# Эксперимент: Koopman Skew-Dilation RNN (#786)

{
  "worked": true,
  "confidence": 9,
  "verdict": "Built a finite-channel Koopman skew-dilation module with symmetric/skew parameter decomposition, block-tridiagonal lift, Cayley update, and a PyTorch toy sequence model. Quantitative predictions manifested: ||L+L^T||∞=0 and projection derivative error=1.1e-16; Cayley relative norm error stayed below 5.6e-16 for gamma∈{0,.25,.5,1,2} and eta∈{.05,.2,.5}; the explicit-Euler one-step norm increment exactly matched eta²||Lz||², while 10-step norm ratio grew from 1.04 at gamma=.25 to 12.95 at gamma=2. The toy task ended at MSE 0.0284 for vanilla RNN versus 0.000106 for the lift, but this is only a promising signal because parameter counts and architecture capacity were not strictly matched.",
  "metrics": {
    "baseline": "Vanilla RNN final toy MSE 0.0283883; explicit Euler 10-step norm ratio at gamma=2: 12.9465",
    "idea": "Cayley relative norm error <=5.6e-16 across 15 scale/timestep settings; Euler observed/predicted one-step increments match to <=5e-16; lift final toy MSE 0.000105752; projected gain remained bounded (reported normalized max 0.7071, corresponding unnormalized gain 1.0)"
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 koopman_experiment.py",
  "files": [
    "koopman_experiment.py",
    "results.json"
  ],
  "limitations": "No sequential MNIST experiment, no long-horizon benchmark, no wall-clock/FLOP comparison, and the toy lift has a larger latent state and more parameters than the baseline. The finite lift and dense linear solve do not test the infinite-dimensional theorem or scalable block-tridiagonal implementation."
}