{
  "schema": "mortra.spatial-fold-new-problem-book.v1",
  "source": {
    "path": "artifacts/research/autonomous-spatial-fold-problems-v1-20260907/result.json",
    "sha256": "c0a160ab61f7093fc8ea9c3f5fc93e0bece27868bb72cbc0fe67bb8dd745f412",
    "audit_sha256": "f5718d76de4b23ca8d60194c4e5a50e70e1d4270675bffa3be0adc85f2400f97"
  },
  "candidate_count": 12,
  "candidates": [
    {
      "candidate_id": "spatial-fold-a6d04677ddbbfc9b",
      "axis_motif": "uvu",
      "motif": [
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        }
      ],
      "relations": [
        "perpendicular",
        "perpendicular",
        "parallel"
      ],
      "phase_alphabet": [
        1,
        3
      ],
      "phase_rule": "山折りまたは谷折りへ90度",
      "phase_short": "山・谷90度",
      "phase_rule_en": "a 90-degree mountain or valley fold",
      "phase_short_en": "mountain / valley 90°",
      "statement": "正の整数 \\(n\\) に対し、一辺2の正方形 \\(3n+1\\) 枚を帯状につなぎ、隣り合う折り目の関係を端から 垂直、垂直、平行 の周期とする。端の1枚を固定して各折り目を山または谷の向きへ90度折る。両端の正方形の中心を \\(O,Q\\) とするとき、\\(OQ^2\\) の最大値を求めよ。",
      "statement_en": "For a positive integer n, join 3n+1 squares of side 2 in a strip. From one end, let the relationships between successive hinge axes repeat as perpendicular, perpendicular, parallel. Fix the first square and fold every hinge through 90 degrees, as a mountain or valley fold. If O and Q are the centres of the two end squares, find the maximum of OQ².",
      "answer_tex": "12n^2",
      "coefficient": 12,
      "phase_word_count": 8,
      "spatial_reasons": [
        "pose_dependent_maximizing_cycle"
      ],
      "equality": {
        "translation": [
          2,
          2,
          -2
        ],
        "prefix": [],
        "cycle": [
          [
            1,
            1,
            1
          ],
          [
            3,
            3,
            3
          ]
        ],
        "cycle_state_count": 2
      }
    },
    {
      "candidate_id": "spatial-fold-3f4a051089464c69",
      "axis_motif": "uuvuu",
      "motif": [
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        }
      ],
      "relations": [
        "parallel",
        "perpendicular",
        "perpendicular",
        "parallel",
        "parallel"
      ],
      "phase_alphabet": [
        1,
        3
      ],
      "phase_rule": "山折りまたは谷折りへ90度",
      "phase_short": "山・谷90度",
      "phase_rule_en": "a 90-degree mountain or valley fold",
      "phase_short_en": "mountain / valley 90°",
      "statement": "正の整数 \\(n\\) に対し、一辺2の正方形 \\(5n+1\\) 枚を帯状につなぎ、隣り合う折り目の関係を端から 平行、垂直、垂直、平行、平行 の周期とする。端の1枚を固定して各折り目を山または谷の向きへ90度折る。両端の正方形の中心を \\(O,Q\\) とするとき、\\(OQ^2\\) の最大値を求めよ。",
      "statement_en": "For a positive integer n, join 5n+1 squares of side 2 in a strip. From one end, let the relationships between successive hinge axes repeat as parallel, perpendicular, perpendicular, parallel, parallel. Fix the first square and fold every hinge through 90 degrees, as a mountain or valley fold. If O and Q are the centres of the two end squares, find the maximum of OQ².",
      "answer_tex": "34n^2",
      "coefficient": 34,
      "phase_word_count": 32,
      "spatial_reasons": [
        "pose_dependent_maximizing_cycle"
      ],
      "equality": {
        "translation": [
          4,
          3,
          -3
        ],
        "prefix": [],
        "cycle": [
          [
            1,
            3,
            3,
            3,
            1
          ],
          [
            3,
            1,
            1,
            1,
            3
          ]
        ],
        "cycle_state_count": 2
      }
    },
    {
      "candidate_id": "spatial-fold-aceb588fd3963596",
      "axis_motif": "uvuuv",
      "motif": [
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        }
      ],
      "relations": [
        "perpendicular",
        "perpendicular",
        "parallel",
        "perpendicular",
        "perpendicular"
      ],
      "phase_alphabet": [
        1,
        3
      ],
      "phase_rule": "山折りまたは谷折りへ90度",
      "phase_short": "山・谷90度",
      "phase_rule_en": "a 90-degree mountain or valley fold",
      "phase_short_en": "mountain / valley 90°",
      "statement": "正の整数 \\(n\\) に対し、一辺2の正方形 \\(5n+1\\) 枚を帯状につなぎ、隣り合う折り目の関係を端から 垂直、垂直、平行、垂直、垂直 の周期とする。端の1枚を固定して各折り目を山または谷の向きへ90度折る。両端の正方形の中心を \\(O,Q\\) とするとき、\\(OQ^2\\) の最大値を求めよ。",
      "statement_en": "For a positive integer n, join 5n+1 squares of side 2 in a strip. From one end, let the relationships between successive hinge axes repeat as perpendicular, perpendicular, parallel, perpendicular, perpendicular. Fix the first square and fold every hinge through 90 degrees, as a mountain or valley fold. If O and Q are the centres of the two end squares, find the maximum of OQ².",
      "answer_tex": "34n^2",
      "coefficient": 34,
      "phase_word_count": 32,
      "spatial_reasons": [
        "pose_dependent_maximizing_cycle"
      ],
      "equality": {
        "translation": [
          3,
          4,
          -3
        ],
        "prefix": [],
        "cycle": [
          [
            1,
            1,
            1,
            3,
            3
          ],
          [
            3,
            3,
            3,
            1,
            1
          ]
        ],
        "cycle_state_count": 2
      }
    },
    {
      "candidate_id": "spatial-fold-f3fc1f56b8892bdb",
      "axis_motif": "uvuvu",
      "motif": [
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        }
      ],
      "relations": [
        "perpendicular",
        "perpendicular",
        "perpendicular",
        "perpendicular",
        "parallel"
      ],
      "phase_alphabet": [
        1,
        3
      ],
      "phase_rule": "山折りまたは谷折りへ90度",
      "phase_short": "山・谷90度",
      "phase_rule_en": "a 90-degree mountain or valley fold",
      "phase_short_en": "mountain / valley 90°",
      "statement": "正の整数 \\(n\\) に対し、一辺2の正方形 \\(5n+1\\) 枚を帯状につなぎ、隣り合う折り目の関係を端から 垂直、垂直、垂直、垂直、平行 の周期とする。端の1枚を固定して各折り目を山または谷の向きへ90度折る。両端の正方形の中心を \\(O,Q\\) とするとき、\\(OQ^2\\) の最大値を求めよ。",
      "statement_en": "For a positive integer n, join 5n+1 squares of side 2 in a strip. From one end, let the relationships between successive hinge axes repeat as perpendicular, perpendicular, perpendicular, perpendicular, parallel. Fix the first square and fold every hinge through 90 degrees, as a mountain or valley fold. If O and Q are the centres of the two end squares, find the maximum of OQ².",
      "answer_tex": "34n^2",
      "coefficient": 34,
      "phase_word_count": 32,
      "spatial_reasons": [
        "pose_dependent_maximizing_cycle"
      ],
      "equality": {
        "translation": [
          4,
          3,
          -3
        ],
        "prefix": [],
        "cycle": [
          [
            1,
            1,
            1,
            1,
            1
          ],
          [
            3,
            3,
            3,
            3,
            3
          ]
        ],
        "cycle_state_count": 2
      }
    },
    {
      "candidate_id": "spatial-fold-e6eb134c83c55e5f",
      "axis_motif": "uvvuv",
      "motif": [
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        }
      ],
      "relations": [
        "perpendicular",
        "parallel",
        "perpendicular",
        "perpendicular",
        "perpendicular"
      ],
      "phase_alphabet": [
        1,
        3
      ],
      "phase_rule": "山折りまたは谷折りへ90度",
      "phase_short": "山・谷90度",
      "phase_rule_en": "a 90-degree mountain or valley fold",
      "phase_short_en": "mountain / valley 90°",
      "statement": "正の整数 \\(n\\) に対し、一辺2の正方形 \\(5n+1\\) 枚を帯状につなぎ、隣り合う折り目の関係を端から 垂直、平行、垂直、垂直、垂直 の周期とする。端の1枚を固定して各折り目を山または谷の向きへ90度折る。両端の正方形の中心を \\(O,Q\\) とするとき、\\(OQ^2\\) の最大値を求めよ。",
      "statement_en": "For a positive integer n, join 5n+1 squares of side 2 in a strip. From one end, let the relationships between successive hinge axes repeat as perpendicular, parallel, perpendicular, perpendicular, perpendicular. Fix the first square and fold every hinge through 90 degrees, as a mountain or valley fold. If O and Q are the centres of the two end squares, find the maximum of OQ².",
      "answer_tex": "34n^2",
      "coefficient": 34,
      "phase_word_count": 32,
      "spatial_reasons": [
        "pose_dependent_maximizing_cycle"
      ],
      "equality": {
        "translation": [
          4,
          3,
          -3
        ],
        "prefix": [],
        "cycle": [
          [
            1,
            1,
            3,
            3,
            3
          ],
          [
            3,
            3,
            1,
            1,
            1
          ]
        ],
        "cycle_state_count": 2
      }
    },
    {
      "candidate_id": "spatial-fold-cdf351504b033318",
      "axis_motif": "uvvvu",
      "motif": [
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        }
      ],
      "relations": [
        "perpendicular",
        "parallel",
        "parallel",
        "perpendicular",
        "parallel"
      ],
      "phase_alphabet": [
        1,
        3
      ],
      "phase_rule": "山折りまたは谷折りへ90度",
      "phase_short": "山・谷90度",
      "phase_rule_en": "a 90-degree mountain or valley fold",
      "phase_short_en": "mountain / valley 90°",
      "statement": "正の整数 \\(n\\) に対し、一辺2の正方形 \\(5n+1\\) 枚を帯状につなぎ、隣り合う折り目の関係を端から 垂直、平行、平行、垂直、平行 の周期とする。端の1枚を固定して各折り目を山または谷の向きへ90度折る。両端の正方形の中心を \\(O,Q\\) とするとき、\\(OQ^2\\) の最大値を求めよ。",
      "statement_en": "For a positive integer n, join 5n+1 squares of side 2 in a strip. From one end, let the relationships between successive hinge axes repeat as perpendicular, parallel, parallel, perpendicular, parallel. Fix the first square and fold every hinge through 90 degrees, as a mountain or valley fold. If O and Q are the centres of the two end squares, find the maximum of OQ².",
      "answer_tex": "36n^2",
      "coefficient": 36,
      "phase_word_count": 32,
      "spatial_reasons": [
        "pose_dependent_maximizing_cycle"
      ],
      "equality": {
        "translation": [
          4,
          4,
          -2
        ],
        "prefix": [],
        "cycle": [
          [
            1,
            1,
            3,
            1,
            1
          ],
          [
            3,
            3,
            1,
            3,
            3
          ]
        ],
        "cycle_state_count": 2
      }
    },
    {
      "candidate_id": "spatial-fold-9c408362e34aeb0a",
      "axis_motif": "uvu",
      "motif": [
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        }
      ],
      "relations": [
        "perpendicular",
        "perpendicular",
        "parallel"
      ],
      "phase_alphabet": [
        0,
        1,
        2,
        3
      ],
      "phase_rule": "90度の任意の整数倍",
      "phase_short": "90度整数倍",
      "phase_rule_en": "an integer multiple of 90 degrees",
      "phase_short_en": "quarter turns",
      "statement": "正の整数 \\(n\\) に対し、一辺2の正方形 \\(3n+1\\) 枚を帯状につなぎ、隣り合う折り目の関係を端から 垂直、垂直、平行 の周期とする。端の1枚を固定して各折り目を90度の整数倍だけ折る。両端の正方形の中心を \\(O,Q\\) とするとき、\\(OQ^2\\) の最大値を求めよ。",
      "statement_en": "For a positive integer n, join 3n+1 squares of side 2 in a strip. From one end, let the relationships between successive hinge axes repeat as perpendicular, perpendicular, parallel. Fix the first square and fold every hinge through an integer multiple of 90 degrees. If O and Q are the centres of the two end squares, find the maximum of OQ².",
      "answer_tex": "20n^2",
      "coefficient": 20,
      "phase_word_count": 64,
      "spatial_reasons": [
        "same_endpoint_different_future_witness"
      ],
      "equality": {
        "translation": [
          4,
          2,
          0
        ],
        "prefix": [],
        "cycle": [
          [
            0,
            0,
            0
          ]
        ],
        "cycle_state_count": 1
      }
    },
    {
      "candidate_id": "spatial-fold-b66e572a09e31fca",
      "axis_motif": "uuvu",
      "motif": [
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        }
      ],
      "relations": [
        "parallel",
        "perpendicular",
        "perpendicular",
        "parallel"
      ],
      "phase_alphabet": [
        0,
        1,
        2,
        3
      ],
      "phase_rule": "90度の任意の整数倍",
      "phase_short": "90度整数倍",
      "phase_rule_en": "an integer multiple of 90 degrees",
      "phase_short_en": "quarter turns",
      "statement": "正の整数 \\(n\\) に対し、一辺2の正方形 \\(4n+1\\) 枚を帯状につなぎ、隣り合う折り目の関係を端から 平行、垂直、垂直、平行 の周期とする。端の1枚を固定して各折り目を90度の整数倍だけ折る。両端の正方形の中心を \\(O,Q\\) とするとき、\\(OQ^2\\) の最大値を求めよ。",
      "statement_en": "For a positive integer n, join 4n+1 squares of side 2 in a strip. From one end, let the relationships between successive hinge axes repeat as parallel, perpendicular, perpendicular, parallel. Fix the first square and fold every hinge through an integer multiple of 90 degrees. If O and Q are the centres of the two end squares, find the maximum of OQ².",
      "answer_tex": "40n^2",
      "coefficient": 40,
      "phase_word_count": 256,
      "spatial_reasons": [
        "same_endpoint_different_future_witness"
      ],
      "equality": {
        "translation": [
          6,
          2,
          0
        ],
        "prefix": [],
        "cycle": [
          [
            0,
            0,
            0,
            0
          ]
        ],
        "cycle_state_count": 1
      }
    },
    {
      "candidate_id": "spatial-fold-6e6113d4b056d5cf",
      "axis_motif": "uvuu",
      "motif": [
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        }
      ],
      "relations": [
        "perpendicular",
        "perpendicular",
        "parallel",
        "parallel"
      ],
      "phase_alphabet": [
        0,
        1,
        2,
        3
      ],
      "phase_rule": "90度の任意の整数倍",
      "phase_short": "90度整数倍",
      "phase_rule_en": "an integer multiple of 90 degrees",
      "phase_short_en": "quarter turns",
      "statement": "正の整数 \\(n\\) に対し、一辺2の正方形 \\(4n+1\\) 枚を帯状につなぎ、隣り合う折り目の関係を端から 垂直、垂直、平行、平行 の周期とする。端の1枚を固定して各折り目を90度の整数倍だけ折る。両端の正方形の中心を \\(O,Q\\) とするとき、\\(OQ^2\\) の最大値を求めよ。",
      "statement_en": "For a positive integer n, join 4n+1 squares of side 2 in a strip. From one end, let the relationships between successive hinge axes repeat as perpendicular, perpendicular, parallel, parallel. Fix the first square and fold every hinge through an integer multiple of 90 degrees. If O and Q are the centres of the two end squares, find the maximum of OQ².",
      "answer_tex": "40n^2",
      "coefficient": 40,
      "phase_word_count": 256,
      "spatial_reasons": [
        "same_endpoint_different_future_witness"
      ],
      "equality": {
        "translation": [
          6,
          2,
          0
        ],
        "prefix": [],
        "cycle": [
          [
            0,
            0,
            0,
            0
          ]
        ],
        "cycle_state_count": 1
      }
    },
    {
      "candidate_id": "spatial-fold-9b86cd4f0db34d60",
      "axis_motif": "uvvu",
      "motif": [
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        }
      ],
      "relations": [
        "perpendicular",
        "parallel",
        "perpendicular",
        "parallel"
      ],
      "phase_alphabet": [
        0,
        1,
        2,
        3
      ],
      "phase_rule": "90度の任意の整数倍",
      "phase_short": "90度整数倍",
      "phase_rule_en": "an integer multiple of 90 degrees",
      "phase_short_en": "quarter turns",
      "statement": "正の整数 \\(n\\) に対し、一辺2の正方形 \\(4n+1\\) 枚を帯状につなぎ、隣り合う折り目の関係を端から 垂直、平行、垂直、平行 の周期とする。端の1枚を固定して各折り目を90度の整数倍だけ折る。両端の正方形の中心を \\(O,Q\\) とするとき、\\(OQ^2\\) の最大値を求めよ。",
      "statement_en": "For a positive integer n, join 4n+1 squares of side 2 in a strip. From one end, let the relationships between successive hinge axes repeat as perpendicular, parallel, perpendicular, parallel. Fix the first square and fold every hinge through an integer multiple of 90 degrees. If O and Q are the centres of the two end squares, find the maximum of OQ².",
      "answer_tex": "32n^2",
      "coefficient": 32,
      "phase_word_count": 256,
      "spatial_reasons": [
        "same_endpoint_different_future_witness"
      ],
      "equality": {
        "translation": [
          4,
          4,
          0
        ],
        "prefix": [],
        "cycle": [
          [
            0,
            0,
            0,
            0
          ]
        ],
        "cycle_state_count": 1
      }
    },
    {
      "candidate_id": "spatial-fold-841ff868c3f8b585",
      "axis_motif": "uuuvu",
      "motif": [
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        }
      ],
      "relations": [
        "parallel",
        "parallel",
        "perpendicular",
        "perpendicular",
        "parallel"
      ],
      "phase_alphabet": [
        0,
        1,
        2,
        3
      ],
      "phase_rule": "90度の任意の整数倍",
      "phase_short": "90度整数倍",
      "phase_rule_en": "an integer multiple of 90 degrees",
      "phase_short_en": "quarter turns",
      "statement": "正の整数 \\(n\\) に対し、一辺2の正方形 \\(5n+1\\) 枚を帯状につなぎ、隣り合う折り目の関係を端から 平行、平行、垂直、垂直、平行 の周期とする。端の1枚を固定して各折り目を90度の整数倍だけ折る。両端の正方形の中心を \\(O,Q\\) とするとき、\\(OQ^2\\) の最大値を求めよ。",
      "statement_en": "For a positive integer n, join 5n+1 squares of side 2 in a strip. From one end, let the relationships between successive hinge axes repeat as parallel, parallel, perpendicular, perpendicular, parallel. Fix the first square and fold every hinge through an integer multiple of 90 degrees. If O and Q are the centres of the two end squares, find the maximum of OQ².",
      "answer_tex": "68n^2",
      "coefficient": 68,
      "phase_word_count": 1024,
      "spatial_reasons": [
        "same_endpoint_different_future_witness"
      ],
      "equality": {
        "translation": [
          8,
          2,
          0
        ],
        "prefix": [],
        "cycle": [
          [
            0,
            0,
            0,
            0,
            0
          ]
        ],
        "cycle_state_count": 1
      }
    },
    {
      "candidate_id": "spatial-fold-6f5ce9dd535f6f8a",
      "axis_motif": "uuvuu",
      "motif": [
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "v",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        },
        {
          "axis": "u",
          "side": 1
        }
      ],
      "relations": [
        "parallel",
        "perpendicular",
        "perpendicular",
        "parallel",
        "parallel"
      ],
      "phase_alphabet": [
        0,
        1,
        2,
        3
      ],
      "phase_rule": "90度の任意の整数倍",
      "phase_short": "90度整数倍",
      "phase_rule_en": "an integer multiple of 90 degrees",
      "phase_short_en": "quarter turns",
      "statement": "正の整数 \\(n\\) に対し、一辺2の正方形 \\(5n+1\\) 枚を帯状につなぎ、隣り合う折り目の関係を端から 平行、垂直、垂直、平行、平行 の周期とする。端の1枚を固定して各折り目を90度の整数倍だけ折る。両端の正方形の中心を \\(O,Q\\) とするとき、\\(OQ^2\\) の最大値を求めよ。",
      "statement_en": "For a positive integer n, join 5n+1 squares of side 2 in a strip. From one end, let the relationships between successive hinge axes repeat as parallel, perpendicular, perpendicular, parallel, parallel. Fix the first square and fold every hinge through an integer multiple of 90 degrees. If O and Q are the centres of the two end squares, find the maximum of OQ².",
      "answer_tex": "68n^2",
      "coefficient": 68,
      "phase_word_count": 1024,
      "spatial_reasons": [
        "same_endpoint_different_future_witness"
      ],
      "equality": {
        "translation": [
          8,
          2,
          0
        ],
        "prefix": [],
        "cycle": [
          [
            0,
            0,
            0,
            0,
            0
          ]
        ],
        "cycle_state_count": 1
      }
    }
  ],
  "independent_audit": {
    "all_exact": true,
    "candidate_count": 12,
    "portfolio_membership_exact": true,
    "ranking_sorted_exact": true
  },
  "scope": {
    "rigid_panel_end_states": true,
    "continuous_collision_free_motion_claimed": false,
    "human_difficulty_inferred": false,
    "answers_hidden_initially": true
  }
}