{
  "checks": [
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      "name": "identity_nu_0.0",
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      "name": "replacement_nu_0.17",
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      "name": "identity_nu_1.0",
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    },
    {
      "name": "replacement_nu_1.0",
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    {
      "name": "unital_gamma_0.03",
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      "name": "direct_matrix_0.55_0.3_1.0",
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      "name": "thermal_symmetry_0.9_0.03",
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      "passed": true
    },
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      "passed": true
    },
    {
      "name": "direct_matrix_0.9_0.03_0.23",
      "absolute_error": 1.942890293094024e-16,
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      "passed": true
    },
    {
      "name": "direct_matrix_0.9_0.03_0.5",
      "absolute_error": 5.551115123125783e-17,
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      "passed": true
    },
    {
      "name": "direct_matrix_0.9_0.03_0.79",
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      "passed": true
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    {
      "name": "direct_matrix_0.9_0.03_1.0",
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      "tolerance": 1e-11,
      "passed": true
    },
    {
      "name": "grid_allowance_0.02_0.08",
      "absolute_error": 0,
      "tolerance": 1e-12,
      "passed": true
    },
    {
      "name": "grid_allowance_0.5_0.27",
      "absolute_error": 0,
      "tolerance": 1e-12,
      "passed": true
    },
    {
      "name": "grid_allowance_0.97_0.9",
      "absolute_error": 0,
      "tolerance": 1e-12,
      "passed": true
    }
  ],
  "random_ensemble_checks": [
    {
      "gamma": 0.2,
      "nu": 0.1,
      "ensembles_sampled": 500,
      "best_random_holevo_bits": 0.7004853214037154,
      "scalar_benchmark_bits": 0.7197487555790375
    },
    {
      "gamma": 0.55,
      "nu": 0.3,
      "ensembles_sampled": 500,
      "best_random_holevo_bits": 0.35062874758183393,
      "scalar_benchmark_bits": 0.3632124111712466
    },
    {
      "gamma": 0.9,
      "nu": 0.03,
      "ensembles_sampled": 500,
      "best_random_holevo_bits": 0.10724079474433024,
      "scalar_benchmark_bits": 0.11299317683667698
    }
  ],
  "theorem_verification": "not_run",
  "certified_numeric_error_bound": "not_established",
  "caveat": "Random ensembles and finite grid comparisons are diagnostics, not an optimality proof."
}