{
  "description": "Exact rational construction proving a(14)=715.",
  "partition": [
    [
      1,
      2,
      3,
      4,
      5,
      6,
      7
    ],
    [
      8,
      9,
      10,
      11,
      12,
      13,
      14
    ]
  ],
  "source_slice": 0,
  "source_mask_indices": [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
  ],
  "max_margin_lp": {
    "normalized_slack": 7.884262168368212e-09,
    "coordinate_bounds": [
      -100,
      100
    ],
    "rationalization_max_denominator": 1000000,
    "exact_minimum_active_form": "5951992613034111836675767919322159818920590967181362510204/3405191526285972615816352212341245555454222143259149494524787825935307095743875"
  },
  "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": [
        "24999999/25000001",
        "-10000/25000001"
      ],
      "v": [
        "24999999/25000001",
        "10000/25000001"
      ]
    },
    {
      "apex": [
        "-374371/453933",
        "-857/714195"
      ],
      "radius": "1594451/873802",
      "u": [
        "20811843/20811845",
        "9124/20811845"
      ],
      "v": [
        "2601480/2601481",
        "2281/2601481"
      ]
    },
    {
      "apex": [
        "-511095/619526",
        "856/69043"
      ],
      "radius": "1449297/794128",
      "u": [
        "2033451/2033501",
        "-14260/2033501"
      ],
      "v": [
        "18138885/18139277",
        "-119252/18139277"
      ]
    },
    {
      "apex": [
        "-725585/880073",
        "-237/65846"
      ],
      "radius": "1732181/949419",
      "u": [
        "5202957/5202965",
        "9124/5202965"
      ],
      "v": [
        "20811819/20811869",
        "45620/20811869"
      ]
    },
    {
      "apex": [
        "-752881/913426",
        "-742/309289"
      ],
      "radius": "185881/101895",
      "u": [
        "1665312/1665313",
        "1825/1665313"
      ],
      "v": [
        "6796445/6796453",
        "10428/6796453"
      ]
    },
    {
      "apex": [
        "-424833/515381",
        "-5975/996103"
      ],
      "radius": "521905/286082",
      "u": [
        "20811795/20811893",
        "63868/20811893"
      ],
      "v": [
        "5202945/5202977",
        "18248/5202977"
      ]
    },
    {
      "apex": [
        "-818863/993918",
        "-1958/408139"
      ],
      "radius": "1474792/808601",
      "u": [
        "2752277/2752285",
        "6636/2752285"
      ],
      "v": [
        "12068760/12068809",
        "34391/12068809"
      ]
    },
    {
      "apex": [
        "388121/388120",
        "-395/886188"
      ],
      "radius": "827/977821",
      "u": [
        "7057500/7057549",
        "26299/7057549"
      ],
      "v": [
        "-18248/5202977",
        "5202945/5202977"
      ]
    },
    {
      "apex": [
        "913993/913218",
        "-313/706778"
      ],
      "radius": "76/63513",
      "u": [
        "20811763/20811925",
        "82116/20811925"
      ],
      "v": [
        "-20040993/28234145",
        "19887824/28234145"
      ]
    },
    {
      "apex": [
        "967529/963152",
        "-135/278528"
      ],
      "radius": "1236/266579",
      "u": [
        "2601468/2601493",
        "11405/2601493"
      ],
      "v": [
        "-90830541/92530709",
        "17656300/92530709"
      ]
    },
    {
      "apex": [
        "231827/229393",
        "-563/940630"
      ],
      "radius": "10451/977888",
      "u": [
        "11419360/11419481",
        "52569/11419481"
      ],
      "v": [
        "-10405831117/10451566165",
        "976685244/10451566165"
      ]
    },
    {
      "apex": [
        "459885/445933",
        "-845/851937"
      ],
      "radius": "15811/502389",
      "u": [
        "20811723/20811965",
        "100364/20811965"
      ],
      "v": [
        "-44243517485/44286911413",
        "1960021188/44286911413"
      ]
    },
    {
      "apex": [
        "675811/674430",
        "-395/899577"
      ],
      "radius": "2203/994716",
      "u": [
        "18679603/18679765",
        "77796/18679765"
      ],
      "v": [
        "-1336217/1443865",
        "547056/1443865"
      ]
    },
    {
      "apex": [
        "2539472/900291",
        "-28846/821019"
      ],
      "radius": "1610366/883705",
      "u": [
        "15736989/15737189",
        "79340/15737189"
      ],
      "v": [
        "-1572614008/1572913217",
        "30678495/1572913217"
      ]
    }
  ],
  "claimed_pair_crossing_matrix": [
    [
      0,
      7,
      7,
      7,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      0,
      7,
      7,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      0,
      7,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      0,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      7,
      0,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      7,
      7,
      0,
      7,
      8,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      7,
      7,
      7,
      0,
      8,
      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,
      8,
      7,
      0,
      7,
      7,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      0,
      7,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      0,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      7,
      0,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      7,
      7,
      0,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      7,
      7,
      7,
      0
    ]
  ],
  "claimed_total_crossings": 686,
  "claimed_regions": 715,
  "conclusion": "This exact rational colored-K7,7 construction has 686 globally distinct proper crossings and 715 regions. Together with the unconditional Mantel upper bound, it proves a(14)=715."
}