{
  "description": "Exact rational construction proving a(12)=523.",
  "partition": [
    [
      1,
      2,
      3,
      4,
      5,
      6
    ],
    [
      7,
      8,
      9,
      10,
      11,
      12
    ]
  ],
  "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": 2.4646768945228207e-08,
    "coordinate_bounds": [
      -100,
      100
    ],
    "rationalization_max_denominator": 10000,
    "exact_minimum_active_form": "43187077147945076037628787084477036021990028794/338107131498484208514356270849305895820236227692030268260594232455"
  },
  "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": [
        "36645/2984237",
        "-2984012/2984237"
      ],
      "v": [
        "36645/2984237",
        "2984012/2984237"
      ]
    },
    {
      "apex": [
        "-33/4555",
        "-4093/5573"
      ],
      "radius": "517/2293",
      "u": [
        "101520/1168609",
        "-1164191/1168609"
      ],
      "v": [
        "1599/142049",
        "142040/142049"
      ]
    },
    {
      "apex": [
        "-62/7661",
        "-5541/6184"
      ],
      "radius": "476/8677",
      "u": [
        "3284437/8808925",
        "-8173716/8808925"
      ],
      "v": [
        "83760/7795489",
        "7795039/7795489"
      ]
    },
    {
      "apex": [
        "-11/2078",
        "-1247/2528"
      ],
      "radius": "2506/5309",
      "u": [
        "212499/5690549",
        "-5686580/5690549"
      ],
      "v": [
        "61351/5213401",
        "5213040/5213401"
      ]
    },
    {
      "apex": [
        "70/6383",
        "-8501/8991"
      ],
      "radius": "19/9539",
      "u": [
        "5628000/8039809",
        "-5741441/8039809"
      ],
      "v": [
        "244300/23873621",
        "23872371/23873621"
      ]
    },
    {
      "apex": [
        "85/7611",
        "-8615/9098"
      ],
      "radius": "4/3353",
      "u": [
        "17023515/17024237",
        "-156788/17024237"
      ],
      "v": [
        "24893/2560645",
        "2560524/2560645"
      ]
    },
    {
      "apex": [
        "10/1331",
        "-19802/9903"
      ],
      "radius": "7173/6814",
      "u": [
        "86860/18861749",
        "18861549/18861749"
      ],
      "v": [
        "161381/45058669",
        "45058380/45058669"
      ]
    },
    {
      "apex": [
        "44/4095",
        "-8043/4022"
      ],
      "radius": "8051/7647",
      "u": [
        "1303/848905",
        "848904/848905"
      ],
      "v": [
        "3909/7640141",
        "7640140/7640141"
      ]
    },
    {
      "apex": [
        "11/2569",
        "-4391/2196"
      ],
      "radius": "7477/7103",
      "u": [
        "150620/19625189",
        "19624611/19625189"
      ],
      "v": [
        "187925/28252957",
        "28252332/28252957"
      ]
    },
    {
      "apex": [
        "56/9495",
        "-12253/6128"
      ],
      "radius": "9624/9143",
      "u": [
        "117591/19152041",
        "19151680/19152041"
      ],
      "v": [
        "112965/22078157",
        "22077868/22078157"
      ]
    },
    {
      "apex": [
        "77/8434",
        "-9287/4645"
      ],
      "radius": "6282/5969",
      "u": [
        "110097/35862065",
        "35861896/35862065"
      ],
      "v": [
        "48860/23873021",
        "23872971/23873021"
      ]
    },
    {
      "apex": [
        "20/7497",
        "-3055/1528"
      ],
      "radius": "3731/3545",
      "u": [
        "156788/17024237",
        "17023515/17024237"
      ],
      "v": [
        "48860/5968349",
        "5968149/5968349"
      ]
    }
  ],
  "claimed_pair_crossing_matrix": [
    [
      0,
      7,
      7,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      0,
      7,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      0,
      7,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      0,
      7,
      7,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      7,
      0,
      7,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      7,
      7,
      7,
      7,
      7,
      0,
      8,
      8,
      8,
      8,
      8,
      8
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      0,
      7,
      7,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      0,
      7,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      0,
      7,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      0,
      7,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      7,
      0,
      7
    ],
    [
      8,
      8,
      8,
      8,
      8,
      8,
      7,
      7,
      7,
      7,
      7,
      0
    ]
  ],
  "claimed_total_crossings": 498,
  "claimed_regions": 523,
  "conclusion": "This exact rational colored-K6,6 construction has 498 globally distinct proper crossings and 523 regions. Together with the unconditional Mantel upper bound, it proves a(12)=523."
}