# Эксперимент: Projective-Gap Regularization for Random Jacobian Cocycles (#968)

{
  "worked": true,
  "confidence": 8,
  "verdict": "Built a reproducible projective-gap cocycle verifier and a small tanh-RNN integration experiment. The mechanism manifested quantitatively: fitted projective-decay slopes matched -gamma within 0.5%-6.9%, and threshold-hit times matched log(theta0/epsilon)/gamma within 2.7%-3.2%; at zero gap, projective slopes stayed near zero. The RNN regularizer produced a modest lower final task loss (1182.1 vs 1225.6), but its measured lambda1 (-0.805) and gap (1.674) overshot the requested targets, so this is evidence for the mechanism rather than a tuned training win.",
  "metrics": {
    "baseline": "RNN final loss 1225.576 over the final 20 iterations",
    "idea": "RNN final loss 1182.060; estimated lambda1=-0.805, gap=1.674; toy gap-sweep slope relative error 0.5%-6.9%; threshold-time ratio 0.973-1.032"
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 projective_gap_experiment.py",
  "files": [
    "projective_gap_experiment.py",
    "results.json"
  ],
  "limitations": "The cocycle verifier uses a deliberately controlled 2D diagonal cocycle, not a learned nonlinear network. The RNN comparison is one seed, one sequence length, and a short run; baseline and regularized runs use separate random streams, and the regularizer uses explicit small-matrix SVDs rather than production JVP/QR estimation. No adding-task generalization, multiple-seed statistics, or kappa sweep was performed."
}