{
  "description": "Exact rational construction proving a(9)=291.",
  "partition": [
    [
      1,
      2,
      3,
      4
    ],
    [
      5,
      6,
      7,
      8,
      9
    ]
  ],
  "parameterization": "A shape has apex p, positive affine leg scale r, and rational equal-norm direction vectors u,v. Its rays are p+t*u and p+t*v for t>0; its crossbar is p+r*u+t*(v-u), 0<t<r.",
  "shapes": [
    {
      "apex": [
        "0",
        "0"
      ],
      "radius": "1",
      "u": [
        "17877621/181682621",
        "-180800900/181682621"
      ],
      "v": [
        "17877621/181682621",
        "180800900/181682621"
      ]
    },
    {
      "apex": [
        "-33/728",
        "151/297"
      ],
      "radius": "485/986",
      "u": [
        "8493121/89087329",
        "-88681560/89087329"
      ],
      "v": [
        "7996033/27166465",
        "25963056/27166465"
      ]
    },
    {
      "apex": [
        "79/537",
        "-15239/911"
      ],
      "radius": "6345/356",
      "u": [
        "18005955/177460037",
        "-176544188/177460037"
      ],
      "v": [
        "-187663/60929905",
        "60929616/60929905"
      ]
    },
    {
      "apex": [
        "-62/945",
        "773/994"
      ],
      "radius": "199/772",
      "u": [
        "16864417/182775745",
        "-181996056/182775745"
      ],
      "v": [
        "57791559/90128441",
        "69161200/90128441"
      ]
    },
    {
      "apex": [
        "-1081/924",
        "13573/720"
      ],
      "radius": "1533/85",
      "u": [
        "7512827/116235085",
        "-115992036/116235085"
      ],
      "v": [
        "6334180/89490341",
        "-89265891/89490341"
      ]
    },
    {
      "apex": [
        "-731/732",
        "13313/709"
      ],
      "radius": "11219/625",
      "u": [
        "4801380/86649589",
        "-86516461/86649589"
      ],
      "v": [
        "5572860/90525949",
        "-90354251/90525949"
      ]
    },
    {
      "apex": [
        "-1168/883",
        "12469/668"
      ],
      "radius": "13164/737",
      "u": [
        "1367400/18515281",
        "-18464719/18515281"
      ],
      "v": [
        "8252975/103169617",
        "-102838992/103169617"
      ]
    },
    {
      "apex": [
        "-805/544",
        "5607/302"
      ],
      "radius": "10576/595",
      "u": [
        "9692005/116681197",
        "-116277972/116681197"
      ],
      "v": [
        "2556344/28658225",
        "-28543983/28658225"
      ]
    },
    {
      "apex": [
        "37/4",
        "-57257/512"
      ],
      "radius": "37679/4096",
      "u": [
        "-8143/8192",
        "25089/2048"
      ],
      "v": [
        "-8241/8192",
        "25087/2048"
      ]
    }
  ],
  "claimed_pair_crossing_matrix": [
    [
      0,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      0,
      7,
      7,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      0,
      7,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      0,
      8,
      8,
      8,
      8,
      8
    ],
    [
      8,
      8,
      8,
      8,
      0,
      7,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      7,
      0,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      7,
      7,
      0,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      0,
      7
    ],
    [
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      7,
      0
    ]
  ],
  "claimed_total_crossings": 272,
  "claimed_regions": 291,
  "conclusion": "This exact rational colored-K4,5 construction has 272 globally distinct proper crossings and 291 regions. Together with the unconditional Mantel upper bound, it proves a(9)=291."
}