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  "passed": true,
  "checks": [
    {
      "name": "same observed seed reproduces every draw",
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    },
    {
      "name": "observed and simulation streams differ",
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    },
    {
      "name": "repetition exercise contains the required 1000 data samples",
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    },
    {
      "name": "refreshed objective advances stream on consecutive calls at the same sigma",
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      "first_moment": 24.19467532839516,
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      "name": "GMM numerical minimum matches analytic minimum",
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      "absolute_error": 4.7553072590744705e-12
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    {
      "name": "simulation stream 2 exactly reproducible",
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      "absolute_error": 5.550004900101158e-13
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      "name": "SMM stream 2, S=1: mean statistic equals scaling formula",
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    {
      "name": "SMM stream 2, S=1000: mean statistic equals scaling formula",
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    {
      "name": "simulation stream 3 exactly reproducible",
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      "absolute_error": 1.7763568394002505e-15
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    {
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  "formula_maximum_absolute_error": 1.4210854715202004e-14,
  "optimizer_maximum_absolute_error": 5.0321968814159845e-12,
  "simulation_error_check": {
    "method": "exact chi-square distribution of the normal unbiased within-sector variance",
    "repetitions": 20000,
    "seed": 71010,
    "degrees_of_freedom": 400,
    "gmm_sigma_sd_monte_carlo": 0.03514716660100815,
    "gmm_sigma_sd_delta_approximation": 0.035355339059327376,
    "rows": [
      {
        "S": 1,
        "moment_sd_exact": 1.5954096750521478,
        "moment_sd_monte_carlo": 1.6038557701295633,
        "smm_sigma_sd_monte_carlo": 0.05014111173203548,
        "smm_sigma_sd_delta_approximation": 0.05
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      {
        "S": 10,
        "moment_sd_exact": 0.50451283742339,
        "moment_sd_monte_carlo": 0.5032187613064752,
        "smm_sigma_sd_monte_carlo": 0.036826132694745335,
        "smm_sigma_sd_delta_approximation": 0.03708099243547831
      },
      {
        "S": 100,
        "moment_sd_exact": 0.1595409675052148,
        "moment_sd_monte_carlo": 0.1578187046632746,
        "smm_sigma_sd_monte_carlo": 0.0353265055850589,
        "smm_sigma_sd_delta_approximation": 0.035531676008879735
      },
      {
        "S": 1000,
        "moment_sd_exact": 0.05045128374233901,
        "moment_sd_monte_carlo": 0.050690687620568206,
        "smm_sigma_sd_monte_carlo": 0.035162043856875415,
        "smm_sigma_sd_delta_approximation": 0.03537301231164798
      }
    ]
  },
  "task4_repetition_summary": [
    {
      "method": "GMM",
      "mean_estimate": 0.9978190994565969,
      "empirical_sd": 0.035526265062172664,
      "asymptotic_sd": 0.035355339059327376,
      "repetitions": 1000,
      "ratio_empirical_to_asymptotic_sd": 1.0048345174277207
    },
    {
      "method": "SMM · S = 1",
      "mean_estimate": 1.0001833998220964,
      "empirical_sd": 0.04947954359127339,
      "asymptotic_sd": 0.05,
      "repetitions": 1000,
      "ratio_empirical_to_asymptotic_sd": 0.9895908718254678
    },
    {
      "method": "SMM · S = 10",
      "mean_estimate": 0.998503427303039,
      "empirical_sd": 0.03681350738800079,
      "asymptotic_sd": 0.03708099243547832,
      "repetitions": 1000,
      "ratio_empirical_to_asymptotic_sd": 0.9927864647111869
    }
  ],
  "limitations": [
    "This is a new teaching reference; no earlier result table was available to compare with.",
    "Five-firm sample variance uses ddof=1, then averages over sectors.",
    "The large-sample estimator standard deviations are approximations, not exact finite-sample equalities.",
    "PCG64 seeds reproduce this Python version."
  ]
}