{
  "math_check": {
    "prediction": "For alpha=0.5, E_alpha(-((l+1)/tau)^alpha) has log-log tail slope -alpha=-0.5.",
    "observed": -0.49452241095234456,
    "tolerance": 0.08,
    "confirmed": true
  },
  "approximation_check": {
    "prediction": "A positive EMA bank should approximate the target kernel over the fit range and retain old impulses.",
    "relative_rmse": 0.9508585231497676,
    "weights_for_taus_1_4_16_64_256_1024": [0.14153542796733265, 0.8584645720326676, 0.0, 0.0, 0.0, 0.0],
    "confirmed": false,
    "reason": "The specified h_hat has total mass 1, while h_0 is generally below 1 and the target's discrete total mass exceeds 1; simplex-constrained weights therefore cannot match both amplitude and shape."
  },
  "toy_optimizer": {
    "adamw_losses_seeds_0_1_2": [0.0005451923983249815, 0.0010024577645447689, 0.0007329659343301935],
    "mittag_leffler_losses_seeds_0_1_2": [0.003656475218428207, 0.004100033031715034, 0.004673566851021797],
    "confirmed": false
  }
}