{
  "arithmetic": "exact Python integers and modular residues",
  "root": [
    -11,
    21,
    24,
    28
  ],
  "generators": "S1S2, S2S3, S3S4 (rightmost reflection acts first)",
  "orbit_checks": [
    {
      "p": 5,
      "vector_stabilizer_index": 144,
      "nonzero_null_vectors": 144,
      "projective_orbit_size": 36,
      "p_squared": 25,
      "first_coordinate_fiber_counts": {
        "0": 24,
        "1": 30,
        "2": 30,
        "3": 30,
        "4": 30
      },
      "finite_field_formula": 144,
      "comparison_only": "Finite checks; the all-prime claim requires a separate proof."
    },
    {
      "p": 7,
      "vector_stabilizer_index": 300,
      "nonzero_null_vectors": 300,
      "projective_orbit_size": 50,
      "p_squared": 49,
      "first_coordinate_fiber_counts": {
        "0": 48,
        "1": 42,
        "2": 42,
        "3": 42,
        "4": 42,
        "5": 42,
        "6": 42
      },
      "finite_field_formula": 300,
      "comparison_only": "Finite checks; the all-prime claim requires a separate proof."
    },
    {
      "p": 11,
      "vector_stabilizer_index": 1220,
      "nonzero_null_vectors": 1220,
      "projective_orbit_size": 122,
      "p_squared": 121,
      "first_coordinate_fiber_counts": {
        "0": 120,
        "1": 110,
        "2": 110,
        "3": 110,
        "4": 110,
        "5": 110,
        "6": 110,
        "7": 110,
        "8": 110,
        "9": 110,
        "10": 110
      },
      "finite_field_formula": 1220,
      "comparison_only": "Finite checks; the all-prime claim requires a separate proof."
    },
    {
      "p": 13,
      "vector_stabilizer_index": 2352,
      "nonzero_null_vectors": 2352,
      "projective_orbit_size": 196,
      "p_squared": 169,
      "first_coordinate_fiber_counts": {
        "0": 168,
        "1": 182,
        "2": 182,
        "3": 182,
        "4": 182,
        "5": 182,
        "6": 182,
        "7": 182,
        "8": 182,
        "9": 182,
        "10": 182,
        "11": 182,
        "12": 182
      },
      "finite_field_formula": 2352,
      "comparison_only": "Finite checks; the all-prime claim requires a separate proof."
    },
    {
      "p": 17,
      "vector_stabilizer_index": 5184,
      "nonzero_null_vectors": 5184,
      "projective_orbit_size": 324,
      "p_squared": 289,
      "first_coordinate_fiber_counts": {
        "0": 288,
        "1": 306,
        "2": 306,
        "3": 306,
        "4": 306,
        "5": 306,
        "6": 306,
        "7": 306,
        "8": 306,
        "9": 306,
        "10": 306,
        "11": 306,
        "12": 306,
        "13": 306,
        "14": 306,
        "15": 306,
        "16": 306
      },
      "finite_field_formula": 5184,
      "comparison_only": "Finite checks; the all-prime claim requires a separate proof."
    },
    {
      "p": 19,
      "vector_stabilizer_index": 6516,
      "nonzero_null_vectors": 6516,
      "projective_orbit_size": 362,
      "p_squared": 361,
      "first_coordinate_fiber_counts": {
        "0": 360,
        "1": 342,
        "2": 342,
        "3": 342,
        "4": 342,
        "5": 342,
        "6": 342,
        "7": 342,
        "8": 342,
        "9": 342,
        "10": 342,
        "11": 342,
        "12": 342,
        "13": 342,
        "14": 342,
        "15": 342,
        "16": 342,
        "17": 342,
        "18": 342
      },
      "finite_field_formula": 6516,
      "comparison_only": "Finite checks; the all-prime claim requires a separate proof."
    }
  ],
  "group_check_mod_7": {
    "image_order": 58800,
    "minus_identity_in_image": false,
    "SO_order_lower_bound_by_adjoining_central_minus_I": 117600,
    "minus_identity_preserves_form": true,
    "minus_identity_determinant": 1,
    "conclusion": "Image has 58800 elements; adjoining the missing central -I yields 117600 distinct form-preserving determinant-one matrices. Thus full SO_F(F7) has strictly larger order than the printed quotient, independently of a classification theorem."
  }
}