{
  "L_ORB_1_KL_additivity": {
    "KL_joint_monte_carlo": 0.3861605992412522,
    "interpretation": "When vantages are conditionally independent, the KL of the joint distribution equals the sum of per-vantage KLs.  This is the exact algebra behind the linear-in-N error exponent.",
    "lemma": "L_ORB.1 (KL additivity under conditional independence)",
    "mus_p": [
      0.4,
      0.6,
      0.5
    ],
    "mus_q": [
      0.0,
      0.0,
      0.0
    ],
    "n_samples": 1000000,
    "passed": true,
    "per_vantage_D_i": [
      0.08000000000000007,
      0.17999999999999994,
      0.125
    ],
    "rel_error": 0.0030145434837718682,
    "sigma": 1.0,
    "sum_D_i_analytical": 0.385
  },
  "L_ORB_2_finite_n_Chernoff": {
    "D_total_nats": 0.385,
    "alpha": 0.01,
    "interpretation": "(a) The empirical rate -(1/n) log beta_n converges to D_total as n grows (rel.err < 1% at n = 10000).  This is the statement of Stein's Lemma at the operating alpha.  (b) Mills' inequality gives a VALID finite-n upper bound on beta_n at every tested n.  We DO NOT use exp(-n D) as a finite-n bound (this is a Stein asymptotic, not a Chernoff bound).",
    "lemma": "L_ORB.2 (Stein-rate convergence + Mills finite-n upper bound)",
    "mills_valid_at_all_n": true,
    "passed": true,
    "per_vantage_D_i": [
      0.08000000000000002,
      0.18,
      0.125
    ],
    "rate_converges_to_D_total_within_1pct_at_n_1e6": true,
    "rows": [
      {
        "empirical_rate_minus_log_beta_over_n": 0.1302134274405894,
        "gap_from_D_total": -0.2547865725594106,
        "log_beta_exact_gaussian": -1.9532014116088408,
        "log_mills_chernoff_upper_bound": -1.2679334643404296,
        "mills_valid_upper_bound": 1.0,
        "n": 15
      },
      {
        "empirical_rate_minus_log_beta_over_n": 0.1675029882045146,
        "gap_from_D_total": -0.21749701179548542,
        "log_beta_exact_gaussian": -5.025089646135438,
        "log_mills_chernoff_upper_bound": -3.7680943801066475,
        "mills_valid_upper_bound": 1.0,
        "n": 30
      },
      {
        "empirical_rate_minus_log_beta_over_n": 0.19223562141757533,
        "gap_from_D_total": -0.19276437858242468,
        "log_beta_exact_gaussian": -8.842838585208465,
        "log_mills_chernoff_upper_bound": -7.263903868400445,
        "mills_valid_upper_bound": 1.0,
        "n": 46
      },
      {
        "empirical_rate_minus_log_beta_over_n": 0.23597880460617213,
        "gap_from_D_total": -0.14902119539382788,
        "log_beta_exact_gaussian": -23.597880460617212,
        "log_mills_chernoff_upper_bound": -21.485474648693366,
        "mills_valid_upper_bound": 1.0,
        "n": 100
      },
      {
        "empirical_rate_minus_log_beta_over_n": 0.2703807776867517,
        "gap_from_D_total": -0.11461922231324828,
        "log_beta_exact_gaussian": -54.076155537350346,
        "log_mills_chernoff_upper_bound": -51.52987649219563,
        "mills_valid_upper_bound": 1.0,
        "n": 200
      },
      {
        "empirical_rate_minus_log_beta_over_n": 0.32730852900655516,
        "gap_from_D_total": -0.05769147099344485,
        "log_beta_exact_gaussian": -327.30852900655515,
        "log_mills_chernoff_upper_bound": -323.84556070573177,
        "mills_valid_upper_bound": 1.0,
        "n": 1000
      },
      {
        "empirical_rate_minus_log_beta_over_n": 0.3653936444118357,
        "gap_from_D_total": -0.019606355588164315,
        "log_beta_exact_gaussian": -3653.936444118357,
        "log_mills_chernoff_upper_bound": -3649.262896922151,
        "mills_valid_upper_bound": 1.0,
        "n": 10000
      },
      {
        "empirical_rate_minus_log_beta_over_n": 0.3786370692348708,
        "gap_from_D_total": -0.006362930765129227,
        "log_beta_exact_gaussian": -37863.70692348708,
        "log_mills_chernoff_upper_bound": -37857.86375749254,
        "mills_valid_upper_bound": 1.0,
        "n": 100000
      },
      {
        "empirical_rate_minus_log_beta_over_n": 0.382969037330572,
        "gap_from_D_total": -0.0020309626694279825,
        "log_beta_exact_gaussian": -382969.037330572,
        "log_mills_chernoff_upper_bound": -382962.0371196567,
        "mills_valid_upper_bound": 1.0,
        "n": 1000000
      }
    ]
  },
  "L_ORB_3_information_dominance": {
    "D_total_nats": 0.25,
    "alpha": 0.01,
    "indistinguishability_holds_at_v1": true,
    "interpretation": "Vantage 1 sees no signal (D_1 = 0) so its beta = 1 - alpha for every n.  No amount of single-vantage observations can reduce this.  The joint detector achieves exponential decay of beta in n because D_total = 0.25 > 0.  This is the actual operational advantage of multi-vantage, NOT a wall-clock speedup.",
    "joint_strictly_dominates_v1_at_all_n": true,
    "lemma": "L_ORB.3 (Information-theoretic dominance, NOT wall-clock)",
    "passed": true,
    "per_vantage_D_i": [
      0.0,
      0.125,
      0.125
    ],
    "rows": [
      {
        "beta_joint_3_vantage": 0.5359676024285032,
        "beta_single_vantage_v1_blind": 0.99,
        "beta_single_vantage_v2": 0.7719273218172839,
        "n": 10
      },
      {
        "beta_joint_3_vantage": 0.06097558553636404,
        "beta_single_vantage_v1_blind": 0.99,
        "beta_single_vantage_v2": 0.3400726313427965,
        "n": 30
      },
      {
        "beta_joint_3_vantage": 1.0439750471862995e-06,
        "beta_single_vantage_v1_blind": 0.99,
        "beta_single_vantage_v2": 0.0037515118748431297,
        "n": 100
      },
      {
        "beta_joint_3_vantage": 1.3826250178657307e-89,
        "beta_single_vantage_v1_blind": 0.99,
        "beta_single_vantage_v2": 9.578023353478314e-42,
        "n": 1000
      }
    ],
    "scenario": "mu = (0, 0.5, 0.5): rank-1 alternative orthogonal to vantage 1",
    "sigma": 1.0
  },
  "lemmas": [
    "L_ORB.1",
    "L_ORB.2",
    "L_ORB.3"
  ],
  "reference": "docs/MVPS_ORBITAL_PROOF.txt",
  "schema": "com.catellix.mvps.orbital_error_exponent_receipt_v1",
  "sensitivity_D_x_N": [
    {
      "D_nats": 0.005,
      "by_N": {
        "1": {
          "n_samples_to_beta_0.01": 921.0340371976183,
          "speedup_over_single_vantage": 1.0
        },
        "10": {
          "n_samples_to_beta_0.01": 92.10340371976183,
          "speedup_over_single_vantage": 10.0
        },
        "3": {
          "n_samples_to_beta_0.01": 307.01134573253944,
          "speedup_over_single_vantage": 3.0
        },
        "5": {
          "n_samples_to_beta_0.01": 184.20680743952366,
          "speedup_over_single_vantage": 5.0
        }
      }
    },
    {
      "D_nats": 0.01,
      "by_N": {
        "1": {
          "n_samples_to_beta_0.01": 460.51701859880916,
          "speedup_over_single_vantage": 1.0
        },
        "10": {
          "n_samples_to_beta_0.01": 46.051701859880914,
          "speedup_over_single_vantage": 10.0
        },
        "3": {
          "n_samples_to_beta_0.01": 153.50567286626972,
          "speedup_over_single_vantage": 3.0
        },
        "5": {
          "n_samples_to_beta_0.01": 92.10340371976183,
          "speedup_over_single_vantage": 5.0
        }
      }
    },
    {
      "D_nats": 0.05,
      "by_N": {
        "1": {
          "n_samples_to_beta_0.01": 92.10340371976183,
          "speedup_over_single_vantage": 1.0
        },
        "10": {
          "n_samples_to_beta_0.01": 9.210340371976184,
          "speedup_over_single_vantage": 10.0
        },
        "3": {
          "n_samples_to_beta_0.01": 30.701134573253942,
          "speedup_over_single_vantage": 3.0
        },
        "5": {
          "n_samples_to_beta_0.01": 18.420680743952367,
          "speedup_over_single_vantage": 5.0
        }
      }
    },
    {
      "D_nats": 0.1,
      "by_N": {
        "1": {
          "n_samples_to_beta_0.01": 46.051701859880914,
          "speedup_over_single_vantage": 1.0
        },
        "10": {
          "n_samples_to_beta_0.01": 4.605170185988092,
          "speedup_over_single_vantage": 10.0
        },
        "3": {
          "n_samples_to_beta_0.01": 15.350567286626971,
          "speedup_over_single_vantage": 3.0
        },
        "5": {
          "n_samples_to_beta_0.01": 9.210340371976184,
          "speedup_over_single_vantage": 5.0
        }
      }
    },
    {
      "D_nats": 0.5,
      "by_N": {
        "1": {
          "n_samples_to_beta_0.01": 9.210340371976184,
          "speedup_over_single_vantage": 1.0
        },
        "10": {
          "n_samples_to_beta_0.01": 0.9210340371976183,
          "speedup_over_single_vantage": 10.0
        },
        "3": {
          "n_samples_to_beta_0.01": 3.0701134573253945,
          "speedup_over_single_vantage": 3.0
        },
        "5": {
          "n_samples_to_beta_0.01": 1.8420680743952367,
          "speedup_over_single_vantage": 5.0
        }
      }
    }
  ],
  "verdict": "PASS"
}