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        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -5,
        -6
      ],
      "determinant": -64,
      "smith_factors": [
        2,
        2,
        16
      ],
      "inertia_positive_negative_zero": [
        20,
        1,
        0
      ],
      "isotropic_vector": [
        -1,
        0,
        -1,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0
      ],
      "matching_exception_smith_factors": [],
      "reflection_quotient_status": "infinite conditional on cited Nikulin classification"
    },
    {
      "family": 2,
      "n": 23,
      "chosen_orbit_operator": "-A",
      "finite_generator_order_bound": 44,
      "orbit_determinant": "64",
      "original_B_orbit_lattice_transition_determinant": "-1",
      "gram_first_row": [
        -2,
        -3,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -5,
        -6
      ],
      "determinant": -64,
      "smith_factors": [
        2,
        2,
        16
      ],
      "inertia_positive_negative_zero": [
        22,
        1,
        0
      ],
      "isotropic_vector": [
        -1,
        0,
        -1,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0
      ],
      "matching_exception_smith_factors": [],
      "reflection_quotient_status": "infinite conditional on cited Nikulin classification"
    },
    {
      "family": 2,
      "n": 25,
      "chosen_orbit_operator": "-A",
      "finite_generator_order_bound": 48,
      "orbit_determinant": "-64",
      "original_B_orbit_lattice_transition_determinant": "1",
      "gram_first_row": [
        -2,
        -3,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -5,
        -6
      ],
      "determinant": -64,
      "smith_factors": [
        2,
        2,
        16
      ],
      "inertia_positive_negative_zero": [
        24,
        1,
        0
      ],
      "isotropic_vector": [
        -1,
        0,
        -1,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0
      ],
      "matching_exception_smith_factors": [],
      "reflection_quotient_status": "infinite conditional on cited Nikulin classification"
    },
    {
      "family": 2,
      "n": 27,
      "chosen_orbit_operator": "-A",
      "finite_generator_order_bound": 52,
      "orbit_determinant": "64",
      "original_B_orbit_lattice_transition_determinant": "-1",
      "gram_first_row": [
        -2,
        -3,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -5,
        -6
      ],
      "determinant": -64,
      "smith_factors": [
        2,
        2,
        16
      ],
      "inertia_positive_negative_zero": [
        26,
        1,
        0
      ],
      "isotropic_vector": [
        -1,
        0,
        -1,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0
      ],
      "matching_exception_smith_factors": [],
      "reflection_quotient_status": "infinite conditional on cited Nikulin classification"
    },
    {
      "family": 2,
      "n": 29,
      "chosen_orbit_operator": "-A",
      "finite_generator_order_bound": 56,
      "orbit_determinant": "-64",
      "original_B_orbit_lattice_transition_determinant": "1",
      "gram_first_row": [
        -2,
        -3,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -4,
        -5,
        -6
      ],
      "determinant": -64,
      "smith_factors": [
        2,
        2,
        16
      ],
      "inertia_positive_negative_zero": [
        28,
        1,
        0
      ],
      "isotropic_vector": [
        -1,
        0,
        -1,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0
      ],
      "matching_exception_smith_factors": [],
      "reflection_quotient_status": "infinite conditional on cited Nikulin classification"
    }
  ],
  "classification_candidates": [
    {
      "name": "U+A3",
      "rank": 5,
      "determinant": -4,
      "smith_factors": [
        4
      ]
    },
    {
      "name": "U+A1+A2",
      "rank": 5,
      "determinant": -6,
      "smith_factors": [
        6
      ]
    },
    {
      "name": "U+A1+2A2",
      "rank": 7,
      "determinant": -18,
      "smith_factors": [
        3,
        6
      ]
    },
    {
      "name": "U+2A1+A3",
      "rank": 7,
      "determinant": -16,
      "smith_factors": [
        2,
        2,
        4
      ]
    },
    {
      "name": "U+A2+A3",
      "rank": 7,
      "determinant": -12,
      "smith_factors": [
        12
      ]
    },
    {
      "name": "U+A1+A4",
      "rank": 7,
      "determinant": -10,
      "smith_factors": [
        10
      ]
    },
    {
      "name": "U+A5",
      "rank": 7,
      "determinant": -6,
      "smith_factors": [
        6
      ]
    },
    {
      "name": "U+D5",
      "rank": 7,
      "determinant": -4,
      "smith_factors": [
        4
      ]
    },
    {
      "name": "U+A7",
      "rank": 9,
      "determinant": -8,
      "smith_factors": [
        8
      ]
    },
    {
      "name": "U+A3+D4",
      "rank": 9,
      "determinant": -16,
      "smith_factors": [
        2,
        2,
        4
      ]
    },
    {
      "name": "U+A2+D5",
      "rank": 9,
      "determinant": -12,
      "smith_factors": [
        12
      ]
    },
    {
      "name": "U+D7",
      "rank": 9,
      "determinant": -4,
      "smith_factors": [
        4
      ]
    },
    {
      "name": "U+A1+E6",
      "rank": 9,
      "determinant": -6,
      "smith_factors": [
        6
      ]
    },
    {
      "name": "U+A1+E8 (extra 2-elementary candidate)",
      "rank": 11,
      "determinant": -2,
      "smith_factors": [
        2
      ]
    },
    {
      "name": "U+A3+E8",
      "rank": 13,
      "determinant": -4,
      "smith_factors": [
        4
      ]
    },
    {
      "name": "U(4)+3A1",
      "rank": 5,
      "determinant": -128,
      "smith_factors": [
        2,
        2,
        2,
        4,
        4
      ]
    },
    {
      "name": "<-6>+2A2",
      "rank": 5,
      "determinant": -54,
      "smith_factors": [
        3,
        3,
        6
      ]
    },
    {
      "name": "<-4>+D4",
      "rank": 5,
      "determinant": -16,
      "smith_factors": [
        2,
        2,
        4
      ]
    },
    {
      "name": "<-8>+D4",
      "rank": 5,
      "determinant": -32,
      "smith_factors": [
        2,
        2,
        8
      ]
    },
    {
      "name": "<-16>+D4",
      "rank": 5,
      "determinant": -64,
      "smith_factors": [
        2,
        2,
        16
      ]
    }
  ],
  "n5_family2_isometry": {
    "gram": [
      [
        -2,
        -3,
        -4,
        -5,
        -6
      ],
      [
        -3,
        -2,
        -3,
        -4,
        -5
      ],
      [
        -4,
        -3,
        -2,
        -3,
        -4
      ],
      [
        -5,
        -4,
        -3,
        -2,
        -3
      ],
      [
        -6,
        -5,
        -4,
        -3,
        -2
      ]
    ],
    "basis_change": [
      [
        -5,
        -1,
        -1,
        1,
        1
      ],
      [
        8,
        1,
        2,
        -2,
        -1
      ],
      [
        0,
        0,
        0,
        1,
        -1
      ],
      [
        0,
        0,
        0,
        0,
        1
      ],
      [
        3,
        1,
        0,
        -1,
        -1
      ]
    ],
    "basis_change_determinant": -1,
    "transformed_gram": [
      [
        -16,
        0,
        0,
        0,
        0
      ],
      [
        0,
        2,
        -1,
        -1,
        -1
      ],
      [
        0,
        -1,
        2,
        0,
        0
      ],
      [
        0,
        -1,
        0,
        2,
        0
      ],
      [
        0,
        -1,
        0,
        0,
        2
      ]
    ],
    "target": "<-16> orthogonal_sum D4",
    "implication": "The lattice is reflective by Nikulin 1979 Theorem 4. The infinite-reflection-quotient certificate fails; thinness of this hypergeometric subgroup remains unresolved."
  },
  "uniform_scope": "Uniform arguments are supplied in headline-audit.md; the finite cases alone do not prove all n."
}