{
  "description": "Exact rational construction proving a(13)=615.",
  "partition": [
    [
      1,
      2,
      3,
      4,
      5,
      6
    ],
    [
      7,
      8,
      9,
      10,
      11,
      12,
      13
    ]
  ],
  "source_slice": 3,
  "source_mask_indices": [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
  ],
  "max_margin_lp": {
    "normalized_slack": 1.1553855037958177e-07,
    "coordinate_bounds": [
      -100,
      100
    ],
    "rationalization_max_denominator": 100000,
    "exact_minimum_active_form": "3056720135122785518910165559614168373266143334784/75672345794804345208289929808881561256311703556904850589838571661875"
  },
  "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": [
        "9108299/9108349",
        "-30180/9108349"
      ],
      "v": [
        "9108299/9108349",
        "30180/9108349"
      ]
    },
    {
      "apex": [
        "-994437/79609",
        "3814/53565"
      ],
      "radius": "1303367/96605",
      "u": [
        "5310500/5310581",
        "-29331/5310581"
      ],
      "v": [
        "19114263/19114505",
        "-96184/19114505"
      ]
    },
    {
      "apex": [
        "-1231528/98615",
        "3509/68383"
      ],
      "radius": "695068/51531",
      "u": [
        "8779333/8779405",
        "-35556/8779405"
      ],
      "v": [
        "315843/315845",
        "-1124/315845"
      ]
    },
    {
      "apex": [
        "-409669/93188",
        "617/27402"
      ],
      "radius": "342708/63509",
      "u": [
        "6285013/6285085",
        "-30084/6285085"
      ],
      "v": [
        "5419559/5419609",
        "-23280/5419609"
      ]
    },
    {
      "apex": [
        "-145865/93407",
        "-314/21935"
      ],
      "radius": "146059/57018",
      "u": [
        "5419559/5419609",
        "23280/5419609"
      ],
      "v": [
        "6285013/6285085",
        "30084/6285085"
      ]
    },
    {
      "apex": [
        "-87391/56027",
        "-1176/94679"
      ],
      "radius": "217139/84826",
      "u": [
        "315843/315845",
        "1124/315845"
      ],
      "v": [
        "8779333/8779405",
        "35556/8779405"
      ]
    },
    {
      "apex": [
        "230128/64957",
        "3837/93977"
      ],
      "radius": "198106/77589",
      "u": [
        "-201870649945/201900883033",
        "-3493888992/201900883033"
      ],
      "v": [
        "11021368/11021657",
        "-79815/11021657"
      ]
    },
    {
      "apex": [
        "27956/26959",
        "156/40981"
      ],
      "radius": "475/12591",
      "u": [
        "-1330316479/1354662721",
        "-255673920/1354662721"
      ],
      "v": [
        "3828104/3828185",
        "-24903/3828185"
      ]
    },
    {
      "apex": [
        "106594/86575",
        "108/16667"
      ],
      "radius": "19529/83890",
      "u": [
        "-15921399615/15935518913",
        "-670669312/15935518913"
      ],
      "v": [
        "18386719/18387169",
        "-128640/18387169"
      ]
    },
    {
      "apex": [
        "51313/50966",
        "263/76773"
      ],
      "radius": "485/50586",
      "u": [
        "-51081521/71818321",
        "-50483160/71818321"
      ],
      "v": [
        "13380843/13381085",
        "-80476/13381085"
      ]
    },
    {
      "apex": [
        "40503/37351",
        "142/31919"
      ],
      "radius": "7561/88952",
      "u": [
        "-11680/11729",
        "-1071/11729"
      ],
      "v": [
        "19748911/19749361",
        "-133320/19749361"
      ]
    },
    {
      "apex": [
        "69503/69501",
        "261/77344"
      ],
      "radius": "658/98379",
      "u": [
        "-29331/5310581",
        "-5310500/5310581"
      ],
      "v": [
        "12019989/12020189",
        "-69340/12020189"
      ]
    },
    {
      "apex": [
        "92388/90859",
        "287/80991"
      ],
      "radius": "137/7531",
      "u": [
        "-40058947/43249285",
        "-16302804/43249285"
      ],
      "v": [
        "408317/408325",
        "-2556/408325"
      ]
    }
  ],
  "claimed_pair_crossing_matrix": [
    [
      0,
      7,
      7,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      0,
      7,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      0,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      0,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      7,
      0,
      7,
      8,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      7,
      7,
      0,
      8,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      0,
      7,
      7,
      7,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      0,
      7,
      7,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      0,
      7,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      0,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      7,
      0,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      7,
      7,
      0,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      7,
      7,
      7,
      0
    ]
  ],
  "claimed_total_crossings": 588,
  "claimed_regions": 615,
  "conclusion": "This exact rational colored-K6,7 construction has 588 globally distinct proper crossings and 615 regions. Together with the unconditional Mantel upper bound, it proves a(13)=615."
}