{
  "description": "Illustrative calculations with explicitly different observation models",
  "receptor_protocol": "fixed complete cycle count, known rates, no spatial rebinding",
  "receptor_bias_correction": "(n-1)/n for both estimators; n>2",
  "receptor_results": [
    {
      "n": 10,
      "repeats": 200000,
      "seed": 20260918,
      "c0": 2.0,
      "k_on": 0.5,
      "k_off": 0.5,
      "mean_duration": 30.0,
      "raw_mle_relative_bias_theory": 0.1111111111111111,
      "raw_mle_relative_bias_mc": 0.10935253627148134,
      "unbound": {
        "relative_mean_mc": 0.9984172826443333,
        "relative_variance_mc": 0.1234710046212879,
        "relative_variance_theory": 0.125,
        "relative_sd_theory": 0.3535533905932738,
        "variance_relative_difference": -0.012231963029696757
      },
      "occupancy": {
        "relative_mean_mc": 0.998839532739552,
        "relative_variance_mc": 0.2364298264558988,
        "relative_variance_theory": 0.2375,
        "relative_sd_theory": 0.48733971724044817,
        "variance_relative_difference": -0.004505993869899694
      }
    },
    {
      "n": 50,
      "repeats": 200000,
      "seed": 20260958,
      "c0": 2.0,
      "k_on": 0.5,
      "k_off": 0.5,
      "mean_duration": 150.0,
      "raw_mle_relative_bias_theory": 0.02040816326530612,
      "raw_mle_relative_bias_mc": 0.020853072241859616,
      "unbound": {
        "relative_mean_mc": 1.0004360107970223,
        "relative_variance_mc": 0.02092258935317943,
        "relative_variance_theory": 0.020833333333333332,
        "relative_sd_theory": 0.14433756729740643,
        "variance_relative_difference": 0.00428428895261268
      },
      "occupancy": {
        "relative_mean_mc": 1.0006498723881394,
        "relative_variance_mc": 0.04119962072147765,
        "relative_variance_theory": 0.04125,
        "relative_sd_theory": 0.203100960115899,
        "variance_relative_difference": -0.001221315842966142
      }
    },
    {
      "n": 100,
      "repeats": 200000,
      "seed": 20261008,
      "c0": 2.0,
      "k_on": 0.5,
      "k_off": 0.5,
      "mean_duration": 300.0,
      "raw_mle_relative_bias_theory": 0.010101010101010102,
      "raw_mle_relative_bias_mc": 0.01033113029513566,
      "unbound": {
        "relative_mean_mc": 1.0002278189921845,
        "relative_variance_mc": 0.010201877686243133,
        "relative_variance_theory": 0.01020408163265306,
        "relative_sd_theory": 0.10101525445522107,
        "variance_relative_difference": -0.00021598674817280283
      },
      "occupancy": {
        "relative_mean_mc": 1.0002492486018009,
        "relative_variance_mc": 0.02024155991920832,
        "relative_variance_theory": 0.020306122448979592,
        "relative_sd_theory": 0.14249955245185716,
        "variance_relative_difference": -0.003179461265117922
      }
    },
    {
      "n": 500,
      "repeats": 200000,
      "seed": 20261408,
      "c0": 2.0,
      "k_on": 0.5,
      "k_off": 0.5,
      "mean_duration": 1500.0,
      "raw_mle_relative_bias_theory": 0.002004008016032064,
      "raw_mle_relative_bias_mc": 0.0018708447413795515,
      "unbound": {
        "relative_mean_mc": 0.9998671030518969,
        "relative_variance_mc": 0.002013855571839406,
        "relative_variance_theory": 0.002008032128514056,
        "relative_sd_theory": 0.04481107149482208,
        "variance_relative_difference": 0.002900074776024164
      },
      "occupancy": {
        "relative_mean_mc": 0.999795421386791,
        "relative_variance_mc": 0.004005420128196907,
        "relative_variance_theory": 0.0040120481927710845,
        "relative_sd_theory": 0.06334073091440519,
        "variance_relative_difference": -0.0016520401190892597
      }
    }
  ],
  "spatial_checks": {
    "shell_quadrature_checks": [
      {
        "inner_radius_fraction": 0.0,
        "energy_quad": 1.2,
        "energy_closed": 1.2,
        "variance_coefficient": 0.6
      },
      {
        "inner_radius_fraction": 0.2,
        "energy_quad": 1.1956295525494278,
        "energy_closed": 1.1956295525494276,
        "variance_coefficient": 0.5978147762747138
      },
      {
        "inner_radius_fraction": 0.7,
        "energy_quad": 1.0970163257646839,
        "energy_closed": 1.0970163257646837,
        "variance_coefficient": 0.5485081628823418
      },
      {
        "inner_radius_fraction": 0.95,
        "energy_quad": 1.016655132074283,
        "energy_closed": 1.0166551320742834,
        "variance_coefficient": 0.5083275660371417
      },
      {
        "inner_radius_fraction": 0.999,
        "energy_quad": 1.000333333244489,
        "energy_closed": 1.0003333332443778,
        "variance_coefficient": 0.5001666666221889
      }
    ],
    "bounded_geometry_checks": [
      {
        "span_fraction": 0.1,
        "interval_spectral": 0.8099999999999921,
        "interval_closed": 0.81,
        "two_points_spectral": 0.7299939208505539,
        "two_points_closed": 0.73
      },
      {
        "span_fraction": 0.25,
        "interval_spectral": 0.5624999999999987,
        "interval_closed": 0.5625,
        "two_points_spectral": 0.43749392085056404,
        "two_points_closed": 0.4375
      },
      {
        "span_fraction": 0.5,
        "interval_spectral": 0.24999999999999967,
        "interval_closed": 0.25,
        "two_points_spectral": 0.24999392085056527,
        "two_points_closed": 0.25
      },
      {
        "span_fraction": 0.75,
        "interval_spectral": 0.06249999999999987,
        "interval_closed": 0.0625,
        "two_points_spectral": 0.43749392085056404,
        "two_points_closed": 0.4375
      },
      {
        "span_fraction": 0.9,
        "interval_spectral": 0.009999999999999934,
        "interval_closed": 0.009999999999999995,
        "two_points_spectral": 0.7299939208505536,
        "two_points_closed": 0.73
      }
    ],
    "same_width_0_02_center": 0.9603999999997945,
    "same_width_0_02_wall": 3.841599999999896,
    "wall_to_center_ratio": 4.000000000000748,
    "bounded_finite_time": [
      {
        "D_T_over_L2": 0.01,
        "scaled_variance": 0.12259248027391315
      },
      {
        "D_T_over_L2": 0.1,
        "scaled_variance": 0.4360023030445989
      },
      {
        "D_T_over_L2": 1,
        "scaled_variance": 0.5496093749999986
      },
      {
        "D_T_over_L2": 10,
        "scaled_variance": 0.5612109374999985
      },
      {
        "D_T_over_L2": 100,
        "scaled_variance": 0.5623710937499986
      }
    ],
    "surface_to_volume_variance_ratio": 0.8333333333333334,
    "surface_to_volume_sd_ratio": 0.9128709291752769
  },
  "illustrative_physical_example": {
    "c_nM": 1.0,
    "c_molecules_per_um3": 0.602214076,
    "D_um2_per_s": 100.0,
    "radius_um": 1.0,
    "duration_s": 1.0,
    "unperturbed_mean_count_in_ball": 2.522548422733287,
    "absorber_arrival_rate_per_s": 756.7645268199861,
    "absorber_relative_sd": 0.03635127238061431,
    "uniform_monitor_relative_sd": 0.05631514901742755,
    "surface_monitor_relative_sd": 0.051408462410183264,
    "diffusion_time_a2_over_D_s": 0.01
  },
  "limits": [
    "No fit to measured cell parameters or claim of cellular optimality.",
    "Bounded monitors: fixed N, 1D reflecting domain, uniform/point weights.",
    "Shell monitors: 3D unbounded Poisson bath, passive permeable observation.",
    "The spatial and receptor calculations are distinct physical models."
  ]
}