{
 "artifacts": [
  {
   "name": "bench_experiment.py",
   "url": "https://synthcore.org/code/1078/bench_experiment.py"
  },
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1078/bench_report.json"
  },
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/1078/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1078/report.md"
  },
  {
   "name": "report_bench_2026-09-02T001424.md",
   "url": "https://synthcore.org/code/1078/report_bench_2026-09-02T001424.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1078/results.json"
  }
 ],
 "category": "regularization",
 "description": "Add a curvature-margin regularizer to a neural latent-state estimator or world model so that every initial-state direction is sufficiently constrained by the observation history and prior. The regularizer targets the smallest posterior-curvature eigenvalue, not total information, making the estimator resistant to systematic transition-model mismatch in poorly observed latent directions.",
 "download_zip": "https://synthcore.org/download/1078",
 "formulas_latex": [
  "$$H(z)=\\Sigma_0^{-1}+\\sum_{k=0}^{N-1}J_k(z)^\\top J_k(z),\\qquad J_k(z)=\\frac{\\partial r_k(z)}{\\partial z}$$",
  "$$\\Delta z\\approx-H^{-1}b,\\qquad \\|\\Delta z\\|_2\\leq\\frac{\\|b\\|_2}{\\lambda_{\\min}(H)}=\\frac{\\|b\\|_2}{m}$$",
  "$$\\mathcal{L}_{\\mathrm{total}}=\\mathcal{L}_{\\mathrm{task}}+\\beta\\,\\phi_\\tau(H),\\qquad \\phi_\\tau(H)=-\\tau\\log\\operatorname{tr}\\left[\\exp\\left(-H/\\tau\\right)\\right]$$",
  "$$\\phi_\\tau(H)\\xrightarrow[\\tau\\to0]{}\\lambda_{\\min}(H),\\qquad \\operatorname{tr}(H)\\text{ may increase while }\\lambda_{\\min}(H)\\text{ remains small}$$"
 ],
 "id": 2724,
 "implementation": "1. Exact integration point: use a recurrent state-space model or transformer world model with latent initial state z, transition map x_(k+1) = f_theta(x_k, u_k), and observation decoder y_k = g_theta(x_k). For each training window of N observations, infer z with the model's MAP encoder or an iterative refinement step. Define r_k(z) = R^(-1/2)(y_k - y_hat_k(z)), where R is a fixed or learned observation-noise covariance. 2. Pseudocode: unroll the model from z; compute r_k; obtain J_k with Jacobian-vector products; construct H = Sigma0^(-1) + sum_k J_k.T @ J_k; estimate its smallest eigenvalue using Lanczos; minimize task_loss + beta * phi_tau(H), where phi_tau is the soft minimum above. Initially stop gradients through H to avoid expensive second-order autodiff, then enable them for joint training. 3. Computed from the paper's mechanism: the posterior-curvature decomposition, the inverse-smallest-eigenvalue sensitivity bound, and the distinction between weakest-direction and aggregate information. Estimated empirically: J_k, R, the mismatch vector b, and actual oracle displacement. Lanczos or stochastic low-rank approximations avoid materializing H. 4. First cheap experiment: use a four-dimensional linear dynamical system with one weakly observed latent coordinate, train a GRU or neural state-space model on length-20 partially observed sequences, and compare baseline training, trace(H) regularization, and minimum-eigenvalue regularization. Perturb the transition matrix at test time by controlled Frobenius-norm amounts and recompute MAP initial states. Sweep sensor strength so m spans one decade. The predicted signature is ||Delta z|| proportional to 1/m on a log-log plot with slope approximately -1; minimum-eigenvalue regularization should reduce worst-coordinate error even when trace(H) is matched.",
 "math_summary": "For a horizon of N observations, let z be the inferred latent initial state, r_k(z) be the whitened prediction residual at time k, J_k = d r_k(z) / d z be its Jacobian, and Sigma_0 be the latent prior covariance. The nominal Gauss–Newton posterior curvature is H = Sigma_0^(-1) + sum from k=0 to N-1 of J_k^T J_k. Define the information-curvature margin m = lambda_min(H). If model mismatch produces a first-order gradient perturbation b at the nominal MAP point, then the MAP displacement satisfies approximately Delta z = -H^(-1)b and therefore ||Delta z||_2 \u003c= ||b||_2 / m. Train the neural model to increase m, or a smooth soft-min approximation to it. The mechanism predicts that equal-size mismatch perturbations produce displacement proportional to 1/m, while increasing trace(H) without increasing lambda_min(H) does not improve worst-direction robustness.",
 "math_tags": [
  "control-theory",
  "dynamical-systems",
  "linear-algebra",
  "spectral-theory",
  "statistics",
  "optimization"
 ],
 "ml_areas": [
  "ssm",
  "rnn",
  "world-model",
  "regularization",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.24550",
  "arxiv_url": "https://arxiv.org/abs/2608.24550",
  "summary_what_math_gives_to_ml": "The paper provides a transferable sensitivity mechanism for finite-horizon MAP state estimation under model mismatch: the nominal-to-oracle estimate displacement is controlled by a mismatch injection divided by the weakest posterior-curvature direction. Its key asset is an information-curvature margin, approximately the smallest eigenvalue of the Gauss–Newton posterior Hessian, rather than aggregate Fisher information or trace information. In neural networks, this can become a differentiable robustness objective for latent-state estimators, world models, and partially observed recurrent or state-space architectures. The quantitative prediction is that estimation error under a fixed model perturbation scales approximately as the inverse of this smallest eigenvalue, and that maximizing trace information can fail when one latent direction remains weakly observed.",
  "title": "Partial Observation Amplifies Model Mismatch in MAP Estimation via Information-Curvature Margins",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "generalization",
  "accuracy"
 ],
 "title": "Weakest-Direction Information Margin for Latent-State Training",
 "url": "https://synthcore.org/idea/2724/weakest-direction-information-margin-for-latent-state-traini",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0.0122297231,
    "idea_mean": 0.0122296974,
    "delta_mean": -2.5698682e-8,
    "per_seed_diffs": [
     2.5425106e-7,
     2.8684735e-7,
     -7.8231096e-8,
     -0.0000011939555,
     1.4458783e-7,
     4.5355409e-7,
     5.6996942e-7,
     -6.4261258e-7
    ],
    "idea_wins": 3,
    "n_pairs": 8,
    "p_value": 0.91285,
    "mde": 5.0345351e-7,
    "mde_rel_pct": 0.00411664,
    "verdict": "no measurable effect",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "lr": 0.006
    },
    "sweep": [
     {
      "cfg": {
       "lr": 0.001
      },
      "mean": 0.1492516194
     },
     {
      "cfg": {
       "lr": 0.003
      },
      "mean": 0.03069703
     },
     {
      "cfg": {
       "lr": 0.006
      },
      "mean": 0.0124126468
     }
    ],
    "full": {
     "mean": 0.0122297231,
     "std": 0.0049134211,
     "per_seed": [
      0.0141053172,
      0.0131594678,
      0.0073661068,
      0.0150196953,
      0.0033576291,
      0.0126793338,
      0.0210158844,
      0.0111343507
     ],
     "n": 8
    }
   },
   "idea": {
    "mean": 0.0122296974,
    "std": 0.0049134829,
    "per_seed": [
     0.0141055714,
     0.0131597547,
     0.0073660286,
     0.0150185013,
     0.0033577736,
     0.0126797874,
     0.0210164543,
     0.0111337081
    ],
    "n": 8
   },
   "mechanism_signature": {
    "baseline_min_curvature": 0.0499999635,
    "baseline_observed_perturbation": 0.00007737995,
    "baseline_trace_curvature": 1.3966932297,
    "confirmed": true,
    "idea_min_curvature": 0.0499999709,
    "idea_observed_perturbation": 0.00007729608,
    "idea_trace_curvature": 1.3966948986,
    "observed_perturbation_ratio": 0.998916,
    "predicted_inverse_margin_ratio": 0.9999998
   },
   "custom_track": null
  },
  "mechanism_confirmed": true,
  "peer_reviewed": false,
  "practical_verdict": "no_effect",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 9,
   "verdict": "Built a self-contained linear latent-state MVP implementing posterior curvature, weakest-eigenvalue margin, soft-min approximation, and mismatch-induced MAP displacement. The mechanism manifested: log-log displacement versus margin had slope -1.00, normalized sensitivity was exactly 1, and equal-trace curvature produced a 50x weakest-direction shift difference. Under equal observation-energy allocation, margin optimization increased the minimum curvature from 0.05 to 1.568 and reduced the tested weak-direction shift from 2.00 to 0.0395, while trace stayed 4.1.",
   "metrics": {
    "baseline": "Trace-oriented allocation: trace(H)=4.1, lambda_min(H)=0.05, tested MAP shift=2.000.",
    "idea": "Smooth-min/margin allocation: trace(H)=4.1, lambda_min(H)=1.5678, tested MAP shift=0.03949; math sweep slope=-1.000 and inverse-margin normalized shift in [1.0,1.0]."
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "README.txt",
    "results.json"
   ],
   "limitations": "This is a deterministic two-dimensional linear mechanism verification, not a GRU or neural world-model training study. The baseline is a trace-oriented sensor allocation under a fixed energy budget rather than a trained neural estimator; stochastic observations, explicit transition-matrix mismatch sweeps, learned encoders, CUDA execution, and runtime/FLOP comparisons were not tested."
  },
  "status": "mech_ok_no_baseline",
  "status_label": "Mechanism confirmed, baseline not beaten",
  "updated_at": "2026-09-02T00:14:24",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}