each learned module and rejects updates when the composed shell crosses a predicted instability boundary. The key falsifiable signature is that instability onset should track the shell/passivity margin rather than merely the largest individual block eigenvalue.",
  "title": "Impacts of Heterogeneous Grid-Forming Devices on Power System Dynamics Quantified by DW Shells",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy"
 ],
 "title": "Davis–Wielandt Shell Constraint for Heterogeneous SSMs",
 "url": "https://synthcore.org/idea/2682/davis-wielandt-shell-constraint-for-heterogeneous-ssms",
 "verification": {
  "peer_reviewed": false,
  "stage1_mechanism_check": {
   "worked": false,
   "confidence": 9,
   "verdict": "Built a heterogeneous two-module discrete-time SSM toy with frequency-response indices, Davis–Wielandt shell verification, conservative coupling projection, and rollout sweeps. The shell identity error was 2.7e-15 and the certificate scaled exactly with slope -g^2 (relative error 1.8e-16); the predicted certificate crossing was observed at 0.00347. However, actual dynamical instability occurred near coupling 0.240, about 69x larger than the certified boundary, so the claimed degradation boundary within 20% was not confirmed. The mechanism is mathematically valid but substantially conservative in this setup.",
   "metrics": {
    "baseline": "Unconstrained feedback: instability near alpha=0.240; at alpha=0.5, spectral radius/final norm were 1.642/1.25e12; at alpha=1.0 they were 51.12/6.31e12.",
    "idea": "Projected coupling alpha\u003c=0.00343 maintained p_bound\u003e=0.00133, maximum spectral radius 0.80005, and final rollout norm about 1.06e-29 over 300 steps; exact shell and certificate-slope checks passed."
   },
   "how_to_run": "python3 dw_experiment.py",
   "files": [
    "dw_experiment.py",
    "results.json"
   ],
   "limitations": "Only a small linear two-module toy was tested; there was no GRU/SSM training task, sequential MNIST or copy-memory evaluation, optimizer integration, nonlinear linearization, FLOP/speed measurement, or comparison against spectral normalization. The finite frequency grid and scalar/diagonal module construction also do not establish usefulness for general heterogeneous neural SSMs."
  },
  "status": "mechanism_failed",
  "status_label": "Mechanism failed",
  "updated_at": "2026-09-01T18:49:49",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}