{
  "task_id": "hle-673dda11d3981c54a42874ef",
  "method": "exact_rational_arithmetic",
  "matrix": [
    [
      2,
      1,
      2
    ],
    [
      -2,
      -1,
      -3
    ],
    [
      -2,
      -1,
      -1
    ]
  ],
  "square": [
    [
      -2,
      -1,
      -1
    ],
    [
      4,
      2,
      2
    ],
    [
      0,
      0,
      0
    ]
  ],
  "nilpotency_index": 3,
  "popov": [
    [
      2,
      1,
      0
    ],
    [
      0,
      0,
      1
    ],
    [
      0,
      0,
      0
    ]
  ],
  "popov_convention": "normalized_row_Popov_over_Q_with_rightmost_constant_pivots_and_zero_rows_last",
  "popov_norm_ratio": "sqrt(6)/2",
  "global_rank_two_bound_slack": "x**2/2 - 2*x + 3*y**2/2 + 2",
  "witnesses": [
    {
      "scale": 1,
      "immanants": {
        "trivial": 24,
        "sign": 0,
        "standard": -6
      },
      "largest_immanant": 24,
      "popov_unchanged": true
    },
    {
      "scale": 2,
      "immanants": {
        "trivial": 192,
        "sign": 0,
        "standard": -48
      },
      "largest_immanant": 192,
      "popov_unchanged": true
    },
    {
      "scale": 10,
      "immanants": {
        "trivial": 24000,
        "sign": 0,
        "standard": -6000
      },
      "largest_immanant": 24000,
      "popov_unchanged": true
    },
    {
      "scale": 100,
      "immanants": {
        "trivial": 24000000,
        "sign": 0,
        "standard": -6000000
      },
      "largest_immanant": 24000000,
      "popov_unchanged": true
    }
  ],
  "conclusion": "Equally optimal integer matrices have different, unbounded largest immanants; part 2 needs a normalization or tie-breaking rule.",
  "original_grades_changed": false,
  "all_checks_passed": true
}