{
  "environment": {
    "python": "3.13.7",
    "numpy": "2.1.2",
    "mpmath": "1.3.0"
  },
  "inputs": {
    "N": 4,
    "lambda": "4",
    "samples": 65536,
    "seed": 20261020,
    "rng": "PCG64",
    "batch_size": 2048,
    "decimal_digits": 60,
    "measure": "full Hermitian Gaussian"
  },
  "analytic_checks": {
    "comparisons": 24,
    "decimal_digits": 60,
    "maximum_scaled_residual": "2.83837798869e-61",
    "tolerance": "1.0e-48",
    "method": "N=2 radial quadrature plus independent center integral; N=1 and zero coupling"
  },
  "full_matrix_targets": {
    "U": "1.594227738982689889757477",
    "SU": "1.545178507946101340451239",
    "central_factor_U_over_SU": "1.031743407499102670938748",
    "planar": "1.590636854637329063382254",
    "U_over_planar_minus_one": "0.00225751360839653181",
    "SU_over_planar_minus_one": "-0.0285787083071153848",
    "extra_precision_scaled_change": "1.2567881e-61"
  },
  "sampling": {
    "U": {
      "mean": 1.5942679725850997,
      "standard_error": 0.00188056341986582,
      "target": 1.59422773898269,
      "difference": 4.0233602409678326e-05,
      "difference_in_standard_errors": 0.02139444061532848
    },
    "SU": {
      "mean": 1.5454423547381175,
      "standard_error": 0.0009483637783056316,
      "target": 1.5451785079461013,
      "difference": 0.0002638467920161869,
      "difference_in_standard_errors": 0.27821264165907056
    },
    "paired_U_minus_factor_SU": {
      "mean": -0.00023198858584264317,
      "standard_error": 0.001601098877594659,
      "target": 0.0,
      "difference": -0.00023198858584264317,
      "difference_in_standard_errors": -0.14489335361421343
    },
    "center": {
      "mean": -0.0003455697354143936,
      "standard_error": 0.0009769232619934712,
      "target": 0.0,
      "difference": -0.0003455697354143936,
      "difference_in_standard_errors": -0.3537327330185972
    },
    "center_squared": {
      "mean": 0.06254535110400554,
      "standard_error": 0.0003424406716323151,
      "target": 0.0625,
      "difference": 4.535110400553932e-05,
      "difference_in_standard_errors": 0.13243492307547405
    },
    "normalized_trace_M_squared": {
      "mean": 1.0003154851487532,
      "standard_error": 0.0013794797945912267,
      "target": 1.0,
      "difference": 0.0003154851487532362,
      "difference_in_standard_errors": 0.2286986369718609
    }
  },
  "nested_prefixes": [
    {
      "samples": 16384,
      "U": {
        "mean": 1.5991263928470265,
        "standard_error": 0.003776109084277301,
        "target": 1.59422773898269,
        "difference": 0.004898653864336522,
        "difference_in_standard_errors": 1.2972755169423453
      },
      "SU": {
        "mean": 1.5458683279567633,
        "standard_error": 0.0019036026999760022,
        "target": 1.5451785079461013,
        "difference": 0.000689820010661979,
        "difference_in_standard_errors": 0.36237604131927065
      }
    },
    {
      "samples": 65536,
      "U": {
        "mean": 1.5942679725850997,
        "standard_error": 0.00188056341986582,
        "target": 1.59422773898269,
        "difference": 4.0233602409678326e-05,
        "difference_in_standard_errors": 0.02139444061532848
      },
      "SU": {
        "mean": 1.5454423547381175,
        "standard_error": 0.0009483637783056316,
        "target": 1.5451785079461013,
        "difference": 0.0002638467920161869,
        "difference_in_standard_errors": 0.27821264165907056
      }
    }
  ],
  "interpretation": "Standard errors estimate sampling noise for independent matrices. Prefix errors need not decrease monotonically. Exact/planar differences are approximation effects. No statistical test proves localization."
}