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    ]
  },
  "exact_grid": {
    "x_formula": "2*pi*j/q",
    "j": [
      0,
      1,
      2,
      3,
      4,
      5,
      6,
      7,
      8,
      9,
      10,
      11,
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      14,
      15,
      16,
      17,
      18,
      19,
      20,
      21,
      22,
      23,
      24
    ],
    "values": [
      12.0,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5,
      -0.5
    ],
    "negative_depth": "1/2",
    "mean_square": "N/2 = 6"
  },
  "analytic_witness": {
    "x_formula": "3*pi/q",
    "value_formula": "-1/2 - 1/(2*sin(3*pi/(2*q)))",
    "x_decimal_approx": 0.37699111843077515,
    "value_decimal_approx": -3.16835570614623,
    "status": "exact formula; decimal is approximate; not asserted to be the exact minimizer"
  },
  "dense_sample": {
    "M": 8192,
    "negative_depth_approx": 3.230103268317106,
    "mathematical_discretization_bound": "sum(a*a)*pi*pi/(2*M*M)",
    "bound_decimal_approx": 4.7797283982545755e-05,
    "status": "discretization bound applies in exact arithmetic; computed float values are not an interval-arithmetic certificate"
  }
}