{
 "artifacts": [
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1212/report.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1212/results.json"
  },
  {
   "name": "run_experiment.py",
   "url": "https://synthcore.org/code/1212/run_experiment.py"
  }
 ],
 "category": "dynamics",
 "description": "Train a recurrent or neural state-space model on fixed-initial-state subsequences, but select the training horizon and burn-in from an empirically estimated turnpike bound instead of choosing them arbitrarily. If the cumulative discrepancy between fixed-initial-state and free-initial-state optima is bounded, the average discrepancy decreases as 1/N, allowing shorter windows while preserving the long-horizon optimum.",
 "download_zip": "https://synthcore.org/download/1212",
 "formulas_latex": [
  "$$x_{k+1}=f_θ(x_k,u_k),\\quad y_k=h_θ(x_k,u_k),\\quad J_N(θ,x_0)=Σ_{k=0}^{N-1}ℓ_k(y_k(x_0,θ,U))$$",
  "$$J_N^{free}(θ)=inf_{x_0∈X}J_N(θ,x_0)$$",
  "$$Σ_{k=0}^{N-1}d(y_k^{fix,N},y_k^{free,N})≤C\\quad⇒\\quad(1/N)Σ_{k=0}^{N-1}d(y_k^{fix,N},y_k^{free,N})≤C/N$$",
  "$$N_{train}=ceil(C_hat/ε)$$"
 ],
 "id": 3037,
 "implementation": "Integrate this into truncated backpropagation through time for a GRU, RNN, or neural state-space model. For each minibatch sequence, use the current fixed initial state supplied by an encoder, a zero state, or a cached state. Optimize the normal rollout loss over a candidate horizon N. Every K optimizer steps, create an auxiliary free-initial-state problem by treating x₀ as a differentiable variable and taking 5 to 20 gradient steps on the same sequence loss. Roll out both trajectories with identical inputs, compute C_hat_N = Σₖ ||yₖ^fix − yₖ^free||₂, and update an exponential moving estimate of C_hat. Select the shortest horizon satisfying C_hat/N ≤ ε, or discard the first estimated burn-in steps when computing the loss. The paper provides the cumulative-turnpike mechanism; C_hat, the free initial state, and burn-in are estimated empirically. First experiment: train a GRU and neural state-space model on Duffing-oscillator or Lorenz-63 sequences, comparing full BPTT, fixed windows, and the adaptive rule. Sweep N and plot the average fixed/free gap. The predicted signature is a log-log slope near −1 for average gap versus N, with the selected N achieving gap below ε within roughly 20 percent. If C_hat grows proportionally with N, the assumed turnpike regime is absent.",
 "math_summary": "The model is xₖ₊₁ = fθ(xₖ,uₖ) and yₖ = hθ(xₖ,uₖ). For a horizon N, the fixed-initial-state objective is J_N(θ,x₀) = Σₖ₌₀ᴺ⁻¹ ℓₖ(yₖ(x₀,θ,U)), while the free-initial-state objective is J_N^free(θ) = inf over x₀ of J_N(θ,x₀). Let yₖ^fix,N be an optimum with a prescribed initial state and yₖ^free,N be the closest free-initial-state optimum. The cumulative turnpike property is Σₖ₌₀ᴺ⁻¹ d(yₖ^fix,N,yₖ^free,N) ≤ C, where C does not depend on N. Therefore the mean discrepancy is at most C/N. Estimate C and the transient length from paired fixed/free rollouts, then choose N so that the predicted mean gap is below a target ε.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "optimization"
 ],
 "ml_areas": [
  "ssm",
  "rnn",
  "world-model",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2609.02071",
  "arxiv_url": "https://arxiv.org/abs/2609.02071",
  "summary_what_math_gives_to_ml": "The paper offers a transferable turnpike mechanism for simulation-error training of nonlinear state-space models. Under reachability, incremental output stability, convex stage costs, and a suitable optimality condition, trajectories optimized from a fixed initial state approach the best trajectory obtained when the initial state is free. This supports shorter recurrent training windows and provides a quantitative prediction: if cumulative deviation is bounded independently of horizon, the average fixed-versus-free trajectory gap decays as O(1/N). The equivalence with strict dissipativity and value-function coercivity also suggests a trainable storage-function regularizer for neural state-space models.",
  "title": "Turnpike properties in nonlinear system identification",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 7,
  "usefulness": 8
 },
 "solves": [
  "speedup",
  "stability",
  "sample-efficiency"
 ],
 "title": "Turnpike-Calibrated Short-Window Training",
 "url": "https://synthcore.org/idea/3037/turnpike-calibrated-short-window-training",
 "verification": {
  "peer_reviewed": false,
  "stage1_mechanism_check": {
   "worked": false,
   "confidence": 9,
   "verdict": "Built a reproducible scalar state-space verification, fixed/free initial-state calibration, horizon sweep, and training comparison in run_experiment.py. The analytic toy system clearly shows bounded cumulative discrepancy and mean gap scaling near 1/N (log-log slope -0.959), while the learned-model sweep is approximately decreasing (slope -0.767). However, strict calibration with epsilon=.05 selected the full horizon 40 and gave essentially the same MSE as full training (0.00710 vs 0.00701); relaxed calibration selected 27 but substantially worsened MSE to 0.05865, so no practical speedup with preserved performance was observed.",
   "metrics": {
    "baseline": "Full horizon 40: MSE 0.007013; fixed horizon 8: MSE 0.024971; empirical gap sweep slope -0.767.",
    "idea": "Calibrated epsilon=.05: selected horizon 40, MSE 0.007104, mean fixed/free gap 0.1233. Relaxed epsilon=.20: selected horizon 27, MSE 0.05865, mean gap 0.0972."
   },
   "how_to_run": "python3 run_experiment.py",
   "files": [
    "run_experiment.py",
    "results.json",
    "run.log",
    "run2.log"
   ],
   "limitations": "Only a scalar linear state-space model and one fixed random seed were tested; no GRU, Duffing, Lorenz-63, minibatch dataset, wall-clock/FLOP measurement, multi-sequence calibration, or statistical repeats were included. The free-initial-state optimization is a small approximate inner solve rather than an exact global optimum."
  },
  "status": "mechanism_failed",
  "status_label": "Mechanism failed",
  "updated_at": "2026-09-03T11:58:48",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": false,
    "tested": true
   }
  }
 }
}