{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261104-r06-spanning-tree-0044",
  "pilot_number": "R0035-18444",
  "number": 18444,
  "family_id": "qf-mod35-spantree-n-minus-one-certificate",
  "title": {
    "zh-Hant": "第 18444 題：生成樹",
    "en": "Question 18444: Counting n−1 Edges Alone Does Not Establish a Tree"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261104-r06-spanning-tree-0044",
    "batch_id": "batch-20261014-advance-20261104-r06-spanning-tree-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-spanning-tree"
    ],
    "title_zh": "生成樹",
    "stem_zh": "完整finite connected simple parent graph與parent edge subset。操作=spantree-n-minus-one-certificate；資料={\"n\":5,\"edges\":[[0,1],[0,3],[1,2],[2,3],[2,4],[3,4]],\"selected\":[0,2,3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(5, 4, 是, 是, 是, 是, n_minus_one_alone_insufficient)",
      "explanation_zh": "Wave5第四層核對=(5, 4, 是, 是, 是, 是, n_minus_one_alone_insufficient)；依完整明示資料重算；spanning tree含全部vertices且connected acyclic，n-1 alone不足；single vertex合法且不作weighted或minimum claim。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "生成樹"
      ]
    },
    "variant": {
      "family_id": "qf-mod35-spantree-n-minus-one-certificate",
      "seed": 202611046044,
      "parameters": {
        "n": 5,
        "edges": [
          [
            0,
            1
          ],
          [
            0,
            3
          ],
          [
            1,
            2
          ],
          [
            2,
            3
          ],
          [
            2,
            4
          ],
          [
            3,
            4
          ]
        ],
        "selected": [
          0,
          2,
          3,
          5
        ],
        "domain": "Complete_finite_graph_spanning_tree",
        "operation": "spantree-n-minus-one-certificate",
        "semantic_template": "qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,3,5]}"
      }
    },
    "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、sampled rank-nullity、infinite-domain uniformity、term-zero convergence、estimator properties、noncoprime CRT shortcut、MST、mode implication、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261104-r06-spanning-tree-0044",
    "batch_id": "batch-20261014-advance-20261104-r06-spanning-tree-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3143bf8672fb3c1f047cfc2965869d13d71b9945a579418f42eb0fd4bd9a7d59",
      "family_id": "qf-mod35-spantree-n-minus-one-certificate",
      "seed": 202611046044,
      "semantic_template": "qf-mod35-spantree-n-minus-one-certificate/{\"edges\":[[0,1],[0,3],[1,2],[2,3],[2,4],[3,4]],\"n\":5,\"selected\":[0,2,3,5]}"
    },
    "fields": {
      "title": "Counting n−1 Edges Alone Does Not Establish a Tree",
      "stem": "Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)",
        "explanation": "Compare the selected edge count with n−1 and retain both connectivity and acyclicity checks. A cycle on some vertices with an isolated vertex can have n−1 edges and still fail to span as a tree. The exact result, retaining the requested order, is (5, 4, Yes, Yes, Yes, Yes, n_minus_one_alone_insufficient)."
      },
      "skills": [
        "Counting n−1 Edges Alone Does Not Establish a Tree"
      ],
      "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": "完整finite connected simple parent graph與parent edge subset。操作=spantree-n-minus-one-certificate；資料={\"n\":5,\"edges\":[[0,1],[0,3],[1,2],[2,3],[2,4],[3,4]],\"selected\":[0,2,3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the complete finite undirected loop-free simple parent graph on Fin(5)={0,…,4}, canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]], and distinct parent-edge selection=[0, 2, 3, 5], sorted increasingly. The selected subgraph retains every parent vertex, including isolated vertices. Audit parent connectivity, selected connectivity and acyclicity separately. A spanning tree is connected and acyclic on that full carrier; n−1 selected edges alone do not suffice. No edge weights or minimum-weight objective are supplied. Give (vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=spantree-n-minus-one-certificate;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[vertex_count, selected_edge_count, connected, acyclic, edge_count_is_n_minus_one, spanning_tree, boundary]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題的歷史介面標記保留在快照；本頁直接列原始欄位順序，不需JavaScript或外部表。欄位描述不自動成為數學證據，No標記須依本題的聲稱範圍解讀。",
        "en": "Keep the historical interface marker in the original snapshot while giving ordered fields here,without JavaScript or an external table.Labels are not mathematical evidence,and each No marker retains the operation's claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "complete original vertex carrier Fin(5);canonical original edges=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]];raw selected edge indices=[0, 2, 3, 5]",
      "after": "sorted unique ORIGINAL selection=[0, 2, 3, 5];selected edges=[[0, 1], [1, 2], [2, 3], [3, 4]];ALL parent vertices retained;no10-edge cap or weights",
      "relation": "implies",
      "rule_id": "spantree_original_graph_whole_carrier_not_matching_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本題頂點上界12但沒有maximum-matching的10邊上限。只選原邊、保留全部頂點及孤立點，選邊排序不改原索引。沒有權重或最小化目標。",
        "en": "This family bounds vertices by12 but has no maximum-matching10-edge cap.Select only original edges,retain every vertex including isolates,and normalize selection order without changing original indices.No weights or minimization objective are supplied."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "parent graph:actual edge index=0;edge=[0, 1];reached from=0;new vertex=1",
      "after": "original-edge path from0=[0, 1];reached vertices=[0, 1]",
      "relation": "implies",
      "rule_id": "spantree_every_discovered_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。",
        "en": "Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "parent graph:actual edge index=1;edge=[0, 3];reached from=0;new vertex=3",
      "after": "original-edge path from0=[0, 3];reached vertices=[0, 1, 3]",
      "relation": "implies",
      "rule_id": "spantree_every_discovered_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。",
        "en": "Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "parent graph:actual edge index=2;edge=[1, 2];reached from=1;new vertex=2",
      "after": "original-edge path from0=[0, 1, 2];reached vertices=[0, 1, 2, 3]",
      "relation": "implies",
      "rule_id": "spantree_every_discovered_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。",
        "en": "Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "parent graph:actual edge index=5;edge=[3, 4];reached from=3;new vertex=4",
      "after": "original-edge path from0=[0, 3, 4];reached vertices=[0, 1, 2, 3, 4]",
      "relation": "implies",
      "rule_id": "spantree_every_discovered_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。",
        "en": "Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "parent graph:complete carrier Fin(5);ALL graph edges=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]];reached=[0, 1, 2, 3, 4];unreached=[]",
      "after": "every graph edge stays within reached or its complement;connected=是",
      "relation": "implies",
      "rule_id": "spantree_reachability_and_closed_cut_completeness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "queue耗盡後，沒有邊跨越已達與未達集合；有未達點便無從0的路徑，無未達點則路徑見證覆蓋全集。單點空邊圖由0自己覆蓋而connected。",
        "en": "After the queue is exhausted,no edge crosses the reached/unreached cut.Unreached vertices therefore have no path from0;otherwise path witnesses cover the carrier.A one-vertex empty-edge graph is connected via its initial vertex."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "selected spanning subgraph:actual edge index=0;edge=[0, 1];reached from=0;new vertex=1",
      "after": "original-edge path from0=[0, 1];reached vertices=[0, 1]",
      "relation": "implies",
      "rule_id": "spantree_every_discovered_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。",
        "en": "Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "selected spanning subgraph:actual edge index=1;edge=[1, 2];reached from=1;new vertex=2",
      "after": "original-edge path from0=[0, 1, 2];reached vertices=[0, 1, 2]",
      "relation": "implies",
      "rule_id": "spantree_every_discovered_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。",
        "en": "Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "selected spanning subgraph:actual edge index=2;edge=[2, 3];reached from=2;new vertex=3",
      "after": "original-edge path from0=[0, 1, 2, 3];reached vertices=[0, 1, 2, 3]",
      "relation": "implies",
      "rule_id": "spantree_every_discovered_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。",
        "en": "Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "selected spanning subgraph:actual edge index=3;edge=[3, 4];reached from=3;new vertex=4",
      "after": "original-edge path from0=[0, 1, 2, 3, 4];reached vertices=[0, 1, 2, 3, 4]",
      "relation": "implies",
      "rule_id": "spantree_every_discovered_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個新點都有由真實邊串成的0起點路徑，不用頂點數或邊數猜connected。",
        "en": "Each newly reached vertex has an actual-edge path from0;do not guess connectivity from counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "selected spanning subgraph:complete carrier Fin(5);ALL graph edges=[[0, 1], [1, 2], [2, 3], [3, 4]];reached=[0, 1, 2, 3, 4];unreached=[]",
      "after": "every graph edge stays within reached or its complement;connected=是",
      "relation": "implies",
      "rule_id": "spantree_reachability_and_closed_cut_completeness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "queue耗盡後，沒有邊跨越已達與未達集合；有未達點便無從0的路徑，無未達點則路徑見證覆蓋全集。單點空邊圖由0自己覆蓋而connected。",
        "en": "After the queue is exhausted,no edge crosses the reached/unreached cut.Unreached vertices therefore have no path from0;otherwise path witnesses cover the carrier.A one-vertex empty-edge graph is connected via its initial vertex."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "original selected edge index=0;edge=[0, 1];previous forest component roots=[0, 1]",
      "after": "closes cycle=否;actual cycle witness=[]",
      "relation": "implies",
      "rule_id": "spantree_every_edge_cycle_or_forest_extension",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩端原已在同forest分量時，舊forest路徑加此真邊成cycle；否則連兩分量仍無cycle。逐全部選邊，不能只以n−1數量判。",
        "en": "If endpoints were already in the same forest component,the previous forest path plus this actual edge gives a cycle.Otherwise joining two components preserves acyclicity.Check every selected edge,not merely n−1 counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "original selected edge index=2;edge=[1, 2];previous forest component roots=[0, 2]",
      "after": "closes cycle=否;actual cycle witness=[]",
      "relation": "implies",
      "rule_id": "spantree_every_edge_cycle_or_forest_extension",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩端原已在同forest分量時，舊forest路徑加此真邊成cycle；否則連兩分量仍無cycle。逐全部選邊，不能只以n−1數量判。",
        "en": "If endpoints were already in the same forest component,the previous forest path plus this actual edge gives a cycle.Otherwise joining two components preserves acyclicity.Check every selected edge,not merely n−1 counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "original selected edge index=3;edge=[2, 3];previous forest component roots=[0, 3]",
      "after": "closes cycle=否;actual cycle witness=[]",
      "relation": "implies",
      "rule_id": "spantree_every_edge_cycle_or_forest_extension",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩端原已在同forest分量時，舊forest路徑加此真邊成cycle；否則連兩分量仍無cycle。逐全部選邊，不能只以n−1數量判。",
        "en": "If endpoints were already in the same forest component,the previous forest path plus this actual edge gives a cycle.Otherwise joining two components preserves acyclicity.Check every selected edge,not merely n−1 counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "original selected edge index=5;edge=[3, 4];previous forest component roots=[0, 4]",
      "after": "closes cycle=否;actual cycle witness=[]",
      "relation": "implies",
      "rule_id": "spantree_every_edge_cycle_or_forest_extension",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩端原已在同forest分量時，舊forest路徑加此真邊成cycle；否則連兩分量仍無cycle。逐全部選邊，不能只以n−1數量判。",
        "en": "If endpoints were already in the same forest component,the previous forest path plus this actual edge gives a cycle.Otherwise joining two components preserves acyclicity.Check every selected edge,not merely n−1 counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "same complete vertex carrier Fin(5);parent connected=是;selected connected=是;all cycle witnesses=[]",
      "after": "selected acyclic=是;spanning tree validity=是",
      "relation": "implies",
      "rule_id": "spantree_connectivity_acyclicity_full_vertices_conjunction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "parent若不連通便無spanning tree；parent連通時一次選取失敗不否定其他選取存在。這裡同時核全部頂點、selected connected與acyclic，孤立點沒有被刪掉。",
        "en": "A disconnected parent has no spanning tree,but failure of one selection in a connected parent does not refute existence of another.This audit retains all vertices and conjoins selected connectivity and acyclicity without dropping isolates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "spanning tree vertex count minus one := sub(5, 1)",
      "after": "spanning tree vertex count minus one = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確計算，保留索引、分母、正負號及所屬載體。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 spanning tree vertex count minus one，不以小數近似替代。",
          "en": "Recompute spanning tree vertex count minus one with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "1"
        ],
        "output": "4",
        "label": "spanning tree vertex count minus one"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "ACTUAL vertex count=5;selected edge count=4;n-1=4;selected connected=是;acyclic=是",
      "after": "edge count matches=是;tree=是;n_minus_one_alone_insufficient",
      "relation": "implies",
      "rule_id": "spantree_edge_count_not_sufficient_alone",
      "conditions": [],
      "explanation": {
        "zh-Hant": "某些cycle加孤立點也可有n−1邊；本欄只比較數量，真正tree仍要完整connected/acyclic證書。",
        "en": "A cycle plus an isolate can also have n−1 edges.This field compares counts only;tree validity still needs complete connectivity and acyclicity certificates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      5,
      4,
      true,
      true,
      true,
      true,
      "n_minus_one_alone_insufficient"
    ]
  },
  "conclusion": {
    "canonical_value": "(5, 4, 是, 是, 是, 是, n_minus_one_alone_insufficient)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261104-r06-spanning-tree-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261104-r06-spanning-tree-foundations/questions.jsonl",
      "batch_sha256": "57d1689f51e99f5f643e93bda9acb1314ed45bbfc9bbac858eeedbed25268ad9",
      "original_line": 44,
      "original_line_sha256": "45026b72dedb7d5a79d18dd734692c9e2d2692b702e4afe509b117d1b3ad4cab",
      "question_compact_sha256": "3143bf8672fb3c1f047cfc2965869d13d71b9945a579418f42eb0fd4bd9a7d59",
      "english_row_compact_sha256": "44ed0bebf153f896516e20e71a4a094a6517821d33ac141062b12f3cee0656ab",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261104-r06-spanning-tree-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261104-r06-spanning-tree-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-spanning-tree": {
        "sha256": "f93b1e5d271c80d9a571d19d5ddba8bceb178dce9bc9c8e6f3b1b0279a0d3833",
        "english_sha256": "32cad5f8907e94e691500db0fe35e01257bbe42816f694ffb01b2277a26d725d",
        "original_path": "public/data/mko/spanning-tree.json",
        "english_path": "public/data/i18n/editions/mko-spanning-tree.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "9adff0bbee39d9c3dec346304161ea9633576ccd5ca954f6b64ad482d2533a8a",
    "original_candidate_sha256": "31af4bfedd598243831f3270b045fc0dcafbb3babcbecc1326c3d07991fb817f",
    "original_candidate_line_sha256": "31af4bfedd598243831f3270b045fc0dcafbb3babcbecc1326c3d07991fb817f"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-008-p36",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "713db6309b22bf8b31f0253d4aaa7aca9011ab908587ae0e9d4a365edce1f9bc",
      "question_first_witness_sha256": "0338a3db344ab237113fc63be29153c3684da96e388109e9dcd913a0a693aed2"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 19,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "113a5c1ef8c05729fdbb3bc7604e44e4df4a4c390c482a6965dc7bbf96764d6e",
      "arithmetic_steps": 1
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}