{
  "description": "Exact rational construction proving a(10)=361.",
  "partition": [
    [
      1,
      2,
      3,
      4,
      5
    ],
    [
      6,
      7,
      8,
      9,
      10
    ]
  ],
  "source_slice": 3,
  "source_mask_indices": [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
  ],
  "max_margin_lp": {
    "normalized_slack": 1.4697459133905334e-07,
    "coordinate_bounds": [
      -100,
      100
    ],
    "rationalization_max_denominator": 100000,
    "exact_minimum_active_form": "393831843566098008574280841782400/12304071066045487221161065437439365906245880370433"
  },
  "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": [
        "23833915/23833933",
        "-29292/23833933"
      ],
      "v": [
        "23833915/23833933",
        "29292/23833933"
      ]
    },
    {
      "apex": [
        "31/93203",
        "287/77860"
      ],
      "radius": "48505/48521",
      "u": [
        "4137131/4137181",
        "-20340/4137181"
      ],
      "v": [
        "16548599/16548649",
        "-40680/16548649"
      ]
    },
    {
      "apex": [
        "31103/34215",
        "27/15106"
      ],
      "radius": "7516/82633",
      "u": [
        "24466085/24479533",
        "-811308/24479533"
      ],
      "v": [
        "4289480/4289561",
        "-26361/4289561"
      ]
    },
    {
      "apex": [
        "181/96861",
        "2583/72650"
      ],
      "radius": "71137/71228",
      "u": [
        "1415764/1416725",
        "-52173/1416725"
      ],
      "v": [
        "9515072/9520697",
        "-327225/9520697"
      ]
    },
    {
      "apex": [
        "125/55188",
        "-366/99523"
      ],
      "radius": "81439/81623",
      "u": [
        "16548599/16548649",
        "40680/16548649"
      ],
      "v": [
        "4137131/4137181",
        "20340/4137181"
      ]
    },
    {
      "apex": [
        "93368/92357",
        "-316/98143"
      ],
      "radius": "957/80645",
      "u": [
        "3269064/3269185",
        "28127/3269185"
      ],
      "v": [
        "-29697304/32035145",
        "12013353/32035145"
      ]
    },
    {
      "apex": [
        "83851/83473",
        "-202/62111"
      ],
      "radius": "498/78041",
      "u": [
        "1838711/1838761",
        "13560/1838761"
      ],
      "v": [
        "-37175192/52221833",
        "36675945/52221833"
      ]
    },
    {
      "apex": [
        "55129/55128",
        "-137/42057"
      ],
      "radius": "331/73769",
      "u": [
        "4289480/4289561",
        "26361/4289561"
      ],
      "v": [
        "-20340/4137181",
        "4137131/4137181"
      ]
    },
    {
      "apex": [
        "86858/84883",
        "-301/93971"
      ],
      "radius": "1090/45563",
      "u": [
        "517132/517157",
        "5085/517157"
      ],
      "v": [
        "-100204200/101975761",
        "18925489/101975761"
      ]
    },
    {
      "apex": [
        "149089/74780",
        "-3149/36743"
      ],
      "radius": "35218/35259",
      "u": [
        "817191/817241",
        "9040/817241"
      ],
      "v": [
        "-8220977855/8252325217",
        "718606008/8252325217"
      ]
    }
  ],
  "claimed_pair_crossing_matrix": [
    [
      0,
      7,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      0,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      0,
      7,
      7,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      0,
      7,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      7,
      0,
      8,
      8,
      8,
      8,
      8
    ],
    [
      8,
      8,
      8,
      8,
      8,
      0,
      7,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      7,
      0,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      0,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      0,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      7,
      0
    ]
  ],
  "claimed_total_crossings": 340,
  "claimed_regions": 361,
  "conclusion": "This exact rational colored-K5,5 construction has 340 globally distinct proper crossings and 361 regions. Together with the unconditional Mantel upper bound, it proves a(10)=361."
}