1{
 2  "math_check": {
 3    "prediction": "For alpha=0.5, E_alpha(-((l+1)/tau)^alpha) has log-log tail slope -alpha=-0.5.",
 4    "observed": -0.49452241095234456,
 5    "tolerance": 0.08,
 6    "confirmed": true
 7  },
 8  "approximation_check": {
 9    "prediction": "A positive EMA bank should approximate the target kernel over the fit range and retain old impulses.",
10    "relative_rmse": 0.9508585231497676,
11    "weights_for_taus_1_4_16_64_256_1024": [0.14153542796733265, 0.8584645720326676, 0.0, 0.0, 0.0, 0.0],
12    "confirmed": false,
13    "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."
14  },
15  "toy_optimizer": {
16    "adamw_losses_seeds_0_1_2": [0.0005451923983249815, 0.0010024577645447689, 0.0007329659343301935],
17    "mittag_leffler_losses_seeds_0_1_2": [0.003656475218428207, 0.004100033031715034, 0.004673566851021797],
18    "confirmed": false
19  }
20}