outputs/claim5.json

{
  "claim": "5 - adaptive threshold shadow price (Eq 21/27); Table 6 peak tau0=0.6",
  "threshold_formula": "tau_t = clip(tau0 + k_used/(2*K_max) + l_used/(2*L_max), 0, 1)",
  "threshold_config": {
    "tau0": 0.2,
    "K_max": 0.02,
    "L_max": 20.0
  },
  "threshold_dynamics": [
    {
      "t": 0,
      "k_used": 0.0,
      "l_used": 0.0,
      "tau_t": 0.2
    },
    {
      "t": 1,
      "k_used": 0.0016,
      "l_used": 1.4,
      "tau_t": 0.275
    },
    {
      "t": 2,
      "k_used": 0.0032,
      "l_used": 2.8,
      "tau_t": 0.35
    },
    {
      "t": 3,
      "k_used": 0.0048,
      "l_used": 4.2,
      "tau_t": 0.425
    },
    {
      "t": 4,
      "k_used": 0.0064,
      "l_used": 5.6,
      "tau_t": 0.5
    },
    {
      "t": 5,
      "k_used": 0.008,
      "l_used": 7.0,
      "tau_t": 0.575
    },
    {
      "t": 6,
      "k_used": 0.0096,
      "l_used": 8.4,
      "tau_t": 0.65
    },
    {
      "t": 7,
      "k_used": 0.0112,
      "l_used": 9.8,
      "tau_t": 0.725
    },
    {
      "t": 8,
      "k_used": 0.0128,
      "l_used": 11.2,
      "tau_t": 0.8
    },
    {
      "t": 9,
      "k_used": 0.0144,
      "l_used": 12.6,
      "tau_t": 0.875
    },
    {
      "t": 10,
      "k_used": 0.016,
      "l_used": 14.0,
      "tau_t": 0.95
    }
  ],
  "threshold_monotone_rising": true,
  "table6": [
    {
      "tau0": 1.0,
      "accuracy": 25.54,
      "c_calc": 0.0,
      "u_calc": null,
      "c_reported": null,
      "u_reported": null
    },
    {
      "tau0": 0.9,
      "accuracy": 35.51,
      "c_calc": 0.2,
      "c_reported": 0.2,
      "u_calc": 0.4985,
      "u_reported": 0.4985
    },
    {
      "tau0": 0.8,
      "accuracy": 42.89,
      "c_calc": 0.2915,
      "c_reported": 0.2915,
      "u_calc": 0.5952,
      "u_reported": 0.5952
    },
    {
      "tau0": 0.7,
      "accuracy": 44.51,
      "c_calc": 0.3115,
      "c_reported": 0.3115,
      "u_calc": 0.609,
      "u_reported": 0.609
    },
    {
      "tau0": 0.6,
      "accuracy": 47.85,
      "c_calc": 0.3525,
      "c_reported": 0.3525,
      "u_calc": 0.6329,
      "u_reported": 0.6329
    },
    {
      "tau0": 0.5,
      "accuracy": 51.62,
      "c_calc": 0.4145,
      "c_reported": 0.4145,
      "u_calc": 0.6292,
      "u_reported": 0.6292
    },
    {
      "tau0": 0.4,
      "accuracy": 53.29,
      "c_calc": 0.491,
      "c_reported": 0.491,
      "u_calc": 0.5652,
      "u_reported": 0.5652
    },
    {
      "tau0": 0.3,
      "accuracy": 54.13,
      "c_calc": 0.614,
      "c_reported": 0.614,
      "u_calc": 0.4656,
      "u_reported": 0.4656
    },
    {
      "tau0": 0.2,
      "accuracy": 55.14,
      "c_calc": 0.675,
      "c_reported": 0.675,
      "u_calc": 0.4385,
      "u_reported": 0.4385
    },
    {
      "tau0": 0.1,
      "accuracy": 55.41,
      "c_calc": 0.7355,
      "c_reported": 0.7355,
      "u_calc": 0.4061,
      "u_reported": 0.4061
    },
    {
      "tau0": 0.0,
      "accuracy": 57.28,
      "c_calc": 0.776,
      "c_reported": 0.776,
      "u_calc": 0.409,
      "u_reported": 0.409
    }
  ],
  "table6_all_rows_match": true,
  "utility_peak": {
    "tau0": 0.6,
    "u": 0.6329
  },
  "honest_notes": [
    "Threshold formula is Eq 21/27, not Eq 13 (Eq 13 = calibrated-utility bandit).",
    "Utility peak is tau0=0.6 (u=0.6329), not tau0=0.5 (u=0.6292); brief off by one."
  ],
  "figures": [
    "figs/claim5_tau_vs_usage.png",
    "figs/claim5_u_vs_tau0.png"
  ],
  "verdict": "PASS: tau_t rises monotonically with budget usage (shadow price); all 10 finite Table-6 rows reproduced exactly; utility peaks at tau0=0.6."
}