{
  "description": "Exact rational rank certificate for the 144 cyclic Freudenthal quadrics",
  "indexing": "All indices are zero-based; a monomial [i,j] means variables[i]*variables[j].",
  "variables": [
    "a0",
    "a1",
    "a2",
    "a3",
    "a4",
    "a5",
    "a6",
    "a7",
    "a8",
    "a9",
    "a10",
    "a11",
    "a12",
    "a13",
    "a14",
    "a15",
    "a16",
    "a17",
    "x0",
    "x1",
    "x2",
    "x3",
    "x4",
    "x5",
    "x6",
    "x7",
    "x8",
    "x9",
    "x10",
    "x11",
    "x12",
    "x13",
    "x14",
    "x15",
    "x16",
    "x17",
    "y0",
    "y1",
    "y2",
    "y3",
    "y4",
    "y5",
    "y6",
    "y7",
    "y8",
    "y9",
    "y10",
    "y11",
    "y12",
    "y13",
    "y14",
    "y15",
    "y16",
    "y17",
    "z1",
    "z2"
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        "1"
      ],
      [
        131,
        "-1/2"
      ],
      [
        132,
        "-1/2"
      ]
    ],
    [
      [
        127,
        "-1"
      ],
      [
        128,
        "1"
      ],
      [
        130,
        "-2"
      ],
      [
        132,
        "-1/2"
      ]
    ],
    [
      [
        126,
        "-1"
      ]
    ],
    [
      [
        127,
        "-1"
      ]
    ],
    [
      [
        128,
        "-1"
      ]
    ],
    [
      [
        126,
        "-1"
      ],
      [
        128,
        "1"
      ],
      [
        129,
        "-2"
      ]
    ],
    [
      [
        127,
        "-1"
      ],
      [
        129,
        "1"
      ],
      [
        130,
        "-2"
      ]
    ],
    [
      [
        126,
        "-1"
      ],
      [
        129,
        "-1"
      ],
      [
        130,
        "1"
      ],
      [
        131,
        "-1/2"
      ]
    ],
    [
      [
        126,
        "1"
      ],
      [
        127,
        "-1"
      ],
      [
        128,
        "-1"
      ],
      [
        129,
        "1"
      ],
      [
        130,
        "-1"
      ],
      [
        131,
        "-1/2"
      ],
      [
        132,
        "-1/2"
      ]
    ],
    [
      [
        126,
        "-1"
      ],
      [
        127,
        "1"
      ],
      [
        129,
        "-2"
      ],
      [
        130,
        "1"
      ],
      [
        131,
        "-1/2"
      ],
      [
        132,
        "-1/2"
      ]
    ],
    [
      [
        127,
        "-1"
      ],
      [
        128,
        "1"
      ],
      [
        130,
        "-2"
      ],
      [
        132,
        "-1/2"
      ]
    ]
  ],
  "verification": "The distinct leading pivot monomials prove independence. The two exact change-of-span arrays prove equality of spans over Q. Scalar extension gives the same rank over C."
}