{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261116-r06-topological-ordering-dag-0003",
  "pilot_number": "P250-0203",
  "number": 203,
  "family_id": "qf-mod47-toposort-order-audit",
  "title": {
    "zh-Hant": "第 203 題：有向無環圖的拓撲排序",
    "en": "Question 203: Vertex-Set Equality Is Not Yet a Complete Valid Topological Permutation"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261116-r06-topological-ordering-dag-0003",
    "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-order-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([v0, v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], 是, [[[v0, v1], 0, 1, 是], [[v0, v2], 0, 2, 是]], 是, invalid_candidate_does_not_itself_prove_a_cycle)",
      "explanation_zh": "Wave6第十層核對=([v0, v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], 是, [[[v0, v1], 0, 1, 是], [[v0, v2], 0, 2, 是]], 是, invalid_candidate_does_not_itself_prove_a_cycle)；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": 1,
      "skills": [
        "有向無環圖的拓撲排序"
      ]
    },
    "variant": {
      "family_id": "qf-mod47-toposort-order-audit",
      "seed": 202611166003,
      "parameters": {
        "vertices": [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5"
        ],
        "edges": [
          [
            "v0",
            "v1"
          ],
          [
            "v0",
            "v2"
          ]
        ],
        "candidate": [
          "v0",
          "v1",
          "v2",
          "v3",
          "v4",
          "v5"
        ],
        "domain": "Complete_finite_topological_ordering",
        "operation": "toposort-order-audit",
        "semantic_template": "qf-mod47-toposort-order-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"]],\"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-0003",
    "batch_id": "batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "6761578cbe919287cbfed6db034ef5cd66ec10475bb8a6d42dd67e917739fb72",
      "family_id": "qf-mod47-toposort-order-audit",
      "seed": 202611166003,
      "semantic_template": "qf-mod47-toposort-order-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"
    },
    "fields": {
      "title": "Vertex-Set Equality Is Not Yet a Complete Valid Topological Permutation",
      "stem": "Use unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free directed edges=[[v0, v1], [v0, v2]]. Every edge u→v needs position(u)<position(v) in a complete vertex permutation; labels use source lexical order. Candidate=[v0, v1, v2, v3, v4, v5]. Give (vertices, candidate, same_vertex_set, edge_position_rows, valid_topological_order, boundary).",
      "answer": {
        "kind": "expression",
        "value": "([v0, v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], Yes, [[[v0, v1], 0, 1, Yes], [[v0, v2], 0, 2, Yes]], Yes, invalid_candidate_does_not_itself_prove_a_cycle)",
        "explanation": "The same_vertex_set flag checks the underlying vertex set and may still be Yes for a duplicated candidate. The final validity check additionally requires the exact permutation and every edge order. One invalid candidate does not prove the graph has a directed cycle. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], Yes, [[[v0, v1], 0, 1, Yes], [[v0, v2], 0, 2, Yes]], Yes, invalid_candidate_does_not_itself_prove_a_cycle)."
      },
      "skills": [
        "Vertex-Set Equality Is Not Yet a Complete Valid Topological Permutation"
      ],
      "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-order-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}。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], [v0, v2]]. Every edge u→v needs position(u)<position(v) in a complete vertex permutation; labels use source lexical order. Candidate=[v0, v1, v2, v3, v4, v5]. Give (vertices, candidate, same_vertex_set, edge_position_rows, valid_topological_order, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "V=[v0, v1, v2, v3, v4, v5]; candidate=[v0, v1, v2, v3, v4, v5]",
      "after": "length(candidate)=6, |V|=6",
      "relation": "implies",
      "rule_id": "permutation_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "拓撲次序必須是全部頂點恰出現一次的排列；先核對長度。",
        "en": "A topological order must list every vertex exactly once; first compare the lengths."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "只有底層集合相同還不夠，重複列出頂點仍可能違規。",
          "en": "Equal underlying sets are not enough: duplicate entries may still invalidate the candidate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "count v0 in candidate",
      "after": "count(v0)=1",
      "relation": "implies",
      "rule_id": "permutation_multiplicity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐頂點核對出現次數，不漏頂點也不重複。",
        "en": "Check the multiplicity of each vertex to detect omissions and duplicates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "完整排列要求每個次數都是 1，且沒有域外標籤。",
          "en": "A full permutation requires multiplicity 1 for every vertex and no outside label."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "count v1 in candidate",
      "after": "count(v1)=1",
      "relation": "implies",
      "rule_id": "permutation_multiplicity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐頂點核對出現次數，不漏頂點也不重複。",
        "en": "Check the multiplicity of each vertex to detect omissions and duplicates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "完整排列要求每個次數都是 1，且沒有域外標籤。",
          "en": "A full permutation requires multiplicity 1 for every vertex and no outside label."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "count v2 in candidate",
      "after": "count(v2)=1",
      "relation": "implies",
      "rule_id": "permutation_multiplicity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐頂點核對出現次數，不漏頂點也不重複。",
        "en": "Check the multiplicity of each vertex to detect omissions and duplicates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "完整排列要求每個次數都是 1，且沒有域外標籤。",
          "en": "A full permutation requires multiplicity 1 for every vertex and no outside label."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "count v3 in candidate",
      "after": "count(v3)=1",
      "relation": "implies",
      "rule_id": "permutation_multiplicity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐頂點核對出現次數，不漏頂點也不重複。",
        "en": "Check the multiplicity of each vertex to detect omissions and duplicates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "完整排列要求每個次數都是 1，且沒有域外標籤。",
          "en": "A full permutation requires multiplicity 1 for every vertex and no outside label."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "count v4 in candidate",
      "after": "count(v4)=1",
      "relation": "implies",
      "rule_id": "permutation_multiplicity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐頂點核對出現次數，不漏頂點也不重複。",
        "en": "Check the multiplicity of each vertex to detect omissions and duplicates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "完整排列要求每個次數都是 1，且沒有域外標籤。",
          "en": "A full permutation requires multiplicity 1 for every vertex and no outside label."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "count v5 in candidate",
      "after": "count(v5)=1",
      "relation": "implies",
      "rule_id": "permutation_multiplicity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐頂點核對出現次數，不漏頂點也不重複。",
        "en": "Check the multiplicity of each vertex to detect omissions and duplicates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "完整排列要求每個次數都是 1，且沒有域外標籤。",
          "en": "A full permutation requires multiplicity 1 for every vertex and no outside label."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "edge v0→v1",
      "after": "pos(v0)=0; pos(v1)=1; 0<1: 是",
      "relation": "implies",
      "rule_id": "directed_edge_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每條有向邊都要求起點出現在終點之前；位置從 0 開始。",
        "en": "Every directed edge requires its source before its target; positions are zero-based."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "一條邊通過不能代替全部邊通過。",
          "en": "One passing edge cannot stand in for all edges."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "edge v0→v2",
      "after": "pos(v0)=0; pos(v2)=2; 0<2: 是",
      "relation": "implies",
      "rule_id": "directed_edge_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每條有向邊都要求起點出現在終點之前；位置從 0 開始。",
        "en": "Every directed edge requires its source before its target; positions are zero-based."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "一條邊通過不能代替全部邊通過。",
          "en": "One passing edge cannot stand in for all edges."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "permutation=是; every edge passes=是",
      "after": "valid topological order=是",
      "relation": "implies",
      "rule_id": "definition_conjunction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整排列與逐邊限制必須同時成立；這正是本有限圖的拓撲次序定義。",
        "en": "The full permutation and every edge constraint must both hold; this is exactly the definition for this finite graph."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "候選次序失敗不代表圖必有環，可能只是給錯了排列。",
          "en": "A failed candidate does not prove a cycle; the proposed permutation may simply be wrong."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "vertices": [
      "v0",
      "v1",
      "v2",
      "v3",
      "v4",
      "v5"
    ],
    "candidate": [
      "v0",
      "v1",
      "v2",
      "v3",
      "v4",
      "v5"
    ],
    "same_set": true,
    "permutation": true,
    "audits": [
      [
        [
          "v0",
          "v1"
        ],
        0,
        1,
        true
      ],
      [
        [
          "v0",
          "v2"
        ],
        0,
        2,
        true
      ]
    ],
    "valid": true
  },
  "conclusion": {
    "canonical_value": "([v0, v1, v2, v3, v4, v5], [v0, v1, v2, v3, v4, v5], 是, [[[v0, v1], 0, 1, 是], [[v0, v2], 0, 2, 是]], 是, invalid_candidate_does_not_itself_prove_a_cycle)",
    "explanation": {
      "zh-Hant": "本候選是否為有效拓撲次序：是。",
      "en": "Is this candidate a valid topological order? Yes."
    }
  },
  "definition_note": {
    "zh-Hant": "有限有向圖；候選必須是完整頂點排列，且每條邊均滿足起點在終點之前。",
    "en": "For the finite digraph, require a complete vertex permutation and source-before-target order for every edge."
  },
  "source": {
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      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations.en.jsonl",
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    "mko_bindings": {
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    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "fb325b42cfcaa25ddbae992459c2684afd767fe218028bc0b9c0823e67c83b7c",
    "original_candidate_sha256": "724665f1c74aa4bc3f9bfa3fb2809f13c2b9d4872fe1e1100a051259a504f67a",
    "original_candidate_line_sha256": "724665f1c74aa4bc3f9bfa3fb2809f13c2b9d4872fe1e1100a051259a504f67a"
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  "historical_source_state": {
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      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
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      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
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    "scope_note": {
      "zh-Hant": "本稿是來源綁定的教學推導候選；精確重算不等於獨立數學審查或形式化證明。原題、答案、公式、AST 與歷史未被修改。",
      "en": "This is a source-bound pedagogical derivation candidate. Exact recomputation is not independent mathematical review or formal proof. Original questions, answers, formulas, ASTs and history are unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
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    "review_date": "2026-10-24",
    "owner_authorized_reader_publication": true,
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    "review_receipts": {
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    "formalization": "not_formalized",
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}