# Эксперимент: Invariant-Sphere Recurrent State (#1138)

{
  "worked": true,
  "confidence": 7,
  "verdict": "Built a ring-coupled cubic recurrent state and verified its stability numerically. The field had zero positive radial directions across 20,000 probes, drove initial norms spanning 0.66–17.92 to approximately 2.80–2.83, and the exact scalar case converged to radius 1 with measured relaxation slope -2.31 versus the predicted -2.0. In a small delayed-bit task, the sphere RNN achieved test MSE 0.00013 versus 0.152 for the tanh baseline, a promising but preliminary signal rather than a definitive general result.",
  "metrics": {
    "baseline": "tanh RNN: test MSE 0.151553; final hidden norm mean/std 1.16569/0.00737",
    "idea": "Invariant-sphere cubic RNN: test MSE 0.000130; final hidden norm mean/std 0.59144/0.00040; radial positive fraction 0/20000; multidimensional final-radius range 2.7975–2.8275; scalar relaxation slope -2.310"
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 run_experiment.py",
  "files": [
    "run_experiment.py",
    "results.json"
  ],
  "limitations": "Only one seed and one short delayed-bit task were tested; no copy/addition benchmark at lengths 500–1000, equal-FLOP or equal-optimization studies, orthogonal-RNN baseline, radial-penalty ablation, dense-coupling ablation, Jacobian spectral measurements, or systematic c and step-size sweeps were performed. The multidimensional field has a direction-dependent radial coefficient rather than the ideal constant-kappa form."
}