# Эксперимент: Adjoint-Weak Fractional Residuals (#880)

{
  "worked": true,
  "confidence": 9,
  "verdict": "Built `experiment.py`, which uses a Grünwald fractional-derivative matrix and its exact weighted transpose as the adjoint weak kernel. The discrete integration-by-parts identity held to relative error 4.84e-16; weak noise variance was proportional to sigma^2 with coefficient 0.01056–0.01139. Under each grid doubling, weak squared kernel norm ratios were 0.503–0.506 versus the predicted 0.5, while strong pointwise derivative norms grew by 3.031 versus the predicted 2^(2*0.8)=3.031. In the matched coefficient toy, weak noise-induced RMSE was 20.1x, 23.8x, 26.0x, and 47.4x lower than the strong estimate for sigma 0.01, 0.03, 0.1, and 0.3, respectively; the mechanism is therefore a real observed win in this setting.",
  "metrics": {
    "baseline": "Strong local Grünwald fractional derivative: noise variances at sigma=.01/.03/.1/.3 = 1.20, 10.57, 117.54, 1069.14; coefficient RMSE = 0.1474, 0.4890, 1.7304, 10.3612; resolution norm ratios = 3.031.",
    "idea": "Exact-transpose weak projection: noise variances = 1.10e-6, 9.76e-6, 1.14e-4, 9.51e-4; coefficient RMSE = 0.00733, 0.02056, 0.06643, 0.21873; resolution norm ratios = 0.503, 0.504, 0.506, close to predicted 0.5."
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 experiment.py",
  "files": [
    "experiment.py",
    "results.json",
    "run.txt"
  ],
  "limitations": "No neural network was trained and no fractional PDE trajectory benchmark, data-fit tradeoff, FLOP comparison, validation trajectory error, GPU execution, or convergence study was tested. The strong-resolution scaling is specific to the implemented Grünwald discretization, and the coefficient experiment compares noise-induced error relative to each method's own noiseless estimate rather than absolute model-identification bias."
}