# Эксперимент: Fractional Jacobian topology loss (#1264)

{
  "worked": false,
  "confidence": 9,
  "verdict": "Built a self-contained 32x32 PyTorch toy experiment comparing TV regularization with fractional spectral energy plus a wrapped plaquette-winding penalty. The fractional spectrum scaled as k^1.518 versus the predicted k^1.5, and the idea reduced high-frequency spectrum (0.00850 vs 0.01178) and fractional energy (1.627 vs 1.816), while preserving the same two defects. However, it had higher task MSE (0.000312 vs 0.000162), and the winding penalty had effectively zero gradient at a constant field (6.9e-12), so this implementation does not demonstrate a topology-preservation win beyond smoothing.",
  "metrics": {
    "baseline": "TV: task MSE 0.0001623, winding MSE 6.58e-17, 2 defects, high-frequency spectrum 0.01178, fractional energy 1.8164",
    "idea": "Fractional + wrapped Jacobian: task MSE 0.0003125, winding MSE 4.35e-17, 2 defects, high-frequency spectrum 0.00850, fractional energy 1.6272; spectral scaling slope 1.518 versus expected 1.5"
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 experiment.py",
  "files": [
    "experiment.py",
    "results.json"
  ],
  "limitations": "This is direct field optimization rather than a learned neural implicit model or U-Net, uses a periodic 2D toy with an initially correct vortex pair, and does not test defect nucleation, erased-hole recovery, nonperiodic boundaries, Hausdorff distance, or matched FLOPs across a dataset. The wrapped winding surrogate is nondifferentiable and piecewise constant, so it cannot reliably create missing defects from a topology-free initialization."
}