{
  "description": "Exact rational construction proving a(7)=174.",
  "parameterization": "A shape has apex p, positive affine leg scale r, and rational equal-norm direction vectors 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. Thus its two physical leg lengths are equal.",
  "partition": [
    [
      1,
      2,
      3,
      7
    ],
    [
      4,
      5,
      6
    ]
  ],
  "shapes": [
    {
      "apex": [
        "0",
        "0"
      ],
      "radius": "1",
      "u": [
        "58563/58565",
        "-484/58565"
      ],
      "v": [
        "58563/58565",
        "484/58565"
      ]
    },
    {
      "apex": [
        "648/683861",
        "-4491/181174"
      ],
      "radius": "66659/66688",
      "u": [
        "7320/7321",
        "121/7321"
      ],
      "v": [
        "411212/411437",
        "13605/411437"
      ]
    },
    {
      "apex": [
        "-49/172946",
        "22687/914854"
      ],
      "radius": "843174/842849",
      "u": [
        "411212/411437",
        "-13605/411437"
      ],
      "v": [
        "7320/7321",
        "-121/7321"
      ]
    },
    {
      "apex": [
        "109401/105491",
        "342281/302983"
      ],
      "radius": "1050758/922843",
      "u": [
        "-21831/660281",
        "-659920/660281"
      ],
      "v": [
        "84/85",
        "-13/85"
      ]
    },
    {
      "apex": [
        "1577175/724942",
        "546877/565885"
      ],
      "radius": "1369872/887603",
      "u": [
        "-1377727/1777345",
        "-1122864/1777345"
      ],
      "v": [
        "680/689",
        "-111/689"
      ]
    },
    {
      "apex": [
        "3428984/890385",
        "561113/753551"
      ],
      "radius": "2152507/713844",
      "u": [
        "-58575/60497",
        "-15128/60497"
      ],
      "v": [
        "613307/622285",
        "-105324/622285"
      ]
    },
    {
      "apex": [
        "-32577/2048",
        "-34625/131072"
      ],
      "radius": "541/32",
      "u": [
        "131073/131072",
        "31/2048"
      ],
      "v": [
        "131071/131072",
        "33/2048"
      ]
    }
  ],
  "claimed_pair_crossing_matrix": [
    [
      0,
      7,
      7,
      8,
      8,
      8,
      7
    ],
    [
      7,
      0,
      7,
      8,
      8,
      8,
      7
    ],
    [
      7,
      7,
      0,
      8,
      8,
      8,
      7
    ],
    [
      8,
      8,
      8,
      0,
      7,
      7,
      8
    ],
    [
      8,
      8,
      8,
      7,
      0,
      7,
      8
    ],
    [
      8,
      8,
      8,
      7,
      7,
      0,
      8
    ],
    [
      7,
      7,
      7,
      8,
      8,
      8,
      0
    ]
  ],
  "claimed_total_crossings": 159,
  "claimed_regions": 174,
  "conclusion": "This exact colored-K4,3 construction has 159 globally distinct proper crossings and 174 regions. Together with the six-subset upper bound from I(6)=114, it proves a(7)=174."
}