# Эксперимент: Entropy-stable split quadratic layer (#818)

{
  "worked": true,
  "confidence": 8,
  "verdict": "Built and numerically tested the split quadratic spectral derivative with alpha+beta=1, including the entropy-stable (1/3,2/3) choice, padded reference, dense two-point flux, and a toy Burgers rollout. The affine aliasing prediction matched to 8.9e-16, the predicted cancellation roots matched across the index sweep to approximately 1e-16, and the dense flux/FFT identity was checked numerically. Relative to the conservative baseline alpha=1, the entropy-stable split reduced mean absolute energy injection from 0.1317 to 0.1317 (equal here because alpha=1 and 1/3 are symmetric in this metric), reduced padded-reference RMSE from 1.6446 to 1.8598 (worse), but reduced 250-step maximum energy growth from 1.6785x to 1.3644x; thus the mechanism manifested, although it did not dominate all accuracy metrics and the symmetric split was best in this toy.",
  "metrics": {
    "baseline": "Conservative alpha=1,beta=0: formula slope/intercept agree with theory; mean |<u,N>|=0.1317449; padded-reference RMSE=1.6446275; 250-step max energy ratio=1.6785156; divergence step=250 (no threshold divergence).",
    "idea": "Entropy-stable alpha=1/3,beta=2/3: aliasing cancellation root for (i,j,k)=(1,1,4) observed at 0.3333333 versus predicted 0.3333333; max affine residual=8.88e-16; mean |<u,N>|=0.1317449; padded-reference RMSE=1.8597605; 250-step max energy ratio=1.3643754; divergence step=250. Other cancellation roots also had residuals about 1e-16."
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 split_quadratic_experiment.py",
  "files": [
    "split_quadratic_experiment.py",
    "results.json",
    "run_output.txt"
  ],
  "limitations": "This is an analytic/tiny spectral toy, not a trained FNO or PDE dataset experiment. It does not test learned alpha, parameter/FLOP-equalized training, noisy long-horizon supervised rollouts, multiple resolutions, or a rigorous entropy bound; the padded-reference implementation and energy metric are only prototype diagnostics. The rollout compares several fixed split forms and did not produce threshold divergence within 250 steps."
}