{
 "artifacts": [
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1050/bench_report.json"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1050/report.md"
  },
  {
   "name": "report_bench_2026-09-01T195600.md",
   "url": "https://synthcore.org/code/1050/report_bench_2026-09-01T195600.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1050/results.json"
  },
  {
   "name": "stage2_bench.py",
   "url": "https://synthcore.org/code/1050/stage2_bench.py"
  },
  {
   "name": "trusted_reachable_rnn.py",
   "url": "https://synthcore.org/code/1050/trusted_reachable_rnn.py"
  }
 ],
 "category": "dynamics",
 "description": "Apply the paper's compositional PAS idea to recurrent or state-space networks by propagating a polytope of possible hidden states and input perturbations over multiple time blocks. Instead of validating one hidden trajectory at a time, maintain a trusted convex family and re-linearize only when its nonlinear-fidelity tolerance is exceeded. This creates a runtime monitor and adaptive horizon mechanism for long-sequence inference, forecasting, and learned world models.",
 "download_zip": "https://synthcore.org/download/1050",
 "formulas_latex": [
  "$$h_{k+1}=f_\\phi(h_k,u_k),\\qquad h_k(\\gamma)=h_k^*+R_k\\gamma,\\qquad u_k(\\gamma)=u_k^*+U_k\\gamma.$$",
  "$$R_{k+1}=A_kR_k+C_kU_k,\\qquad A_k=\\left.\\frac{\\partial f_\\phi}{\\partial h}\\right|_{(h_k^*,u_k^*)},\\quad C_k=\\left.\\frac{\\partial f_\\phi}{\\partial u}\\right|_{(h_k^*,u_k^*)}.$$",
  "$$\\mathcal{H}_k=\\{h_k^*+R_k\\gamma\\mid \\gamma\\in\\mathcal{A}\\},\\qquad \\mathcal{A}=\\{\\gamma:G\\gamma\\le h\\}.$$",
  "$$v(\\gamma)=\\max_{0\\le k\u003cH}\\left\\|f_\\phi(h_k^*+R_k\\gamma,u_k^*+U_k\\gamma)-h_{k+1}^*-R_{k+1}\\gamma\\right\\|_2,$$"
 ],
 "id": 2718,
 "implementation": "Use this as an inference-time monitor or training regularizer for an RNN, GRU, neural ODE discretization, or diagonal state-space model. Select a nominal input sequence and compute a block rollout \\((h_k^*,u_k^*)\\) of length \\(H\\). Let \\(\\gamma\\) contain perturbations to the initial hidden state and a low-rank set of input directions, initialized as a box \\(|\\gamma_i|\\le r_i\\). Compute \\(A_k\\) and \\(C_k\\) with automatic differentiation or Jacobian-vector products, then recursively form \\(R_k\\). Evaluate the true recurrence at polytope vertices, random interior points, and adversarial points found by maximizing observed violation. Shrink the polytope until probes satisfy \\(v(\\gamma)\\le\\varepsilon\\); use the accepted terminal polytope as the next block's initial uncertainty set, or re-center and re-linearize when its volume becomes too small. During training, add \\(\\lambda\\max(0,v_{\\max}-\\varepsilon)^2\\) to the task loss. At inference, trigger re-linearization, shorter blocks, or conservative input selection when trusted volume collapses. First test a small GRU on permuted-MNIST or a nonlinear state-space forecasting dataset against ordinary truncated BPTT and an unmonitored GRU at equal compute. The predicted signature is a critical horizon \\(H_c(r,\\varepsilon)\\): below it, affine multi-step error stays under \\(\\varepsilon\\); above it, error increases rapidly. Re-centering should extend the usable horizon and produce repeatable boundaries across sequences.",
 "math_summary": "For a recurrent map \\(h_{k+1}=f_\\phi(h_k,u_k)\\), let \\((h_k^*,u_k^*)\\) be a nominal rollout and let \\(\\gamma\\) parameterize perturbations to initial state and inputs. The affine family is \\(h_k(\\gamma)=h_k^*+R_k\\gamma\\), \\(u_k(\\gamma)=u_k^*+U_k\\gamma\\), with \\(R_{k+1}=A_kR_k+C_kU_k\\), where \\(A_k=\\partial f_\\phi/\\partial h\\) and \\(C_k=\\partial f_\\phi/\\partial u\\), evaluated on the nominal rollout. For \\(\\mathcal{A}=\\{\\gamma:G\\gamma\\le h\\}\\), the predicted reachable hidden set is \\(\\mathcal{H}_k=\\{h_k^*+R_k\\gamma:\\gamma\\in\\mathcal{A}\\}\\). Define one-step nonlinear violation by \\(v_k(\\gamma)=\\|f_\\phi(h_k^*+R_k\\gamma,u_k^*+U_k\\gamma)-h_{k+1}^*-R_{k+1}\\gamma\\|_2\\), and retain points satisfying \\(\\max_{k\u003cH}v_k(\\gamma)\\le\\varepsilon\\). The terminal trusted polytope is composed with the next block and the map is re-linearized when its volume or fidelity becomes insufficient.",
 "math_tags": [
  "control-theory",
  "dynamical-systems",
  "geometry",
  "linear-algebra",
  "numerical-analysis"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "world-model",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.24019",
  "arxiv_url": "https://arxiv.org/abs/2608.24019",
  "summary_what_math_gives_to_ml": "The paper provides a constructive mechanism for converting locally linearized trajectories of nonlinear underactuated systems into reusable convex action polytopes, while filtering those polytopes through an explicit nonlinear dynamics-violation metric. Its transferable asset is not merely linearization, but extraction of a trusted convex inner approximation whose points remain within a prescribed local-fidelity tolerance, followed by composition of reachable families rather than individual samples. In neural networks, the same mechanism can define trusted polytopes of parameter updates or hidden-state interventions around a nominal computation, enabling multiple safe candidate updates to be optimized jointly. The key falsifiable signature is a sharp boundary in update radius or rollout horizon where the nonlinear violation tolerance is first exceeded.",
  "title": "Trusted Polytopic Action Sets for Fast Planning in Underactuated Systems",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 7,
  "novelty": 8,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy",
  "scalability"
 ],
 "title": "Composed Trusted Reachable Families for Recurrent Networks",
 "url": "https://synthcore.org/idea/2718/composed-trusted-reachable-families-for-recurrent-networks",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
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    "idea_wins": 4,
    "n_pairs": 8,
    "p_value": 0.2468,
    "mde": 0.0006552640413442428,
    "mde_rel_pct": 31.116820060824228,
    "verdict": "no significant win",
    "system_worked": false
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    "method": "multi_step_affine_reachability_penalty",
    "signature": {
     "confirmed": true,
     "observed_log_slope": 1.9886342514929614,
     "observed_violation": [
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      0.00013634731294587255
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     "predicted_slope": 2,
     "radii": [
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   "custom_track": null
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  "mechanism_confirmed": false,
  "peer_reviewed": false,
  "practical_verdict": "inconclusive",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 9,
   "verdict": "Built a scalar nonlinear recurrent trusted-reachable-family monitor with affine Jacobian propagation, box probing, tolerance-based horizon checks, and adaptive radius selection. The mechanism manifested exactly: violation scaled as radius^2 (observed slope 2.000 vs predicted 2), measured critical horizons matched the analytical boundary for all tested radii (8/5/2), and trusted radius decayed with observed log slope -0.22314 per step, matching -log(1.25). The monitor reduced H=10 violation from 0.02665 at fixed radius 0.04 to 0.00100 at radius 0.00775; this is a safety/accuracy effect, not evidence of faster task learning.",
   "metrics": {
    "baseline": "Fixed affine family radius 0.04 at lambda=1.25, q=0.3, H=10: max one-step violation 0.0266454, exceeding epsilon=0.001.",
    "idea": "Adaptive trusted radius sqrt(epsilon/(q*lambda^(2(H-1))))=0.007749 at H=10: measured max violation 0.0010000. Radius-scaling slope 2.000; horizon measured/predicted for radii 0.01,0.02,0.04 = 8/8,5/5,2/2; decay slope -0.223144 vs predicted -log(1.25)=-0.223144."
   },
   "how_to_run": "python3 trusted_reachable_rnn.py",
   "files": [
    "trusted_reachable_rnn.py",
    "results.json"
   ],
   "limitations": "Only a scalar quadratic recurrence with zero nominal input and initial-state uncertainty was tested; input-direction uncertainty, multidimensional polytopes, GRUs, composed block re-centering, adversarial probes, training-time regularization, wall-clock overhead, and downstream forecasting/classification accuracy were not tested."
  },
  "status": "failed_benchmark",
  "status_label": "Failed on benchmark",
  "updated_at": "2026-09-01T19:56:00",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}