{
 "artifacts": [
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1139/bench_report.json"
  },
  {
   "name": "observer_experiment.py",
   "url": "https://synthcore.org/code/1139/observer_experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1139/report.md"
  },
  {
   "name": "report_bench_2026-09-02T122319.md",
   "url": "https://synthcore.org/code/1139/report_bench_2026-09-02T122319.md"
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   "name": "results.json",
   "url": "https://synthcore.org/code/1139/results.json"
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  {
   "name": "stage2_bench.py",
   "url": "https://synthcore.org/code/1139/stage2_bench.py"
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 ],
 "category": "dynamics",
 "description": "Turn a latent recurrent model into an observer that continuously corrects its hidden state from noisy or partial observations while certifying both estimation-error convergence and disturbance attenuation. The bounded-real operator inequality becomes a trainable regularizer for a neural correction gain, providing a principled alternative to unconstrained teacher forcing or ad hoc residual correction.",
 "download_zip": "https://synthcore.org/download/1139",
 "formulas_latex": [
  "$$\\dot{\\hat z}=f_\\phi(\\hat z,\\theta,t)+K_\\eta(\\theta,t)(y-C_\\phi\\hat z),\\qquad e=z-\\hat z.$$",
  "$$\\dot e=A_c e+B_cw,\\qquad A_c=A_\\phi-K_\\eta C_\\phi,\\qquad B_c=B_w-K_\\eta D_w.$$",
  "$$\\begin{bmatrix}\\dot P+A_c^\\top P+PA_c+C_e^\\top C_e+2\\alpha P \u0026 PB_c+C_e^\\top D_e\\\\ B_c^\\top P+D_e^\\top C_e \u0026 D_e^\\top D_e-\\gamma^2I\\end{bmatrix}\\preceq0,\\qquad P\\succeq p_{\\min}I.$$",
  "$$\\dot V+2\\alpha V+\\|q\\|^2-\\gamma^2\\|w\\|^2\\le0,\\qquad V(e,t)=e^\\top P(\\theta,t)e.$$"
 ],
 "id": 2939,
 "implementation": "1. Integration point: attach an observer head to a neural state-space model, world model, or recurrent encoder. The predictor evolves $\\hat z$ autonomously, while each observation produces innovation $r_k=y_k-C_\\phi\\hat z_k$ and correction $K_\\eta r_k$. Use diagonal or low-rank $P$ for large latent dimensions. 2. Pseudocode: predict $\\hat z_{k+1}=\\hat z_k+h f_\\phi(\\hat z_k,\\theta_k)$; calculate $r_k$ and correct with $hK_\\eta r_k$; estimate $A_c$ using Jacobian-vector products; construct $B_c,C_e,D_e$ from the assumed noise and output maps; evaluate the largest eigenvalue of the symmetric bounded-real LMI matrix and add its positive part to the loss. Parameterize $P=LL^\\top+p_{\\min}I$ and optimize $K_\\eta$ jointly with the predictor. 3. Computed from the mechanism: the storage inequality and target gain $\\gamma$; estimated empirically: local Jacobians, observation-noise statistics, and minibatch averages over parameter/time points. 4. First experiment: use a partially observed damped oscillator and Lorenz-63, followed by a small video latent model. Compare an equal-parameter GRU with teacher forcing, a learned Kalman-style correction, and this bounded-real observer; ablate the LMI while retaining the correction gain. Evaluate open-loop rollouts under observation noise and impulsive disturbances. 5. Pre-registered signature: with zero disturbance, $V(e_k)$ should decay at least as $e^{-2\\alpha kh}V(e_0)$ after a discretization transient; with disturbances, cumulative output energy should satisfy $\\sum_k\\|q_k\\|^2\\le\\gamma^2\\sum_k\\|w_k\\|^2+V_0$. Reject the transfer if measured induced gain exceeds $1.2\\gamma$, or if the estimated LMI is feasible but held-out error decay fails.",
 "math_summary": "Let the latent plant be $\\dot z=A_\\phi(\\theta,t)z+B_w w$ with observations $y=C_\\phi z+D_w w$. Use the observer $\\dot{\\hat z}=f_\\phi(\\hat z,\\theta,t)+K_\\eta(\\theta,t)(y-C_\\phi\\hat z)$ and error $e=z-\\hat z$. Its local error dynamics are $\\dot e=A_c e+B_cw$, where $A_c=A_\\phi-K_\\eta C_\\phi$ and $B_c=B_w-K_\\eta D_w$. For performance output $q=C_e e+D_e w$, let $P(\\theta,t)=P^\\top\\succ0$ be a storage metric. The bounded-real differential LMI is $\\begin{bmatrix}\\dot P+A_c^\\top P+PA_c+C_e^\\top C_e+2\\alpha P \u0026 PB_c+C_e^\\top D_e\\\\ B_c^\\top P+D_e^\\top C_e \u0026 D_e^\\top D_e-\\gamma^2I\\end{bmatrix}\\preceq0$. It implies $\\dot V+2\\alpha V+\\|q\\|^2-\\gamma^2\\|w\\|^2\\le0$ for $V=e^\\top Pe$, giving decay rate $\\alpha$ and induced disturbance gain at most $\\gamma$.",
 "math_tags": [
  "control-theory",
  "dynamical-systems",
  "linear-algebra",
  "optimization",
  "statistics"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "world-model",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.28349",
  "arxiv_url": "https://arxiv.org/abs/2608.28349",
  "summary_what_math_gives_to_ml": "The paper provides a transferable robust-stability mechanism: parameter-dependent Lyapunov operators satisfying differential operator LMIs certify uniform exponential stability for non-autonomous evolution systems, while Galerkin projections turn infinite-dimensional inequalities into finite-dimensional synthesis constraints. This can be transferred to parameter-conditioned neural ODEs, state-space models, or recurrent networks by treating hidden-state Jacobians as time- and parameter-dependent generators and learning a positive-definite metric that contracts them. Its H-infinity extension also supplies a concrete disturbance-attenuation certificate for neural observers operating with noisy or partial observations.",
  "title": "Operator-Theoretic Stability and Observer Synthesis for Parameter-Dependent Vlasov--Maxwell Dynamics",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 7,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy"
 ],
 "title": "H-Infinity Disturbance-Attenuating Latent Observer",
 "url": "https://synthcore.org/idea/2939/h-infinity-disturbance-attenuating-latent-observer",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0.0005194859513721894,
    "idea_mean": 0.0005194859513721894,
    "delta_mean": 0,
    "per_seed_diffs": [
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     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    "idea_wins": 0,
    "n_pairs": 8,
    "p_value": 1,
    "mde": 0,
    "mde_rel_pct": 0,
    "verdict": "no measurable effect",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "lr": 0.006
    },
    "sweep": [
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      "cfg": {
       "lr": 0.001
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      "mean": 0.0015310924500226974
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     {
      "cfg": {
       "lr": 0.003
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      "mean": 0.0008559521666029468
     },
     {
      "cfg": {
       "lr": 0.006
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      "mean": 0.0005687399534508586
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    ],
    "full": {
     "mean": 0.0005194859513721894,
     "std": 0.0001693891077925739,
     "per_seed": [
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      0.0002958264376502484,
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     ],
     "n": 8
    }
   },
   "idea": {
    "mean": 0.0005194859513721894,
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    "per_seed": [
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   },
   "mechanism_signature": {
    "baseline_observed_mean_local_gain": 0.431841354817152,
    "baseline_observed_p95_local_gain": 0.4626914465799927,
    "baseline_task_mse_at_same_lr": 0.0005194859513721894,
    "claim": "bounded-real attenuation reduces local input-disturbance gain",
    "confirmed": false,
    "gamma": 0.5,
    "idea_lr": 0.006,
    "idea_observed_mean_local_gain": 0.431841354817152,
    "idea_observed_p95_local_gain": 0.4626914465799927,
    "idea_task_mse_at_same_lr": 0.0005194859513721894
   },
   "custom_track": null
  },
  "mechanism_confirmed": false,
  "peer_reviewed": false,
  "practical_verdict": "no_effect",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 8,
   "verdict": "Built a scalar H-infinity latent observer MVP with numerical storage-metric search, bounded-real LMI evaluation, exact disturbance-transfer verification, and time-domain dissipation checks. The feasible observer gain K=1.7804 had LMI maximum eigenvalue -0.00098, exact H-infinity gain 0.8166 below gamma=0.9, and zero-disturbance V decay rate 4.36 versus the required 0.4. The unconstrained high-gain baseline K=10 converged faster but was not LMI-feasible and exceeded the target gain with exact H-infinity gain 0.9615, demonstrating the claimed stability/attenuation tradeoff on this toy system.",
   "metrics": {
    "baseline": "K=10.0; LMI max eigenvalue=0.07708 (infeasible); exact H-infinity gain=0.96154 \u003e gamma=0.9; observed V decay rate=20.81",
    "idea": "K=1.78043, P=0.50642; LMI max eigenvalue=-0.000981 (feasible); exact H-infinity gain=0.81655 \u003c gamma=0.9; observed V decay rate=4.362 \u003e 2*alpha=0.4; random-input energy ratio=0.01979"
   },
   "how_to_run": "python3 observer_experiment.py",
   "files": [
    "observer_experiment.py",
    "results.json"
   ],
   "limitations": "Only a scalar linear continuous-time error system was tested; no neural predictor, learned gain, GRU comparison, Lorenz-63 system, partial-observation training, parameter/time-varying P, or large-scale computational overhead was evaluated. The gain and storage metric were selected by grid search rather than jointly learned."
  },
  "status": "failed_benchmark",
  "status_label": "Failed on benchmark",
  "updated_at": "2026-09-02T12:30:10",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": false,
    "tested": true
   },
   "practical_benchmark": {
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    "tested": true,
    "verdict": "no_effect"
   },
   "toy_mechanism_gate": {
    "confirmed": true,
    "tested": true
   }
  }
 }
}