{
  "scope": "All exceptional finite types; classical ranks through 8; D4 triality; A1+A1 and A2+A2 component exchanges; rank zero",
  "arithmetic": "Exact Python integers and Fraction Gaussian elimination; no floating point",
  "convention": "a_ij=alpha_i(h_j); internal indices 0-based, report vertices 1-based",
  "method": "Enumerate involutive directed Cartan automorphisms and invariant black subsets; compute opposition and H_X; test fixed white parity; quotient by all directed diagram automorphisms",
  "proof_dependency": "RT-LIE-17 Section8.5 Definition8.4, RT-LIE-08 Lemma4.1 and length/chamber Theorem5.1",
  "not_a_proof_replacement": "Existence, maximal-split reconstruction, completeness and real-isomorphism injectivity are proved universally in RT-LIE-17 Section8.5, not inferred from these counts",
  "checks_passed": true,
  "types_checked": 35,
  "generator_sha256": "1622742f0a0de5b9788bc614aa7efe5c05059fd15460c237331b07934929c0d4",
  "results": [
    {
      "label": "A1",
      "column_coroot_cartan": [
        [
          2
        ]
      ],
      "rank": 1,
      "full_roots_count": 2,
      "diagram_automorphisms": 1,
      "diagram_involutions": 1,
      "admissible_pairs_before_diagram_isomorphism": 2,
      "isomorphism_classes": 2,
      "expected_count_check": 2,
      "rejected_opposition_pairs": 0,
      "rejected_fixed_white_parity_pairs": 0,
      "representatives": [
        {
          "black_vertices": [],
          "tau": [
            1
          ],
          "white_orbits": [
            [
              1
            ]
          ],
          "black_opposition": {},
          "H_X_simple_coroot_coefficients": {},
          "white_alpha_H_X": {
            "1": 0
          },
          "split_rank": 1,
          "black_roots_count": 0,
          "compact_fixed_dimension": 1,
          "killing_signature_positive_negative": [
            2,
            1
          ]
        },
        {
          "black_vertices": [
            1
          ],
          "tau": [
            1
          ],
          "white_orbits": [],
          "black_opposition": {
            "1": 1
          },
          "H_X_simple_coroot_coefficients": {
            "1": 1
          },
          "white_alpha_H_X": {},
          "split_rank": 0,
          "black_roots_count": 2,
          "compact_fixed_dimension": 3,
          "killing_signature_positive_negative": [
            0,
            3
          ]
        }
      ]
    },
    {
      "label": "A2",
      "column_coroot_cartan": [
        [
          2,
          -1
        ],
        [
          -1,
          2
        ]
      ],
      "rank": 2,
      "full_roots_count": 6,
      "diagram_automorphisms": 2,
      "diagram_involutions": 2,
      "admissible_pairs_before_diagram_isomorphism": 3,
      "isomorphism_classes": 3,
      "expected_count_check": 3,
      "rejected_opposition_pairs": 1,
      "rejected_fixed_white_parity_pairs": 2,
      "representatives": [
        {
          "black_vertices": [],
          "tau": [
            1,
            2
          ],
          "white_orbits": [
            [
              1
            ],
            [
              2
            ]
          ],
          "black_opposition": {},
          "H_X_simple_coroot_coefficients": {},
          "white_alpha_H_X": {
            "1": 0,
            "2": 0
          },
          "split_rank": 2,
          "black_roots_count": 0,
          "compact_fixed_dimension": 3,
          "killing_signature_positive_negative": [
            5,
            3
          ]
        },
        {
          "black_vertices": [],
          "tau": [
            2,
            1
          ],
          "white_orbits": [
            [
              1,
              2
            ]
          ],
          "black_opposition": {},
          "H_X_simple_coroot_coefficients": {},
          "white_alpha_H_X": {
            "1": 0,
            "2": 0
          },
          "split_rank": 1,
          "black_roots_count": 0,
          "compact_fixed_dimension": 4,
          "killing_signature_positive_negative": [
            4,
            4
          ]
        },
        {
          "black_vertices": [
            1,
            2
          ],
          "tau": [
            2,
            1
          ],
          "white_orbits": [],
          "black_opposition": {
            "1": 2,
            "2": 1
          },
          "H_X_simple_coroot_coefficients": {
            "1": 2,
            "2": 2
          },
          "white_alpha_H_X": {},
          "split_rank": 0,
          "black_roots_count": 6,
          "compact_fixed_dimension": 8,
          "killing_signature_positive_negative": [
            0,
            8
          ]
        }
      ]
    },
    {
      "label": "A3",
      "column_coroot_cartan": [
        [
          2,
          -1,
          0
        ],
        [
          -1,
          2,
          -1
        ],
        [
          0,
          -1,
          2
        ]
      ],
      "rank": 3,
      "full_roots_count": 12,
      "diagram_automorphisms": 2,
      "diagram_involutions": 2,
      "admissible_pairs_before_diagram_isomorphism": 5,
      "isomorphism_classes": 5,
      "expected_count_check": 5,
      "rejected_opposition_pairs": 4,
      "rejected_fixed_white_parity_pairs": 3,
      "representatives": [
        {
          "black_vertices": [],
          "tau": [
            1,
            2,
            3
          ],
          "white_orbits": [
            [
              1
            ],
            [
              2
            ],
            [
              3
            ]
          ],
          "black_opposition": {},
          "H_X_simple_coroot_coefficients": {},
          "white_alpha_H_X": {
            "1": 0,
            "2": 0,
            "3": 0
          },
          "split_rank": 3,
          "black_roots_count": 0,
          "compact_fixed_dimension": 6,
          "killing_signature_positive_negative": [
            9,
            6
          ]
        },
        {
          "black_vertices": [],
          "tau": [
            3,
            2,
            1
          ],
          "white_orbits": [
            [
              1,
              3
            ],
            [
              2
            ]
          ],
          "black_opposition": {},
          "H_X_simple_coroot_coefficients": {},
          "white_alpha_H_X": {
            "1": 0,
            "2": 0,
            "3": 0
          },
          "split_rank": 2,
          "black_roots_count": 0,
          "compact_fixed_dimension": 7,
          "killing_signature_positive_negative": [
            8,
            7
          ]
        },
        {
          "black_vertices": [
            1,
            2,
            3
          ],
          "tau": [
            3,
            2,
            1
          ],
          "white_orbits": [],
          "black_opposition": {
            "1": 3,
            "2": 2,
            "3": 1
          },
          "H_X_simple_coroot_coefficients": {
            "1": 3,
            "2": 4,
            "3": 3
          },
          "white_alpha_H_X": {},
          "split_rank": 0,
          "black_roots_count": 12,
          "compact_fixed_dimension": 15,
          "killing_signature_positive_negative": [
            0,
            15
          ]
        },
        {
          "black_vertices": [
            1,
            3
          ],
          "tau": [
            1,
            2,
            3
          ],
          "white_orbits": [
            [
              2
            ]
          ],
          "black_opposition": {
            "1": 1,
            "3": 3
          },
          "H_X_simple_coroot_coefficients": {
            "1": 1,
            "3": 1
          },
          "white_alpha_H_X": {
            "2": -2
          },
          "split_rank": 1,
          "black_roots_count": 4,
          "compact_fixed_dimension": 10,
          "killing_signature_positive_negative": [
            5,
            10
          ]
        },
        {
          "black_vertices": [
            2
          ],
          "tau": [
            3,
            2,
            1
          ],
          "white_orbits": [
            [
              1,
              3
            ]
          ],
          "black_opposition": {
            "2": 2
          },
          "H_X_simple_coroot_coefficients": {
            "2": 1
          },
          "white_alpha_H_X": {
            "1": -1,
            "3": -1
          },
          "split_rank": 1,
          "black_roots_count": 2,
          "compact_fixed_dimension": 9,
          "killing_signature_positive_negative": [
            6,
            9
          ]
        }
      ]
    },
    {
      "label": "A4",
      "column_coroot_cartan": [
        [
          2,
          -1,
          0,
          0
        ],
        [
          -1,
          2,
          -1,
          0
        ],
        [
          0,
          -1,
          2,
          -1
        ],
        [
          0,
          0,
          -1,
          2
        ]
      ],
      "rank": 4,
      "full_roots_count": 20,
      "diagram_automorphisms": 2,
      "diagram_involutions": 2,
      "admissible_pairs_before_diagram_isomorphism": 4,
      "isomorphism_classes": 4,
      "expected_count_check": 4,
      "rejected_opposition_pairs": 9,
      "rejected_fixed_white_parity_pairs": 7,
      "representatives": [
        {
          "black_vertices": [],
          "tau": [
            1,
            2,
            3,
            4
          ],
          "white_orbits": [
            [
              1
            ],
            [
              2
            ],
            [
              3
            ],
            [
              4
            ]
          ],
          "black_opposition": {},
          "H_X_simple_coroot_coefficients": {},
          "white_alpha_H_X": {
            "1": 0,
            "2": 0,
            "3": 0,
            "4": 0
          },
          "split_rank": 4,
          "black_roots_count": 0,
          "compact_fixed_dimension": 10,
          "killing_signature_positive_negative": [
            14,
            10
          ]
        },
        {
          "black_vertices": [],
          "tau": [
            4,
            3,
            2,
            1
          ],
          "white_orbits": [
            [
              1,
              4
            ],
            [
              2,
              3
            ]
          ],
          "black_opposition": {},
          "H_X_simple_coroot_coefficients": {},
          "white_alpha_H_X": {
            "1": 0,
            "2": 0,
            "3": 0,
            "4": 0
          },
          "split_rank": 2,
          "black_roots_count": 0,
          "compact_fixed_dimension": 12,
          "killing_signature_positive_negative": [
            12,
            12
          ]
        },
        {
          "black_vertices": [
            1,
            2,
            3,
            4
          ],
          "tau": [
            4,
            3,
            2,
            1
          ],
          "white_orbits": [],
          "black_opposition": {
            "1": 4,
            "2": 3,
            "3": 2,
            "4": 1
          },
          "H_X_simple_coroot_coefficients": {
            "1": 4,
            "2": 6,
            "3": 6,
            "4": 4
          },
          "white_alpha_H_X": {},
          "split_rank": 0,
          "black_roots_count": 20,
          "compact_fixed_dimension": 24,
          "killing_signature_positive_negative": [
            0,
            24
          ]
        },
        {
          "black_vertices": [
            2,
            3
          ],
          "tau": [
            4,
            3,
            2,
            1
          ],
          "white_orbits": [
            [
              1,
              4
            ]
          ],
          "black_opposition": {
            "2": 3,
            "3": 2
          },
          "H_X_simple_coroot_coefficients": {
            "2": 2,
            "3": 2
          },
          "white_alpha_H_X": {
            "1": -2,
            "4": -2
          },
          "split_rank": 1,
          "black_roots_count": 6,
          "compact_fixed_dimension": 16,
          "killing_signature_positive_negative": [
            8,
            16
          ]
        }
      ]
    },
    {
      "label": "A5",
      "column_coroot_cartan": [
        [
          2,
          -1,
          0,
          0,
          0
        ],
        [
          -1,
          2,
          -1,
          0,
          0
        ],
        [
          0,
          -1,
          2,
          -1,
          0
        ],
        [
          0,
          0,
          -1,
          2,
          -1
        ],
        [
          0,
          0,
          0,
          -1,
          2
        ]
      ],
      "rank": 5,
      "full_roots_count": 30,
      "diagram_automorphisms": 2,
      "diagram_involutions": 2,
      "admissible_pairs_before_diagram_isomorphism": 6,
      "isomorphism_classes": 6,
      "expected_count_check": 6,
      "rejected_opposition_pairs": 23,
      "rejected_fixed_white_parity_pairs": 11,
      "representatives": [
        {
          "black_vertices": [],
          "tau": [
            1,
            2,
            3,
            4,
            5
          ],
          "white_orbits": [
            [
              1
            ],
            [
              2
            ],
            [
              3
            ],
            [
              4
            ],
            [
              5
            ]
          ],
          "black_opposition": {},
          "H_X_simple_coroot_coefficients": {},
          "white_alpha_H_X": {
            "1": 0,
            "2": 0,
            "3": 0,
            "4": 0,
            "5": 0
          },
          "split_rank": 5,
          "black_roots_count": 0,
          "compact_fixed_dimension": 15,
          "killing_signature_positive_negative": [
            20,
            15
          ]
        },
        {
          "black_vertices": [],
          "tau": [
            5,
            4,
            3,
            2,
            1
          ],
          "white_orbits": [
            [
              1,
              5
            ],
            [
              2,
              4
            ],
            [
              3
            ]
          ],
          "black_opposition": {},
          "H_X_simple_coroot_coefficients": {},
          "white_alpha_H_X": {
            "1": 0,
            "2": 0,
            "3": 0,
            "4": 0,
            "5": 0
          },
          "split_rank": 3,
          "black_roots_count": 0,
          "compact_fixed_dimension": 17,
          "killing_signature_positive_negative": [
            18,
            17
          ]
        },
        {
          "black_vertices": [
            1,
            2,
            3,
            4,
            5
          ],
          "tau": [
            5,
            4,
            3,
            2,
            1
          ],
          "white_orbits": [],
          "black_opposition": {
            "1": 5,
            "2": 4,
            "3": 3,
            "4": 2,
            "5": 1
          },
          "H_X_simple_coroot_coefficients": {
            "1": 5,
            "2": 8,
            "3": 9,
            "4": 8,
            "5": 5
          },
          "white_alpha_H_X": {},
          "split_rank": 0,
          "black_roots_count": 30,
          "compact_fixed_dimension": 35,
          "killing_signature_positive_negative": [
            0,
            35
          ]
        },
        {
          "black_vertices": [
            1,
            3,
            5
          ],
          "tau": [
            1,
            2,
            3,
            4,
            5
          ],
          "white_orbits": [
            [
              2
            ],
            [
              4
            ]
          ],
          "black_opposition": {
            "1": 1,
            "3": 3,
            "5": 5
          },
          "H_X_simple_coroot_coefficients": {
            "1": 1,
            "3": 1,
            "5": 1
          },
          "white_alpha_H_X": {
            "2": -2,
            "4": -2
          },
          "split_rank": 2,
          "black_roots_count": 6,
          "compact_fixed_dimension": 21,
          "killing_signature_positive_negative": [
            14,
            21
          ]
        },
        {
          "black_vertices": [
            2,
            3,
            4
          ],
          "tau": [
            5,
            4,
            3,
            2,
            1
          ],
          "white_orbits": [
            [
              1,
              5
            ]
          ],
          "black_opposition": {
            "2": 4,
            "3": 3,
            "4": 2
          },
          "H_X_simple_coroot_coefficients": {
            "2": 3,
            "3": 4,
            "4": 3
          },
          "white_alpha_H_X": {
            "1": -3,
            "5": -3
          },
          "split_rank": 1,
          "black_roots_count": 12,
          "compact_fixed_dimension": 25,
          "killing_signature_positive_negative": [
            10,
            25
          ]
        },
        {
          "black_vertices": [
            3
          ],
          "tau": [
            5,
            4,
            3,
            2,
            1
          ],
          "white_orbits": [
            [
              1,
              5
            ],
            [
              2,
              4
            ]
          ],
          "black_opposition": {
            "3": 3
          },
          "H_X_simple_coroot_coefficients": {
            "3": 1
          },
          "white_alpha_H_X": {
            "1": 0,
            "2": -1,
            "4": -1,
            "5": 0
          },
          "split_rank": 2,
          "black_roots_count": 2,
          "compact_fixed_dimension": 19,
          "killing_signature_positive_negative": [
            16,
            19
          ]
        }
      ]
    },
    {
      "label": "A6",
      "column_coroot_cartan": [
        [
          2,
          -1,
          0,
          0,
          0,
          0
        ],
        [
          -1,
          2,
          -1,
          0,
          0,
          0
        ],
        [
          0,
          -1,
          2,
          -1,
          0,
          0
        ],
        [
          0,
          0,
          -1,
          2,
          -1,
          0
        ],
        [
          0,
          0,
          0,
          -1,
          2,
          -1
        ],
        [
          0,
          0,
          0,
          0,
          -1,
          2
        ]
      ],
      "rank": 6,
      "full_roots_count": 42,
      "diagram_automorphisms": 2,
      "diagram_involutions": 2,
      "admissible_pairs_before_diagram_isomorphism": 5,
      "isomorphism_classes": 5,
      "expected_count_check": 5,
      "rejected_opposition_pairs": 47,
      "rejected_fixed_white_parity_pairs": 20,
      "representatives": [
        {
          "black_vertices": [],
          "tau": [
            1,
            2,
            3,
            4,
            5,
            6
          ],
          "white_orbits": [
            [
              1
            ],
            [
              2
            ],
            [
              3
            ],
            [
              4
            ],
            [
              5
            ],
            [
              6
            ]
          ],
          "black_opposition": {},
          "H_X_simple_coroot_coefficients": {},
          "white_alpha_H_X": {
            "1": 0,
            "2": 0,
            "3": 0,
            "4": 0,
            "5": 0,
            "6": 0
          },
          "split_rank": 6,
          "black_roots_count": 0,
          "compact_fixed_dimension": 21,
          "killing_signature_positive_negative": [
            27,
            21
          ]
        },
        {
          "black_vertices": [],
          "tau": [
            6,
            5,
            4,
            3,
            2,
            1
          ],
          "white_orbits": [
            [
              1,
              6
            ],
            [
              2,
              5
            ],
            [
              3,
              4
            ]
          ],
          "black_opposition": {},
          "H_X_simple_coroot_coefficients": {},
          "white_alpha_H_X": {
            "1": 0,
            "2": 0,
            "3": 0,
            "4": 0,
            "5": 0,
            "6": 0
          },
          "split_rank": 3,
          "black_roots_count": 0,
          "compact_fixed_dimension": 24,
          "killing_signature_positive_negative": [
            24,
            24
          ]
        },
        {
          "black_vertices": [
            1,
            2,
            3,
            4,
            5,
            6
          ],
          "tau": [
            6,
            5,
            4,
            3,
            2,
            1
          ],
          "white_orbits": [],
          "black_opposition": {
            "1": 6,
            "2": 5,
            "3": 4,
            "4": 3,
            "5": 2,
            "6": 1
          },
          "H_X_simple_coroot_coefficients": {
            "1": 6,
            "2": 10,
            "3": 12,
            "4": 12,
            "5": 10,
            "6": 6
          },
          "white_alpha_H_X": {},
          "split_rank": 0,
          "black_roots_count": 42,
          "compact_fixed_dimension": 48,
          "killing_signature_positive_negative": [
            0,
            48
          ]
        },
        {
          "black_vertices": [
            2,
            3,
            4,
            5
          ],
          "tau": [
            6,
            5,
            4,
            3,
            2,
            1
          ],
          "white_orbits": [
            [
              1,
              6
            ]
          ],
          "black_opposition": {
            "2": 5,
            "3": 4,
            "4": 3,
            "5": 2
          },
          "H_X_simple_coroot_coefficients": {
            "2": 4,
            "3": 6,
            "4": 6,
            "5": 4
          },
          "white_alpha_H_X": {
            "1": -4,
            "6": -4
          },
          "split_rank": 1,
          "black_roots_count": 20,
          "compact_fixed_dimension": 36,
          "killing_signature_positive_negative": [
            12,
            36
          ]
        },
        {
          "black_vertices": [
            3,
            4
          ],
          "tau": [
            6,
            5,
            4,
            3,
            2,
            1
          ],
          "white_orbits": [
            [
              1,
              6
            ],
            [
              2,
              5
            ]
          ],
          "black_opposition": {
            "3": 4,
            "4": 3
          },
          "H_X_simple_coroot_coefficients": {
            "3": 2,
            "4": 2
          },
          "white_alpha_H_X": {
            "1": 0,
            "2": -2,
            "5": -2,
            "6": 0
          },
          "split_rank": 2,
          "black_roots_count": 6,
          "compact_fixed_dimension": 28,
          "killing_signature_positive_negative": [
            20,
            28
          ]
        }
      ]
    },
    {
      "label": "A7",
      "column_coroot_cartan": [
        [
          2,
          -1,
          0,
          0,
          0,
          0,
          0
        ],
        [
          -1,
          2,
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