# Эксперимент: Cohomological Jacobian Flattening (#690)

{
  "worked": true,
  "confidence": 8,
  "verdict": "Built and numerically verified cohomological Jacobian flattening on a nonlinear conjugate expanding map. The exact potential produced max finite-horizon residuals from 4.2e-17 at k=1 to 1.3e-15 at k=32; pointwise Jacobian variance scaled approximately as eps^2 for small eps, with fitted log-log slope 2.60; and scaling the potential by alpha gave residual ratios exactly matching 1-alpha. A Fourier potential reduced one-step residual MSE from 1.61e-3 for the pointwise-constant baseline to 3.42e-11, but no neural-network optimization or long-horizon stability experiment was run, so this validates the mechanism rather than an end-to-end ML win.",
  "metrics": {
    "baseline": "Pointwise constant-log-Jacobian MSE = 1.6076e-3",
    "idea": "Cohomological Fourier-potential MSE = 3.4204e-11; ratio = 2.13e-8; fitted c = 0.029562 vs true log(a) = 0.029559; exact telescoping residual max <= 1.33e-15 through k=32"
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 coh_jacobian_experiment.py",
  "files": [
    "coh_jacobian_experiment.py",
    "results.json"
  ],
  "limitations": "Only a one-dimensional analytically conjugate expanding toy map was tested. The experiment did not train an invertible RNN, estimate Jacobians with autograd in higher dimensions, compare task loss or FLOPs, or measure long-horizon gradient explosion/collapse and rollout horizon."
}