{
  "author": "Reed",
  "scope": "Synthetic probability/geometry illustration; not a theorem or environment reproduction",
  "distribution": "Uniform Lebesgue probability on [0,1]",
  "target": "T(x)=0; procedure is 1 on the stated closed error set and 0 elsewhere",
  "risk_definition": "Pr_X[exists y with d(X,y)<=radius and P(y)!=T(y)]",
  "cases": [
    {
      "case": "small boundary interval",
      "interval": [
        "0",
        "1/100"
      ],
      "radius": "1/20",
      "baseline_exact": "1/100",
      "metrics": {
        "euclidean": {
          "risk_exact": "3/50",
          "risk": 0.06,
          "grid_hits": 600,
          "grid_size": 10000,
          "grid_risk": 0.06
        },
        "discrete": {
          "risk_exact": "1/100",
          "risk": 0.01,
          "grid_hits": 100,
          "grid_size": 10000,
          "grid_risk": 0.01
        }
      }
    },
    {
      "case": "full budget",
      "interval": [
        "0",
        "1/100"
      ],
      "radius": "1",
      "baseline_exact": "1/100",
      "metrics": {
        "euclidean": {
          "risk_exact": "1",
          "risk": 1.0,
          "grid_hits": 10000,
          "grid_size": 10000,
          "grid_risk": 1.0
        },
        "discrete": {
          "risk_exact": "1",
          "risk": 1.0,
          "grid_hits": 10000,
          "grid_size": 10000,
          "grid_risk": 1.0
        }
      }
    },
    {
      "case": "singleton",
      "interval": [
        "0",
        "0"
      ],
      "radius": "1/20",
      "baseline_exact": "0",
      "metrics": {
        "euclidean": {
          "risk_exact": "1/20",
          "risk": 0.05,
          "grid_hits": 500,
          "grid_size": 10000,
          "grid_risk": 0.05
        },
        "discrete": {
          "risk_exact": "0",
          "risk": 0.0,
          "grid_hits": 0,
          "grid_size": 10000,
          "grid_risk": 0.0
        }
      }
    },
    {
      "case": "empty set",
      "interval": null,
      "radius": "1",
      "baseline_exact": "0",
      "metrics": {
        "euclidean": {
          "risk_exact": "0",
          "risk": 0.0,
          "grid_hits": 0,
          "grid_size": 10000,
          "grid_risk": 0.0
        },
        "discrete": {
          "risk_exact": "0",
          "risk": 0.0,
          "grid_hits": 0,
          "grid_size": 10000,
          "grid_risk": 0.0
        }
      }
    },
    {
      "case": "centered interval",
      "interval": [
        "99/200",
        "101/200"
      ],
      "radius": "1/20",
      "baseline_exact": "1/100",
      "metrics": {
        "euclidean": {
          "risk_exact": "11/100",
          "risk": 0.11,
          "grid_hits": 1100,
          "grid_size": 10000,
          "grid_risk": 0.11
        },
        "discrete": {
          "risk_exact": "1/100",
          "risk": 0.01,
          "grid_hits": 100,
          "grid_size": 10000,
          "grid_risk": 0.01
        }
      }
    },
    {
      "case": "zero budget",
      "interval": [
        "0",
        "1/100"
      ],
      "radius": "0",
      "baseline_exact": "1/100",
      "metrics": {
        "euclidean": {
          "risk_exact": "1/100",
          "risk": 0.01,
          "grid_hits": 100,
          "grid_size": 10000,
          "grid_risk": 0.01
        },
        "discrete": {
          "risk_exact": "1/100",
          "risk": 0.01,
          "grid_hits": 100,
          "grid_size": 10000,
          "grid_risk": 0.01
        }
      }
    },
    {
      "case": "right boundary clipping",
      "interval": [
        "99/100",
        "1"
      ],
      "radius": "1/20",
      "baseline_exact": "1/100",
      "metrics": {
        "euclidean": {
          "risk_exact": "3/50",
          "risk": 0.06,
          "grid_hits": 600,
          "grid_size": 10000,
          "grid_risk": 0.06
        },
        "discrete": {
          "risk_exact": "1/100",
          "risk": 0.01,
          "grid_hits": 100,
          "grid_size": 10000,
          "grid_risk": 0.01
        }
      }
    },
    {
      "case": "entire domain",
      "interval": [
        "0",
        "1"
      ],
      "radius": "0",
      "baseline_exact": "1",
      "metrics": {
        "euclidean": {
          "risk_exact": "1",
          "risk": 1.0,
          "grid_hits": 10000,
          "grid_size": 10000,
          "grid_risk": 1.0
        },
        "discrete": {
          "risk_exact": "1",
          "risk": 1.0,
          "grid_hits": 10000,
          "grid_size": 10000,
          "grid_risk": 1.0
        }
      }
    }
  ],
  "sampling_seed": 20260914,
  "sampling": [
    {
      "p": 0.001,
      "n": 100,
      "trials": 20000,
      "zero_hit_trials": 18110,
      "observed_zero_hit_fraction": 0.9055,
      "exact_zero_hit_probability": 0.9047921471137089,
      "monte_carlo_standard_error": 0.0020753712635944613
    },
    {
      "p": 0.01,
      "n": 100,
      "trials": 20000,
      "zero_hit_trials": 7259,
      "observed_zero_hit_fraction": 0.36295,
      "exact_zero_hit_probability": 0.3660323412732292,
      "monte_carlo_standard_error": 0.003406263836047257
    },
    {
      "p": 0.05,
      "n": 100,
      "trials": 20000,
      "zero_hit_trials": 127,
      "observed_zero_hit_fraction": 0.00635,
      "exact_zero_hit_probability": 0.0059205292203339975,
      "monte_carlo_standard_error": 0.0005424701168767349
    }
  ],
  "checks": {
    "exact_scenario_metric_pairs": 16,
    "independent_grid_pairs": 16,
    "iid_sampling_cases": 3,
    "passed": true
  },
  "limitations": [
    "Midpoint grids and floating RNG draws do not prove measure-zero claims.",
    "One-dimensional examples do not test Levy-family or high-dimensional concentration bounds.",
    "The same numeric radius in different metrics is not the same physical edit budget.",
    "No real agent behavior, classifier accuracy or directory impact measured."
  ]
}