# Эксперимент: Polar-Backstepping Policy Residual (#565)

{
  "worked": true,
  "confidence": 8,
  "verdict": "Built a fixed-seed polar backstepping residual-policy MVP with bounded MLP residuals, Lyapunov drift penalty, exact polar dynamics, and three parameter sweeps. All three mechanism predictions were confirmed: speed-error decay matched k_v with identity error below 2.3e-16, steering decay matched k_gamma*v/(|v|+epsilon) with relative error below 6.2e-15, and the epsilon transition matched the predicted 1/r-1/epsilon residual below epsilon while vanishing above it. In the toy policy comparison, the Lyapunov penalty reduced positive-drift states from 13.09% to 10.89% and improved mean drift-plus-lambda-V from -0.135 to -0.206, but increased task MSE from 0.00071 to 0.0611, so the stability effect is real but comes with a clear accuracy tradeoff.",
  "metrics": {
    "baseline": "Unconstrained polar residual: task MSE 0.0007122, positive drift fraction 0.13086, mean (dV+lambda V) -0.13484",
    "idea": "Lyapunov-regularized polar residual: task MSE 0.06113, positive drift fraction 0.10889, mean (dV+lambda V) -0.20551; 16.8% relative reduction in positive-drift events"
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 polar_backstepping_experiment.py",
  "files": [
    "polar_backstepping_experiment.py",
    "results.json",
    "run_output.txt"
  ],
  "limitations": "This is a sampled one-step toy study rather than full bicycle trajectory navigation; no success-rate or long-horizon parking evaluation was performed, and no Cartesian-policy baseline was implemented. The certificate is evaluated on sampled states and uses clipped actions, so it is not a formal global stability guarantee."
}