{
  "status": "PASS",
  "exact_reuse_verification": {
    "status": "PASS",
    "arithmetic": "fractions.Fraction exclusively",
    "period_convention": "All exact fixture angles are measured in units of pi; T=1/2 therefore means physical period pi/2.",
    "lemma_checked": "If each measurable A_j has measure <=2psi, then integral (sum 1_A_j -1)_+ <=2psi*(N-1)_+.",
    "proof": "For N>=1 delete any one index j0: (sum 1_A_j -1)_+ <= sum_{j!=j0}1_A_j pointwise. Integrate and bound each remaining set's measure by 2psi. N=0 is immediate.",
    "counts": {
      "random_families": 600,
      "sharp_identical_families": 24,
      "exhaustive_pointwise_fixtures": 18434
    },
    "strict_random_cases": 549,
    "smallest_random_slack": "0",
    "sharpness_cases": [
      {
        "psi_in_pi_units": "1/16",
        "points": 0,
        "integral_in_pi_units": "0"
      },
      {
        "psi_in_pi_units": "1/16",
        "points": 1,
        "integral_in_pi_units": "0"
      },
      {
        "psi_in_pi_units": "1/16",
        "points": 2,
        "integral_in_pi_units": "1/8"
      },
      {
        "psi_in_pi_units": "1/16",
        "points": 3,
        "integral_in_pi_units": "1/4"
      },
      {
        "psi_in_pi_units": "1/16",
        "points": 8,
        "integral_in_pi_units": "7/8"
      },
      {
        "psi_in_pi_units": "1/16",
        "points": 20,
        "integral_in_pi_units": "19/8"
      },
      {
        "psi_in_pi_units": "1/12",
        "points": 0,
        "integral_in_pi_units": "0"
      },
      {
        "psi_in_pi_units": "1/12",
        "points": 1,
        "integral_in_pi_units": "0"
      },
      {
        "psi_in_pi_units": "1/12",
        "points": 2,
        "integral_in_pi_units": "1/6"
      },
      {
        "psi_in_pi_units": "1/12",
        "points": 3,
        "integral_in_pi_units": "1/3"
      },
      {
        "psi_in_pi_units": "1/12",
        "points": 8,
        "integral_in_pi_units": "7/6"
      },
      {
        "psi_in_pi_units": "1/12",
        "points": 20,
        "integral_in_pi_units": "19/6"
      },
      {
        "psi_in_pi_units": "3/32",
        "points": 0,
        "integral_in_pi_units": "0"
      },
      {
        "psi_in_pi_units": "3/32",
        "points": 1,
        "integral_in_pi_units": "0"
      },
      {
        "psi_in_pi_units": "3/32",
        "points": 2,
        "integral_in_pi_units": "3/16"
      },
      {
        "psi_in_pi_units": "3/32",
        "points": 3,
        "integral_in_pi_units": "3/8"
      },
      {
        "psi_in_pi_units": "3/32",
        "points": 8,
        "integral_in_pi_units": "21/16"
      },
      {
        "psi_in_pi_units": "3/32",
        "points": 20,
        "integral_in_pi_units": "57/16"
      },
      {
        "psi_in_pi_units": "1/10",
        "points": 0,
        "integral_in_pi_units": "0"
      },
      {
        "psi_in_pi_units": "1/10",
        "points": 1,
        "integral_in_pi_units": "0"
      },
      {
        "psi_in_pi_units": "1/10",
        "points": 2,
        "integral_in_pi_units": "1/5"
      },
      {
        "psi_in_pi_units": "1/10",
        "points": 3,
        "integral_in_pi_units": "2/5"
      },
      {
        "psi_in_pi_units": "1/10",
        "points": 8,
        "integral_in_pi_units": "7/5"
      },
      {
        "psi_in_pi_units": "1/10",
        "points": 20,
        "integral_in_pi_units": "19/5"
      }
    ],
    "constant_and_half_factor_fixtures": [
      {
        "T_in_pi_units": "1/2",
        "psi_in_pi_units": "1/16",
        "point_membership_bound_2psi_in_pi_units": "1/8",
        "local_redundancy_is_twice_reward": "2/7",
        "global_redundancy_lower_bound": "8/7",
        "disk_excess_is_half_global_redundancy": "4/7",
        "amplification_T_over_2psi_equals_pi_over_4psi": "4"
      },
      {
        "T_in_pi_units": "1/2",
        "psi_in_pi_units": "1/12",
        "point_membership_bound_2psi_in_pi_units": "1/6",
        "local_redundancy_is_twice_reward": "2/7",
        "global_redundancy_lower_bound": "6/7",
        "disk_excess_is_half_global_redundancy": "3/7",
        "amplification_T_over_2psi_equals_pi_over_4psi": "3"
      },
      {
        "T_in_pi_units": "1/2",
        "psi_in_pi_units": "3/32",
        "point_membership_bound_2psi_in_pi_units": "3/16",
        "local_redundancy_is_twice_reward": "2/7",
        "global_redundancy_lower_bound": "16/21",
        "disk_excess_is_half_global_redundancy": "8/21",
        "amplification_T_over_2psi_equals_pi_over_4psi": "8/3"
      },
      {
        "T_in_pi_units": "1/2",
        "psi_in_pi_units": "1/10",
        "point_membership_bound_2psi_in_pi_units": "1/5",
        "local_redundancy_is_twice_reward": "2/7",
        "global_redundancy_lower_bound": "5/7",
        "disk_excess_is_half_global_redundancy": "5/14",
        "amplification_T_over_2psi_equals_pi_over_4psi": "5/2"
      }
    ],
    "scope": "Finite reproducibility fixtures supplement the displayed elementary universal proof; they do not enumerate opaque barriers."
  },
  "numerical_membership_diagnostic": {
    "status": "DIAGNOSTIC PASS — NOT A PROOF",
    "warning": "Floating-point midpoint samples below are sanity checks only and are not used in any rigorous certificate.",
    "parameters_are_diagnostic_only": {
      "a": 1.0001425248,
      "t": 0.972532002
    },
    "radial_samples": 54,
    "angular_midpoints_per_radius": 32768,
    "maximum_absolute_sampling_error": 3.770831645288908e-05,
    "sampling_error_threshold": 0.00019174760848570513,
    "analytic_formula": "M(rho)=0 for rho<t; 2acos(t/rho) for t<=rho<=1; 2[acos(t/rho)-acos(1/rho)] for 1<rho<=a; zero outside aD.",
    "analytic_maximum": "2acos(t), attained at rho=1. Below 1 M increases; above 1 its derivative is negative because t<1.",
    "rows": [
      {
        "rho": 0.0,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.0208363026,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.0416726052,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.0625089078,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.0833452104,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.104181513,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.1250178156,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.1458541182,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.1666904208,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.18752672339999998,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.208363026,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.2291993286,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.2500356312,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.2708719338,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.2917082364,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.312544539,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.3333808416,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.3542171442,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.37505344679999997,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.3958897494,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.416726052,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.4375623546,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.4583986572,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.4792349598,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.5000712624,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.520907565,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.5417438676,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.5625801702000001,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.5834164728,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.6042527754,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.625089078,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.6459253805999999,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.6667616832,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.6875979858,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.7084342884,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.729270591,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.7501068935999999,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.7709431962000001,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.7917794988,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.8126158013999999,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.833452104,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.8542884065999999,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.8751247092,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.8959610118,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.9167973144,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.9376336169999999,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.9584699196,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.972532002,
        "sampled_membership": 0.0,
        "formula_membership": 0.0,
        "absolute_error": 0.0
      },
      {
        "rho": 0.9793062222,
        "sampled_membership": 0.23537017714120306,
        "formula_membership": 0.23537810239104964,
        "absolute_error": 7.92524984657983e-06
      },
      {
        "rho": 0.986266001,
        "sampled_membership": 0.3341201903613412,
        "formula_membership": 0.3341578986777941,
        "absolute_error": 3.770831645288908e-05
      },
      {
        "rho": 0.99999999,
        "sampled_membership": 0.4698774900892205,
        "formula_membership": 0.46984819646577797,
        "absolute_error": 2.9293623442505634e-05
      },
      {
        "rho": 1.0,
        "sampled_membership": 0.4698774900892205,
        "formula_membership": 0.46984828002769813,
        "absolute_error": 2.921006152234451e-05
      },
      {
        "rho": 1.00000001,
        "sampled_membership": 0.4695898686914919,
        "formula_membership": 0.4695655208793201,
        "absolute_error": 2.4347812171809924e-05
      },
      {
        "rho": 1.0001425248,
        "sampled_membership": 0.4372803983466506,
        "formula_membership": 0.437272778508761,
        "absolute_error": 7.6198378896052965e-06
      }
    ]
  },
  "symbolic_conclusion": "Integral_alpha R_D(B intersect S_alpha) <=2psi R_D(B); each rotated family has R_D(B intersect S_alpha)>=2R; T=pi/2. Therefore delta>=W+pi/(4psi)*R with psi=acos(t).",
  "certificate_independence": "No numerical disk-bound certificate or optimizer is imported; this file validates only the reuse lemma and its constants.",
  "errors": [],
  "seconds": 0.619033,
  "script_sha256": "1ee59424aff9ada39fbd242b4df288426300ce57e2bc46fa96fd72467b1924ff"
}