{
  "construction": "T(i,b)=(i+1 mod 3,b XOR 1); D=I-T; L=D^T D",
  "full_clock": [
    [
      0,
      0,
      0,
      1,
      0,
      0
    ],
    [
      0,
      0,
      1,
      0,
      0,
      0
    ],
    [
      0,
      0,
      0,
      0,
      0,
      1
    ],
    [
      0,
      0,
      0,
      0,
      1,
      0
    ],
    [
      0,
      1,
      0,
      0,
      0,
      0
    ],
    [
      1,
      0,
      0,
      0,
      0,
      0
    ]
  ],
  "full_mismatch_operator": [
    [
      2,
      0,
      0,
      -1,
      0,
      -1
    ],
    [
      0,
      2,
      -1,
      0,
      -1,
      0
    ],
    [
      0,
      -1,
      2,
      0,
      0,
      -1
    ],
    [
      -1,
      0,
      0,
      2,
      -1,
      0
    ],
    [
      0,
      -1,
      0,
      -1,
      2,
      0
    ],
    [
      -1,
      0,
      -1,
      0,
      0,
      2
    ]
  ],
  "pair_even_embedding": [
    [
      1,
      0,
      0
    ],
    [
      1,
      0,
      0
    ],
    [
      0,
      1,
      0
    ],
    [
      0,
      1,
      0
    ],
    [
      0,
      0,
      1
    ],
    [
      0,
      0,
      1
    ]
  ],
  "pair_odd_embedding": [
    [
      1,
      0,
      0
    ],
    [
      -1,
      0,
      0
    ],
    [
      0,
      1,
      0
    ],
    [
      0,
      -1,
      0
    ],
    [
      0,
      0,
      1
    ],
    [
      0,
      0,
      -1
    ]
  ],
  "even_block": [
    [
      2,
      -1,
      -1
    ],
    [
      -1,
      2,
      -1
    ],
    [
      -1,
      -1,
      2
    ]
  ],
  "odd_block": [
    [
      2,
      1,
      1
    ],
    [
      1,
      2,
      1
    ],
    [
      1,
      1,
      2
    ]
  ],
  "odd_local_edge_blocks": [
    [
      [
        1,
        1,
        0
      ],
      [
        1,
        1,
        0
      ],
      [
        0,
        0,
        0
      ]
    ],
    [
      [
        0,
        0,
        0
      ],
      [
        0,
        1,
        1
      ],
      [
        0,
        1,
        1
      ]
    ],
    [
      [
        1,
        0,
        1
      ],
      [
        0,
        0,
        0
      ],
      [
        1,
        0,
        1
      ]
    ]
  ],
  "even_spectrum": [
    0,
    3,
    3
  ],
  "odd_spectrum": [
    1,
    1,
    4
  ],
  "all_triangle_signings": [
    {
      "edge_flips": [
        0,
        0,
        0
      ],
      "odd_block": [
        [
          2,
          -1,
          -1
        ],
        [
          -1,
          2,
          -1
        ],
        [
          -1,
          -1,
          2
        ]
      ],
      "kernel_dimension": 1
    },
    {
      "edge_flips": [
        0,
        0,
        1
      ],
      "odd_block": [
        [
          2,
          -1,
          1
        ],
        [
          -1,
          2,
          -1
        ],
        [
          1,
          -1,
          2
        ]
      ],
      "kernel_dimension": 0
    },
    {
      "edge_flips": [
        0,
        1,
        0
      ],
      "odd_block": [
        [
          2,
          -1,
          -1
        ],
        [
          -1,
          2,
          1
        ],
        [
          -1,
          1,
          2
        ]
      ],
      "kernel_dimension": 0
    },
    {
      "edge_flips": [
        0,
        1,
        1
      ],
      "odd_block": [
        [
          2,
          -1,
          1
        ],
        [
          -1,
          2,
          1
        ],
        [
          1,
          1,
          2
        ]
      ],
      "kernel_dimension": 1
    },
    {
      "edge_flips": [
        1,
        0,
        0
      ],
      "odd_block": [
        [
          2,
          1,
          -1
        ],
        [
          1,
          2,
          -1
        ],
        [
          -1,
          -1,
          2
        ]
      ],
      "kernel_dimension": 0
    },
    {
      "edge_flips": [
        1,
        0,
        1
      ],
      "odd_block": [
        [
          2,
          1,
          1
        ],
        [
          1,
          2,
          -1
        ],
        [
          1,
          -1,
          2
        ]
      ],
      "kernel_dimension": 1
    },
    {
      "edge_flips": [
        1,
        1,
        0
      ],
      "odd_block": [
        [
          2,
          1,
          -1
        ],
        [
          1,
          2,
          1
        ],
        [
          -1,
          1,
          2
        ]
      ],
      "kernel_dimension": 1
    },
    {
      "edge_flips": [
        1,
        1,
        1
      ],
      "odd_block": [
        [
          2,
          1,
          1
        ],
        [
          1,
          2,
          1
        ],
        [
          1,
          1,
          2
        ]
      ],
      "kernel_dimension": 0
    }
  ],
  "operator_space_dimensions": {
    "clock_commuting_real": 6,
    "clock_commuting_real_symmetric": 4,
    "also_nearest_neighbor": 2,
    "also_annihilates_constant": 1
  },
  "contrast_clock": [
    [
      0,
      -1,
      0
    ],
    [
      0,
      0,
      -1
    ],
    [
      -1,
      0,
      0
    ]
  ],
  "rotation_plane_K": [
    [
      -1,
      -2,
      0
    ],
    [
      0,
      -1,
      -2
    ],
    [
      -2,
      0,
      -1
    ]
  ],
  "rotation_identity": "K^2=-3I+4*ones; on sum(x)=0, J=K/sqrt(3) obeys J^2=-I",
  "conclusions": [
    "The odd contrast f(i,b)=(-1)^b*x_i obtains its sign from seed-pole exchange.",
    "Ordinary clock differences restrict to x_i+x_j; this reproduces the prior signed edge operators.",
    "Local real symmetric clock-stationary operators killing constants form one scale family: a*L.",
    "The real contrast plane supports one intrinsic complex structure; this is not yet a three-complex-dimensional color state space."
  ],
  "scope": [
    "Scalar readouts and the squared-difference observable are the representation under investigation.",
    "Locality, real symmetry, stationarity and the zero-constant condition specify the uniqueness class.",
    "No law of physical expansion, splitting rate, quantum amplitudes, or physical gauge action is claimed by these identities.",
    "A symmetric scalar function of L cannot equal the directed clock: L has paired opposite frequencies while T distinguishes them."
  ]
}