{
  "schema": "exceptional-ordered-trace-exact-certificate/v1",
  "arithmetic": "Rational polynomial reduction modulo the minimal cyclotomic polynomials",
  "row_generators": "Mixed first cup and mixed short-tip projection, together with N",
  "constraint_partition": "Output p2=0; output prefix xi; all other outputs",
  "basis": "Positive diagonal path metric; four endomorphisms of the suffix Wenzl range",
  "checker_sha256": "3C0A86A13A98374313DCACF7B467049F91512C7F6E1D94A217EBE865360B3EA8",
  "field_library_sha256": "809B40E838BDAC649274D2358B985112851B6787169468769099A4D032C0188E",
  "flatness_certificate_sha256": "F6615F05BE85FC368EEC0BF5EA5BCA736809ABCD17AB92005924529476AD3610",
  "results": [
    {
      "graph": "E6",
      "k": 3,
      "branch_endpoint": 2,
      "prefix_paths": [
        [
          0,
          1,
          0,
          1,
          2
        ],
        [
          0,
          1,
          2,
          1,
          2
        ],
        [
          0,
          1,
          2,
          3,
          2
        ],
        [
          0,
          1,
          2,
          5,
          2
        ]
      ],
      "suffix_Wenzl_rank": 2,
      "linear_constraint_rank": 3,
      "equations": 120,
      "all_equations_checked": true,
      "projection_trace": 1,
      "sigma": {
        "0,0": {
          "0": "2",
          "4": "-2",
          "12": "1"
        },
        "0,1": {
          "0": "-2",
          "4": "2",
          "12": "-1"
        },
        "0,2": {
          "0": "-1",
          "4": "1",
          "8": "1",
          "12": "-1"
        },
        "0,3": {
          "0": "3",
          "4": "-3",
          "8": "-1",
          "12": "2"
        },
        "1,0": {
          "0": "5/2",
          "4": "-3",
          "12": "3/2"
        },
        "1,1": {
          "0": "-5/2",
          "4": "3",
          "12": "-3/2"
        },
        "1,2": {
          "0": "-3/2",
          "4": "1",
          "8": "1",
          "12": "-3/2"
        },
        "1,3": {
          "0": "4",
          "4": "-4",
          "8": "-1",
          "12": "3"
        },
        "2,0": {
          "0": "1/2",
          "4": "-1",
          "8": "1",
          "12": "-1/2"
        },
        "2,1": {
          "0": "-1/2",
          "4": "1",
          "8": "-1",
          "12": "1/2"
        },
        "2,2": {
          "0": "-1/2",
          "4": "1",
          "12": "-1/2"
        },
        "2,3": {
          "0": "1",
          "4": "-2",
          "8": "1"
        },
        "3,0": {
          "0": "1",
          "4": "-3/2",
          "8": "1/2",
          "12": "1/2"
        },
        "3,1": {
          "0": "-1",
          "4": "3/2",
          "8": "-1/2",
          "12": "-1/2"
        },
        "3,2": {
          "0": "-1",
          "4": "1/2",
          "8": "1/2",
          "12": "-1/2"
        },
        "3,3": {
          "0": "2",
          "4": "-2",
          "12": "1"
        }
      },
      "ordered_bridge": "U1 U2 ... Uk",
      "inclusion_trace": {
        "2": "-11",
        "10": "22",
        "14": "-19"
      },
      "trace_minus_conjugate": {
        "2": "-11",
        "6": "11",
        "10": "3",
        "14": "-8"
      },
      "nonreal": true
    },
    {
      "graph": "E8",
      "k": 5,
      "branch_endpoint": 4,
      "prefix_paths": [
        [
          0,
          1,
          0,
          1,
          2,
          3,
          4
        ],
        [
          0,
          1,
          2,
          1,
          2,
          3,
          4
        ],
        [
          0,
          1,
          2,
          3,
          2,
          3,
          4
        ],
        [
          0,
          1,
          2,
          3,
          4,
          3,
          4
        ],
        [
          0,
          1,
          2,
          3,
          4,
          5,
          4
        ],
        [
          0,
          1,
          2,
          3,
          4,
          7,
          4
        ]
      ],
      "suffix_Wenzl_rank": 2,
      "linear_constraint_rank": 3,
      "equations": 6880,
      "all_equations_checked": true,
      "projection_trace": 1,
      "sigma": {
        "0,0": {
          "0": "4",
          "8": "-1",
          "12": "-2",
          "16": "-2",
          "20": "-1",
          "24": "1",
          "28": "3"
        },
        "0,1": {
          "0": "-4",
          "8": "1",
          "12": "2",
          "16": "2",
          "20": "1",
          "24": "-1",
          "28": "-3"
        },
        "0,2": {
          "0": "4",
          "8": "-1",
          "12": "-2",
          "16": "-2",
          "20": "-1",
          "24": "1",
          "28": "3"
        },
        "0,3": {
          "0": "-4",
          "8": "1",
          "12": "2",
          "16": "2",
          "20": "1",
          "24": "-1",
          "28": "-3"
        },
        "0,4": {
          "0": "-1",
          "4": "-1",
          "8": "1",
          "12": "1",
          "16": "1",
          "20": "1",
          "28": "-2"
        },
        "0,5": {
          "0": "5",
          "4": "1",
          "8": "-2",
          "12": "-3",
          "16": "-3",
          "20": "-2",
          "24": "1",
          "28": "5"
        },
        "1,0": {
          "0": "5",
          "4": "-1",
          "8": "-1",
          "12": "-2",
          "16": "-3",
          "20": "-1",
          "24": "2",
          "28": "3"
        },
        "1,1": {
          "0": "-5",
          "4": "1",
          "8": "1",
          "12": "2",
          "16": "3",
          "20": "1",
          "24": "-2",
          "28": "-3"
        },
        "1,2": {
          "0": "5",
          "4": "-1",
          "8": "-1",
          "12": "-2",
          "16": "-3",
          "20": "-1",
          "24": "2",
          "28": "3"
        },
        "1,3": {
          "0": "-5",
          "4": "1",
          "8": "1",
          "12": "2",
          "16": "3",
          "20": "1",
          "24": "-2",
          "28": "-3"
        },
        "1,4": {
          "0": "-2",
          "4": "-3",
          "8": "2",
          "12": "2",
          "16": "1",
          "20": "2",
          "28": "-4"
        },
        "1,5": {
          "0": "7",
          "4": "2",
          "8": "-3",
          "12": "-4",
          "16": "-4",
          "20": "-3",
          "24": "2",
          "28": "7"
        },
        "2,0": {
          "0": "3",
          "4": "-1",
          "8": "-1",
          "12": "-1",
          "16": "-1",
          "24": "1",
          "28": "1"
        },
        "2,1": {
          "0": "-3",
          "4": "1",
          "8": "1",
          "12": "1",
          "16": "1",
          "24": "-1",
          "28": "-1"
        },
        "2,2": {
          "0": "3",
          "4": "-1",
          "8": "-1",
          "12": "-1",
          "16": "-1",
          "24": "1",
          "28": "1"
        },
        "2,3": {
          "0": "-3",
          "4": "1",
          "8": "1",
          "12": "1",
          "16": "1",
          "24": "-1",
          "28": "-1"
        },
        "2,4": {
          "0": "-1",
          "4": "-2",
          "8": "1",
          "12": "1",
          "16": "1",
          "20": "2",
          "28": "-3"
        },
        "2,5": {
          "0": "4",
          "4": "1",
          "8": "-2",
          "12": "-2",
          "16": "-2",
          "20": "-2",
          "24": "1",
          "28": "4"
        },
        "3,0": {
          "0": "4",
          "4": "1",
          "8": "-2",
          "12": "-3",
          "16": "-1",
          "20": "-1",
          "28": "4"
        },
        "3,1": {
          "0": "-4",
          "4": "-1",
          "8": "2",
          "12": "3",
          "16": "1",
          "20": "1",
          "28": "-4"
        },
        "3,2": {
          "0": "4",
          "4": "1",
          "8": "-2",
          "12": "-3",
          "16": "-1",
          "20": "-1",
          "28": "4"
        },
        "3,3": {
          "0": "-4",
          "4": "-1",
          "8": "2",
          "12": "3",
          "16": "1",
          "20": "1",
          "28": "-4"
        },
        "3,4": {
          "4": "1",
          "8": "-1",
          "12": "-1",
          "16": "1"
        },
        "3,5": {
          "0": "4",
          "8": "-1",
          "12": "-2",
          "16": "-2",
          "20": "-1",
          "28": "4"
        },
        "4,0": {
          "0": "2",
          "12": "-2",
          "20": "-1",
          "28": "2"
        },
        "4,1": {
          "0": "-2",
          "12": "2",
          "20": "1",
          "28": "-2"
        },
        "4,2": {
          "0": "2",
          "12": "-2",
          "20": "-1",
          "28": "2"
        },
        "4,3": {
          "0": "-2",
          "12": "2",
          "20": "1",
          "28": "-2"
        },
        "4,4": {
          "0": "-1",
          "4": "1",
          "16": "1",
          "24": "-1"
        },
        "4,5": {
          "0": "3",
          "4": "-1",
          "12": "-2",
          "16": "-1",
          "20": "-1",
          "24": "1",
          "28": "2"
        },
        "5,0": {
          "0": "2",
          "12": "-2",
          "16": "-1",
          "28": "2"
        },
        "5,1": {
          "0": "-2",
          "12": "2",
          "16": "1",
          "28": "-2"
        },
        "5,2": {
          "0": "2",
          "12": "-2",
          "16": "-1",
          "28": "2"
        },
        "5,3": {
          "0": "-2",
          "12": "2",
          "16": "1",
          "28": "-2"
        },
        "5,4": {
          "0": "-2",
          "8": "1",
          "16": "1",
          "20": "1",
          "24": "-1",
          "28": "-1"
        },
        "5,5": {
          "0": "4",
          "8": "-1",
          "12": "-2",
          "16": "-2",
          "20": "-1",
          "24": "1",
          "28": "3"
        }
      },
      "ordered_bridge": "U1 U2 ... Uk",
      "inclusion_trace": {
        "2": "77",
        "6": "-141",
        "10": "104",
        "14": "-126",
        "18": "126",
        "22": "-104",
        "26": "141",
        "30": "-77"
      },
      "trace_minus_conjugate": {
        "2": "-76",
        "6": "-15",
        "10": "27",
        "14": "42",
        "18": "49",
        "22": "22",
        "26": "-12",
        "30": "-77"
      },
      "nonreal": true
    }
  ]
}