{
  "timestamp": "2026-10-14T10:26:08+09:00",
  "seed": 20261014,
  "sample_size": 1000000,
  "theta": 1,
  "r_second_derivative_at_zero": -0.5,
  "rice_constant": 0.13651736229720435,
  "method": "independent standard normals; conditional expectation over U given V (Rao-Blackwell); deterministic Gauss-Legendre quadrature (128 vs 64 nodes) cross-check",
  "rows": [
    {
      "mesh": 0.5,
      "crossing_probability_divided_by_mesh": 0.13373640758019378,
      "quadrature_error_estimate": 1.27675647831893e-15,
      "conditional_monte_carlo_mean": 0.13363627826171892,
      "standard_error": 0.00010090320201371295,
      "sample_size": 1000000
    },
    {
      "mesh": 0.2,
      "crossing_probability_divided_by_mesh": 0.13606394868675856,
      "quadrature_error_estimate": 1.4155343563970746e-15,
      "conditional_monte_carlo_mean": 0.13596207476808003,
      "standard_error": 0.00010264644739576756,
      "sample_size": 1000000
    },
    {
      "mesh": 0.1,
      "crossing_probability_divided_by_mesh": 0.13640370084435469,
      "quadrature_error_estimate": 1.3600232051658168e-15,
      "conditional_monte_carlo_mean": 0.13630157247659822,
      "standard_error": 0.0001029024233581357,
      "sample_size": 1000000
    },
    {
      "mesh": 0.05,
      "crossing_probability_divided_by_mesh": 0.13648892762244588,
      "quadrature_error_estimate": 1.304512053934559e-15,
      "conditional_monte_carlo_mean": 0.1363867354390127,
      "standard_error": 0.00010296669716834228,
      "sample_size": 1000000
    },
    {
      "mesh": 0.02,
      "crossing_probability_divided_by_mesh": 0.13651281188341333,
      "quadrature_error_estimate": 1.3600232051658168e-15,
      "conditional_monte_carlo_mean": 0.13641060181702178,
      "standard_error": 0.00010298471399570109,
      "sample_size": 1000000
    },
    {
      "mesh": 0.01,
      "crossing_probability_divided_by_mesh": 0.13651622466282728,
      "quadrature_error_estimate": 3.885780586188048e-16,
      "conditional_monte_carlo_mean": 0.1364140120412074,
      "standard_error": 0.00010298728854912373,
      "sample_size": 1000000
    },
    {
      "mesh": 0.005,
      "crossing_probability_divided_by_mesh": 0.13651707788667264,
      "quadrature_error_estimate": 8.271161533457416e-15,
      "conditional_monte_carlo_mean": 0.1364148646262289,
      "standard_error": 0.00010298793221565028,
      "sample_size": 1000000
    }
  ],
  "limitations": [
    "mesh sign changes undercount crossings at finite mesh",
    "not Euler-product or arithmetic data",
    "reported standard error is Monte Carlo sampling error, not mesh bias",
    "rows share random samples and are correlated"
  ]
}