{
  "scope": "Exact finite checks for n=3,5,7; not general proof or a Lean build.",
  "examples": [
    {
      "s": 1,
      "n": 3,
      "dimension": 1,
      "nodes": [
        1,
        2,
        3
      ],
      "weights": [
        "1/2",
        "-1",
        "1/2"
      ],
      "pattern": [
        [
          2,
          1
        ]
      ],
      "evaluation_determinant": "1",
      "orders_checked": 6,
      "natural_order_image": [
        [
          "1/2"
        ]
      ],
      "natural_order_determinant": "1/2"
    },
    {
      "s": 2,
      "n": 5,
      "dimension": 4,
      "nodes": [
        1,
        2,
        3,
        4,
        5
      ],
      "weights": [
        "1/24",
        "-1/6",
        "1/4",
        "-1/6",
        "1/24"
      ],
      "pattern": [
        [
          2,
          1
        ],
        [
          3,
          1
        ],
        [
          3,
          2
        ],
        [
          4,
          5
        ]
      ],
      "evaluation_determinant": "-4",
      "orders_checked": 120,
      "natural_order_image": [
        [
          "5/144",
          "5/48",
          "5/48",
          "49/144"
        ],
        [
          "5/48",
          "47/144",
          "49/144",
          "55/48"
        ],
        [
          "5/48",
          "49/144",
          "47/144",
          "55/48"
        ],
        [
          "49/144",
          "55/48",
          "55/48",
          "593/144"
        ]
      ],
      "natural_order_determinant": "-1/1492992"
    },
    {
      "s": 3,
      "n": 7,
      "dimension": 9,
      "nodes": [
        1,
        2,
        3,
        4,
        5,
        6,
        7
      ],
      "weights": [
        "1/720",
        "-1/120",
        "1/48",
        "-1/36",
        "1/48",
        "-1/120",
        "1/720"
      ],
      "pattern": [
        [
          2,
          1
        ],
        [
          3,
          1
        ],
        [
          3,
          2
        ],
        [
          4,
          1
        ],
        [
          4,
          2
        ],
        [
          4,
          3
        ],
        [
          5,
          6
        ],
        [
          5,
          7
        ],
        [
          6,
          7
        ]
      ],
      "evaluation_determinant": "-14400",
      "orders_checked": 4,
      "natural_order_image": [
        [
          "7/14400",
          "7/3600",
          "7/864",
          "7/3600",
          "119/14400",
          "49/1350",
          "7/864",
          "49/1350",
          "7219/43200"
        ],
        [
          "7/3600",
          "7/864",
          "7/200",
          "119/14400",
          "49/1350",
          "473/2880",
          "49/1350",
          "7219/43200",
          "8519/10800"
        ],
        [
          "7/864",
          "7/200",
          "3373/21600",
          "49/1350",
          "473/2880",
          "4123/5400",
          "7219/43200",
          "8519/10800",
          "6583/1728"
        ],
        [
          "7/3600",
          "119/14400",
          "49/1350",
          "7/864",
          "49/1350",
          "7219/43200",
          "7/200",
          "473/2880",
          "8519/10800"
        ],
        [
          "119/14400",
          "49/1350",
          "473/2880",
          "49/1350",
          "7219/43200",
          "8519/10800",
          "473/2880",
          "8519/10800",
          "18577/4800"
        ],
        [
          "49/1350",
          "473/2880",
          "4123/5400",
          "7219/43200",
          "8519/10800",
          "6583/1728",
          "8519/10800",
          "18577/4800",
          "209153/10800"
        ],
        [
          "7/864",
          "49/1350",
          "7219/43200",
          "7/200",
          "473/2880",
          "8519/10800",
          "3373/21600",
          "4123/5400",
          "6583/1728"
        ],
        [
          "49/1350",
          "7219/43200",
          "8519/10800",
          "473/2880",
          "8519/10800",
          "18577/4800",
          "4123/5400",
          "6583/1728",
          "209153/10800"
        ],
        [
          "7219/43200",
          "8519/10800",
          "6583/1728",
          "8519/10800",
          "18577/4800",
          "209153/10800",
          "6583/1728",
          "209153/10800",
          "4326571/43200"
        ]
      ],
      "natural_order_determinant": "-1/11501279977342425366528000000"
    }
  ],
  "integer_identities_checked_through": 30
}