{
  "status": "Exact conditional construction; no empirical fitting; not a full seed-only derivation",
  "construction": "L_ij=(e_i-s_ij*e_j)(e_i-s_ij*e_j)^T; G_ij=i*(3*L_ij-2*I)",
  "triangle_sign_census": [
    {
      "edge_signs": [
        -1,
        -1,
        -1
      ],
      "loop_holonomy": -1,
      "traceless_lie_dimension": 8
    },
    {
      "edge_signs": [
        -1,
        -1,
        1
      ],
      "loop_holonomy": 1,
      "traceless_lie_dimension": 4
    },
    {
      "edge_signs": [
        -1,
        1,
        -1
      ],
      "loop_holonomy": 1,
      "traceless_lie_dimension": 4
    },
    {
      "edge_signs": [
        -1,
        1,
        1
      ],
      "loop_holonomy": -1,
      "traceless_lie_dimension": 8
    },
    {
      "edge_signs": [
        1,
        -1,
        -1
      ],
      "loop_holonomy": 1,
      "traceless_lie_dimension": 4
    },
    {
      "edge_signs": [
        1,
        -1,
        1
      ],
      "loop_holonomy": -1,
      "traceless_lie_dimension": 8
    },
    {
      "edge_signs": [
        1,
        1,
        -1
      ],
      "loop_holonomy": -1,
      "traceless_lie_dimension": 8
    },
    {
      "edge_signs": [
        1,
        1,
        1
      ],
      "loop_holonomy": 1,
      "traceless_lie_dimension": 4
    }
  ],
  "color": {
    "dimension": 8,
    "algebra": "su(3)",
    "generators": [
      {
        "real": [
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ],
        "imag": [
          [
            1,
            3,
            0
          ],
          [
            3,
            1,
            0
          ],
          [
            0,
            0,
            -2
          ]
        ]
      },
      {
        "real": [
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ],
        "imag": [
          [
            -2,
            0,
            0
          ],
          [
            0,
            1,
            3
          ],
          [
            0,
            3,
            1
          ]
        ]
      },
      {
        "real": [
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ],
        "imag": [
          [
            1,
            0,
            3
          ],
          [
            0,
            -2,
            0
          ],
          [
            3,
            0,
            1
          ]
        ]
      }
    ],
    "basis": [
      {
        "real": [
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ],
        "imag": [
          [
            1,
            3,
            0
          ],
          [
            3,
            1,
            0
          ],
          [
            0,
            0,
            -2
          ]
        ]
      },
      {
        "real": [
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ],
        "imag": [
          [
            -2,
            0,
            0
          ],
          [
            0,
            1,
            3
          ],
          [
            0,
            3,
            1
          ]
        ]
      },
      {
        "real": [
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ],
        "imag": [
          [
            1,
            0,
            3
          ],
          [
            0,
            -2,
            0
          ],
          [
            3,
            0,
            1
          ]
        ]
      },
      {
        "real": [
          [
            0,
            -1,
            -1
          ],
          [
            1,
            0,
            -1
          ],
          [
            1,
            1,
            0
          ]
        ],
        "imag": [
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ]
      },
      {
        "real": [
          [
            0,
            1,
            -1
          ],
          [
            -1,
            0,
            -1
          ],
          [
            1,
            1,
            0
          ]
        ],
        "imag": [
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ]
      },
      {
        "real": [
          [
            0,
            1,
            1
          ],
          [
            -1,
            0,
            -1
          ],
          [
            -1,
            1,
            0
          ]
        ],
        "imag": [
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ]
      },
      {
        "real": [
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ],
        "imag": [
          [
            -1,
            0,
            1
          ],
          [
            0,
            1,
            1
          ],
          [
            1,
            1,
            0
          ]
        ]
      },
      {
        "real": [
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ],
        "imag": [
          [
            -1,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            1
          ]
        ]
      }
    ],
    "basis_commutator_words": [
      "g0",
      "g1",
      "g2",
      "[g0,g1]",
      "[g0,g2]",
      "[g1,g2]",
      "[[g0,g1],g0]",
      "[[g0,g1],g2]"
    ],
    "raw_coordinate_certificate": [
      [
        1,
        1,
        0,
        3,
        0,
        0,
        0,
        0
      ],
      [
        -2,
        1,
        0,
        0,
        0,
        0,
        0,
        3
      ],
      [
        1,
        -2,
        0,
        0,
        0,
        3,
        0,
        0
      ],
      [
        0,
        0,
        -9,
        0,
        -9,
        0,
        -9,
        0
      ],
      [
        0,
        0,
        9,
        0,
        -9,
        0,
        -9,
        0
      ],
      [
        0,
        0,
        9,
        0,
        9,
        0,
        -9,
        0
      ],
      [
        -54,
        54,
        0,
        0,
        0,
        54,
        0,
        54
      ],
      [
        -54,
        0,
        0,
        0,
        0,
        0,
        0,
        0
      ]
    ],
    "raw_certificate_determinant": 306110016
  },
  "weak": {
    "dimension": 3,
    "algebra": "su(2)",
    "generators": "iX, iZ",
    "one_edge_only_dimension": 1
  },
  "factor_local": {
    "dimension_without_center": 11,
    "with_one_center": 12,
    "algebra": "su(3) + su(2) + u(1)",
    "one_mixed_generator_dimension": 35
  },
  "negative_triangle_spectrum": [
    1,
    1,
    4
  ],
  "same_seed_six_cycle_lift": {
    "common_clock": "T: k -> k+1 mod 6; T^3=M, T^6=I",
    "CRT_role_mirror": [
      3,
      4,
      5,
      0,
      1,
      2
    ],
    "closure_negation": [
      0,
      5,
      4,
      3,
      2,
      1
    ],
    "operators_are_distinct": true,
    "mirror_even_block": {
      "real": [
        [
          2,
          -1,
          -1
        ],
        [
          -1,
          2,
          -1
        ],
        [
          -1,
          -1,
          2
        ]
      ],
      "imag": [
        [
          0,
          0,
          0
        ],
        [
          0,
          0,
          0
        ],
        [
          0,
          0,
          0
        ]
      ]
    },
    "mirror_odd_block": {
      "real": [
        [
          2,
          -1,
          1
        ],
        [
          -1,
          2,
          -1
        ],
        [
          1,
          -1,
          2
        ]
      ],
      "imag": [
        [
          0,
          0,
          0
        ],
        [
          0,
          0,
          0
        ],
        [
          0,
          0,
          0
        ]
      ]
    },
    "mirror_even_spectrum": [
      0,
      3,
      3
    ],
    "mirror_odd_spectrum": [
      1,
      1,
      4
    ],
    "bridge": "odd block is switching-equivalent to the inversion triangle",
    "scope": "collective parity modes across the two roles, not separate spatial and temporal coordinate sets"
  },
  "gauge_symmetry_guard": {
    "uniform_generator_alone_dimension": 1,
    "all_color_generators_commute_with_uniform_energy": false,
    "uniform_energy_commutant_u3_dimension": 5,
    "scope": "A dynamical/operator Lie algebra is not yet a local gauge symmetry. Gauge-covariant link dynamics remains to derive."
  },
  "tensor_representation_center": {
    "kernel": [
      [
        0,
        0,
        0
      ],
      [
        0,
        1,
        3
      ],
      [
        1,
        0,
        4
      ],
      [
        1,
        1,
        1
      ],
      [
        2,
        0,
        2
      ],
      [
        2,
        1,
        5
      ]
    ],
    "cyclic_generator": [
      1,
      1,
      1
    ],
    "quotient": "(SU(3) x SU(2) x U(1))/Z6 for this representation",
    "hypercharge_integer_congruence": "2*color_triality + 3*weak_parity + q = 0 mod 6",
    "charge_residues": {
      "color_doublet": 1,
      "color_singlet": 4,
      "sterile_doublet": 3,
      "sterile_singlet": 0
    },
    "scope": "Congruences only; integer winding shifts q -> q+6m remain, chirality not derived"
  },
  "assumptions_to_derive": [
    "complex amplitude space / nontrivial sign representation of inversion",
    "rank-one edge mismatch energy and commutator closure",
    "independent availability of edge generators, not just their uniform sum",
    "binary contrast observable in addition to binary exchange",
    "factor-local gauge action",
    "one central phase generator and its physical role"
  ],
  "checks": [
    "integer/Fraction arithmetic only",
    "all 8 triangle signings",
    "all pairwise bracket closure",
    "containment of a standard su(3) basis",
    "mixed-generator countercheck",
    "exact triangle polynomial"
  ]
}