{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261116-r06-topological-ordering-dag-0098",
  "pilot_number": "P1000-0848",
  "number": 848,
  "family_id": "qf-mod47-toposort-cycle-boundary",
  "title": {
    "zh-Hant": "第 848 題：有向無環圖的拓撲排序",
    "en": "Question 848: An Explicit Directed Cycle Refutes Every Topological Order"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261116-r06-topological-ordering-dag-0098",
    "batch_id": "batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-topological-ordering-dag"
    ],
    "title_zh": "有向無環圖的拓撲排序",
    "stem_zh": "finite digraph、完整vertex permutation與all-edge order責任。操作=toposort-cycle-boundary；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v0\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v1\",\"v3\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[[v0, v1, v2, v3, v4, v5], [v5], v5]], [v0, v1, v2, v3, v4], [v0, v2, v0], 是, 是, 否, directed_cycle_blocks_every_topological_order_while_multiple_orders_and_empty_edges_are_allowed)",
      "explanation_zh": "Wave6第十層核對=([[[v0, v1, v2, v3, v4, v5], [v5], v5]], [v0, v1, v2, v3, v4], [v0, v2, v0], 是, 是, 否, directed_cycle_blocks_every_topological_order_while_multiple_orders_and_empty_edges_are_allowed)；依完整明示資料獨立重算；全部orders、Kahn source-removal trace與cycle witness交叉核對；invalid candidate不等於cycle；multiple orders與empty-edge graph允許且不混同topological space。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有向無環圖的拓撲排序"
      ]
    },
    "variant": {
      "family_id": "qf-mod47-toposort-cycle-boundary",
      "seed": 202611166098,
      "parameters": {
        "vertices": [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5"
        ],
        "edges": [
          [
            "v0",
            "v1"
          ],
          [
            "v1",
            "v2"
          ],
          [
            "v2",
            "v0"
          ],
          [
            "v0",
            "v2"
          ],
          [
            "v0",
            "v3"
          ],
          [
            "v0",
            "v4"
          ],
          [
            "v1",
            "v3"
          ]
        ],
        "domain": "Complete_finite_topological_ordering",
        "operation": "toposort-cycle-boundary",
        "semantic_template": "qf-mod47-toposort-cycle-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v1\",\"v3\"],[\"v2\",\"v0\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261116-r06-topological-ordering-dag-0098",
    "batch_id": "batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3df457fe27323ef6615f4093908bb7143bd81c6904798c874829dc8e476637d2",
      "family_id": "qf-mod47-toposort-cycle-boundary",
      "seed": 202611166098,
      "semantic_template": "qf-mod47-toposort-cycle-boundary/{\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v1\",\"v3\"],[\"v2\",\"v0\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"
    },
    "fields": {
      "title": "An Explicit Directed Cycle Refutes Every Topological Order",
      "stem": "Use unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free directed edges=[[v0, v1], [v1, v2], [v2, v0], [v0, v2], [v0, v3], [v0, v4], [v1, v3]]. Every edge u→v needs position(u)<position(v) in a complete vertex permutation; labels use source lexical order. Give (source_removal_trace, remaining_vertices, directed_cycle, cycle_detected, no_order_certificate, uniqueness_or_extra_claim, boundary).",
      "answer": {
        "kind": "expression",
        "value": "([[[v0, v1, v2, v3, v4, v5], [v5], v5]], [v0, v1, v2, v3, v4], [v0, v2, v0], Yes, Yes, No, directed_cycle_blocks_every_topological_order_while_multiple_orders_and_empty_edges_are_allowed)",
        "explanation": "Require a genuine directed cycle, not merely a failed candidate. Retain the source-removal prefix, blocked remaining vertices and a closed directed cycle witness. Remaining vertices can include tails outside that cycle. Do not infer uniqueness; multiple orders and edgeless graphs remain allowed when no cycle exists. The exact result, retaining the requested order, is ([[[v0, v1, v2, v3, v4, v5], [v5], v5]], [v0, v1, v2, v3, v4], [v0, v2, v0], Yes, Yes, No, directed_cycle_blocks_every_topological_order_while_multiple_orders_and_empty_edges_are_allowed)."
      },
      "skills": [
        "An Explicit Directed Cycle Refutes Every Topological Order"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "finite digraph、完整vertex permutation與all-edge order責任。操作=toposort-cycle-boundary；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v0\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v1\",\"v3\"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free directed edges=[[v0, v1], [v1, v2], [v2, v0], [v0, v2], [v0, v3], [v0, v4], [v1, v3]]. Every edge u→v needs position(u)<position(v) in a complete vertex permutation; labels use source lexical order. Give (source_removal_trace, remaining_vertices, directed_cycle, cycle_detected, no_order_certificate, uniqueness_or_extra_claim, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "V=[v0, v1, v2, v3, v4, v5]",
      "after": "all directed edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v2, v0]]",
      "relation": "implies",
      "rule_id": "full_directed_graph",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是完整指定的有限有向圖，不是圖論術語的完全圖。",
        "en": "This is a fully specified finite digraph, not necessarily a complete digraph."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "remaining=[v0, v1, v2, v3, v4, v5]; active edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v2, v0]]",
      "after": "indegrees=[[v0, 1], [v1, 1], [v2, 2], [v3, 2], [v4, 1], [v5, 0]]; sources=[v5]",
      "relation": "implies",
      "rule_id": "remaining_indegree",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次都以尚未移除的邊重新計算入度，不以原始入度硬套全部步驟。",
        "en": "Recompute indegrees from remaining edges at every stage, rather than reusing original indegrees."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "zero-indegree choices=[v5]",
      "after": "choose=v5; new prefix=[v5]",
      "relation": "implies",
      "rule_id": "lexicographic_kahn_choice",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按已宣告的字典序挑第一個零入度點，並移除它發出的邊。",
        "en": "Choose the lexicographically first source and remove its outgoing edges."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "remaining=[v0, v1, v2, v3, v4]; active edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v1, v2], [v1, v3], [v2, v0]]",
      "after": "indegrees=[[v0, 1], [v1, 1], [v2, 2], [v3, 2], [v4, 1]]; sources=[]",
      "relation": "implies",
      "rule_id": "remaining_indegree",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次都以尚未移除的邊重新計算入度，不以原始入度硬套全部步驟。",
        "en": "Recompute indegrees from remaining edges at every stage, rather than reusing original indegrees."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "cycle candidate edge v0→v2",
      "after": "edge present: 是",
      "relation": "implies",
      "rule_id": "cycle_edge_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐邊核對回到起點的有向閉路，不只憑Kahn停住的印象宣告。",
        "en": "Check every edge of the directed closed path returning to its start, not merely an impression that Kahn became stuck."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "cycle candidate edge v2→v0",
      "after": "edge present: 是",
      "relation": "implies",
      "rule_id": "cycle_edge_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐邊核對回到起點的有向閉路，不只憑Kahn停住的印象宣告。",
        "en": "Check every edge of the directed closed path returning to its start, not merely an impression that Kahn became stuck."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "[v0, v2, v0]",
      "after": "pos(v0) < pos(v2) < pos(v0) is impossible",
      "relation": "implies",
      "rule_id": "strict_order_cycle_contradiction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若有拓撲次序，沿完整有向環的嚴格位置不等式會要求起點先於自己，矛盾。",
        "en": "A topological order would give a chain of strict position inequalities around the complete directed cycle, forcing the start before itself, a contradiction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "核對完整頂點與邊；一個失敗次序不等於有環，全部合法分支不能漏列。",
          "en": "Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        [
          [
            "v0",
            "v1",
            "v2",
            "v3",
            "v4",
            "v5"
          ],
          [
            "v5"
          ],
          "v5"
        ]
      ],
      [
        "v0",
        "v1",
        "v2",
        "v3",
        "v4"
      ],
      [
        "v0",
        "v2",
        "v0"
      ],
      true,
      true,
      false,
      "directed_cycle_blocks_every_topological_order_while_multiple_orders_and_empty_edges_are_allowed"
    ]
  },
  "conclusion": {
    "canonical_value": "([[[v0, v1, v2, v3, v4, v5], [v5], v5]], [v0, v1, v2, v3, v4], [v0, v2, v0], 是, 是, 否, directed_cycle_blocks_every_topological_order_while_multiple_orders_and_empty_edges_are_allowed)",
    "explanation": {
      "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。",
      "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings."
    }
  },
  "definition_note": {
    "zh-Hant": "Kahn 停住時另列逐邊核對的有向環；環上的嚴格位置不等式使拓撲次序不可能。",
    "en": "When Kahn stops, give an edge-checked directed cycle; strict position inequalities around it make a topological order impossible."
  },
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}