{
  "experiment": "CDR_theta_analysis_2026-04-18",
  "conclusion": "NEW FORMULA VALIDATED: gamma = 1 - T_eval*sqrt(2)/theta",
  "old_formula": {
    "formula": "gamma = C/ln(theta)",
    "C": 9.2103,
    "mean_error_pct": 33.3,
    "status": "REFUTED"
  },
  "new_formula": {
    "formula": "gamma = 1 - T_eval*sqrt(2)/theta",
    "T_eval_e4": 2000,
    "mean_error_pct": 3.0,
    "status": "CONFIRMED"
  },
  "data": [
    {
      "model": "pythia-70m",
      "theta": 10000,
      "gamma_obs": 0.7476,
      "gamma_old": 1.0,
      "gamma_new": 0.7172,
      "err_old_pct": 33.8,
      "err_new_pct": 4.1
    },
    {
      "model": "Llama-3-8B",
      "theta": 500000,
      "gamma_obs": 1.0455,
      "gamma_old": 0.7019,
      "gamma_new": 0.9943,
      "err_old_pct": 32.9,
      "err_new_pct": 4.9
    },
    {
      "model": "Qwen2.5-7B",
      "theta": 1000000,
      "gamma_obs": 0.9967,
      "gamma_old": 0.6667,
      "gamma_new": 0.9972,
      "err_old_pct": 33.1,
      "err_new_pct": 0.0
    }
  ],
  "note": "gamma>1 for Llama-3 likely from long-context fine-tuning (extra positional sharpening)",
  "new_physics": "gamma = 1 - T_eval*sqrt(2)/theta = fraction of RoPE phases NOT aliased at distance T_eval"
}