{
  "description": "Exact rational construction proving a(11)=438.",
  "partition": [
    [
      1,
      2,
      3,
      4,
      5
    ],
    [
      6,
      7,
      8,
      9,
      10,
      11
    ]
  ],
  "source_slice": 8,
  "source_mask_indices": [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
  ],
  "max_margin_lp": {
    "normalized_slack": 5.69698267462876e-08,
    "coordinate_bounds": [
      -100,
      100
    ],
    "rationalization_max_denominator": 10000,
    "exact_minimum_active_form": "536964079038999465692412567859919629681152/996339082498484327588279805977398550610078889535729383795265"
  },
  "parameterization": "Each shape has apex p, positive radius r, and rational unit directions 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": [
        "11696399/11696401",
        "-6840/11696401"
      ],
      "v": [
        "11696399/11696401",
        "6840/11696401"
      ]
    },
    {
      "apex": [
        "1/5059",
        "-9/5131"
      ],
      "radius": "5077/5078",
      "u": [
        "2924099/2924101",
        "3420/2924101"
      ],
      "v": [
        "365512/365513",
        "855/365513"
      ]
    },
    {
      "apex": [
        "2/3915",
        "-29/8269"
      ],
      "radius": "9871/9876",
      "u": [
        "467855/467857",
        "1368/467857"
      ],
      "v": [
        "9666764/9666845",
        "39573/9666845"
      ]
    },
    {
      "apex": [
        "6109/7863",
        "-11/5447"
      ],
      "radius": "1133/5079",
      "u": [
        "9665781/9665981",
        "62180/9665981"
      ],
      "v": [
        "13564293/13564685",
        "103124/13564685"
      ]
    },
    {
      "apex": [
        "6939/8929",
        "-8/4917"
      ],
      "radius": "1600/7179",
      "u": [
        "731021/731029",
        "3420/731029"
      ],
      "v": [
        "116963/116965",
        "684/116965"
      ]
    },
    {
      "apex": [
        "5556/5461",
        "3/2785"
      ],
      "radius": "143/8134",
      "u": [
        "-3765679236/3782628925",
        "-357689077/3782628925"
      ],
      "v": [
        "10254908/10256933",
        "-203805/10256933"
      ]
    },
    {
      "apex": [
        "5065/5059",
        "2/3337"
      ],
      "radius": "5/2981",
      "u": [
        "-254475/359357",
        "-253732/359357"
      ],
      "v": [
        "1299599/1299601",
        "-2280/1299601"
      ]
    },
    {
      "apex": [
        "9905/8108",
        "52/6837"
      ],
      "radius": "73/328",
      "u": [
        "-17225074380/17236763941",
        "-634699741/17236763941"
      ],
      "v": [
        "8601589/8603389",
        "-175980/8603389"
      ]
    },
    {
      "apex": [
        "8615/8561",
        "6/9149"
      ],
      "radius": "64/9919",
      "u": [
        "-2016638759/2054990041",
        "-395160960/2054990041"
      ],
      "v": [
        "467855/467857",
        "-1368/467857"
      ]
    },
    {
      "apex": [
        "1",
        "5/8462"
      ],
      "radius": "11/9355",
      "u": [
        "-6840/11696401",
        "-11696399/11696401"
      ],
      "v": [
        "2924099/2924101",
        "-3420/2924101"
      ]
    },
    {
      "apex": [
        "7917/7894",
        "1/1621"
      ],
      "radius": "29/9185",
      "u": [
        "-187497688/202762337",
        "-77182785/202762337"
      ],
      "v": [
        "365512/365513",
        "-855/365513"
      ]
    }
  ],
  "claimed_pair_crossing_matrix": [
    [
      0,
      7,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      0,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      0,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      0,
      7,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      7,
      0,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      8,
      8,
      8,
      8,
      8,
      0,
      7,
      7,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      7,
      0,
      7,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      0,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      0,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      7,
      0,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      7,
      7,
      0
    ]
  ],
  "claimed_total_crossings": 415,
  "claimed_regions": 438,
  "conclusion": "This exact rational colored-K5,6 construction has 415 globally distinct proper crossings and 438 regions. Together with the unconditional Mantel upper bound, it proves a(11)=438."
}