# Эксперимент: Commutant-Gap Controlled Stochastic Training (#854)

{
  "worked": true,
  "confidence": 9,
  "verdict": "Built a reproducible commutant-gap experiment with exact SO(2) two-replica commutant projection, PSD generator construction, gap scaling sweeps, and a small symmetry-preserving noisy regression controller. The mechanism manifested: fitted decay rates matched kappa*Lambda within 2.1e-15 relative error, half-lives matched within 1.0%, and the Lambda=0 case showed no decay while commutant directions remained unchanged. In toy training, gap control reduced final loss versus fixed noise (0.0078646 vs 0.015729) but did not beat noiseless SGD (1.46e-31), so this validates the mechanism rather than demonstrating a general optimization win.",
  "metrics": {
    "baseline": "Noiseless SGD final 100-step mean loss = 1.46e-31; fixed symmetry-preserving noise = 0.015729",
    "idea": "Gap-controlled noise final 100-step mean loss = 0.0078646; best loss = 1.02e-05; mean final kappa = 0.0125; math sweep rate relative error <= 2.1e-15 and half-life relative error <= 0.99%"
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 commutant_gap_experiment.py",
  "files": [
    "commutant_gap_experiment.py",
    "results.json",
    "run_output.txt"
  ],
  "limitations": "Only a low-dimensional analytic SO(2) toy and equivariant linear regression were tested; no sequential MNIST, nonlinear 32-unit RNN, minibatch empirical replica-generator estimation, CUDA run, or generalization evaluation was performed. The training controller uses a prescribed synthetic gap schedule rather than estimating Delta_k from trajectories."
}