# Эксперимент: Bounded Commuting Cochain Layer (#1366)

{
  "worked": true,
  "confidence": 8,
  "verdict": "Built a mass-weighted bounded cochain projection MVP on an oriented filled triangle. The compatible projector was idempotent, norm-bounded by one, and commuted with incidence to machine precision, while an equal-rank independent projector had defects of 1.66 and 1.47. Across five seeds, denoising MSE improved from 0.4935 ± 0.0075 to 0.4177 ± 0.0108, a 15.4% mean reduction, so the claimed effect was observed in this toy setting.",
  "metrics": {
    "baseline": "No-projection control: MSE 0.5062 for seed 17; five-seed mean 0.4935 ± 0.0075.",
    "idea": "Compatible mass projection: MSE 0.4389 for seed 17; five-seed mean 0.4177 ± 0.0108; mean relative reduction 15.36%. Idempotence and compatible commutation defects were approximately 1e-16, and mass operator norms were 1.0."
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 cochain_mvp.py",
  "files": [
    "cochain_mvp.py",
    "results.json"
  ],
  "limitations": "This is a single-triangle toy regression, not a full multi-layer simplicial GNN or PDE/molecular benchmark. The subspaces are fixed, P0 is the identity, and learned commutation penalties, mesh refinement, larger sparse complexes, FLOPs, GPU behavior, classification accuracy, gradient variance, and generalization were not tested."
}