{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261028-r07-connected-component-0081",
  "pilot_number": "R0026-13656",
  "number": 13656,
  "family_id": "qf-mod28-component-partition",
  "title": {
    "zh-Hant": "第 13656 題：連通分量",
    "en": "Question 13656: Coverage and Disjointness Must Include Isolated Vertices"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261028-r07-connected-component-0081",
    "batch_id": "batch-20261014-advance-20261028-r07-connected-component-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-connected-component"
    ],
    "title_zh": "連通分量",
    "stem_zh": "完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 1, 2, 3, 4, 5], [6]], 是, 是, [[6]], length_zero_paths_include_every_vertex)",
      "explanation_zh": "Wave4第二層核對=([[0, 1, 2, 3, 4, 5], [6]], 是, 是, [[6]], length_zero_paths_include_every_vertex)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；長度零path保證反身且孤立點為單點分量；完整可達性等價類不切換有向強弱連通。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "連通分量"
      ]
    },
    "variant": {
      "family_id": "qf-mod28-component-partition",
      "seed": 202610287081,
      "parameters": {
        "n": 7,
        "edges": [
          [
            0,
            2
          ],
          [
            0,
            4
          ],
          [
            0,
            5
          ],
          [
            1,
            3
          ],
          [
            1,
            4
          ],
          [
            2,
            3
          ],
          [
            2,
            4
          ],
          [
            2,
            5
          ],
          [
            3,
            4
          ]
        ],
        "domain": "Complete_finite_undirected_components",
        "operation": "component-partition",
        "semantic_template": "qf-mod28-component-partition/{\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],\"n\":7}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式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-20261028-r07-connected-component-0081",
    "batch_id": "batch-20261014-advance-20261028-r07-connected-component-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f8bb787b608f4e61f46e4bf0bf4469e430a84085c0679ea32e677d19d6ec229e",
      "family_id": "qf-mod28-component-partition",
      "seed": 202610287081,
      "semantic_template": "qf-mod28-component-partition/{\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],\"n\":7}"
    },
    "fields": {
      "title": "Coverage and Disjointness Must Include Isolated Vertices",
      "stem": "Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.",
      "answer": {
        "kind": "expression",
        "value": "([[0, 1, 2, 3, 4, 5], [6]], Yes, Yes, [[6]], length_zero_paths_include_every_vertex)",
        "explanation": "Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6]], Yes, Yes, [[6]], length_zero_paths_include_every_vertex)."
      },
      "skills": [
        "Coverage and Disjointness Must Include Isolated Vertices"
      ],
      "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": "完整有限無向無loop簡單圖。操作=component-partition；資料={\"n\":7,\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]]}。請依 [components,covers_vertices,disjoint,singletons,boundary] 欄序回答。",
        "en": "Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "full carrier Fin(7);raw undirected edges=[[0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4]]",
      "after": "canonical simple edges=[[0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4]];full adjacency rows=[[2, 4, 5], [3, 4], [0, 3, 4, 5], [1, 2, 4], [0, 1, 2, 3], [0, 2], []]",
      "relation": "implies",
      "rule_id": "component_full_undirected_graph_includes_isolates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用完整載體建立鄰接表，拒絕loop與重複無向邊；沒出現在邊中的頂點仍保留空鄰接列，不切換有向連通語義。",
        "en": "Build adjacency on the full carrier,rejecting loops and duplicate undirected edges.Vertices absent from edges retain empty rows;do not switch to directed connectivity."
      },
      "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": "class representative0:start vertex=0;complete adjacency=[[2, 4, 5], [3, 4], [0, 3, 4, 5], [1, 2, 4], [0, 1, 2, 3], [0, 2], []]",
      "after": "initial visited=[0];path to start=[0];edge length0",
      "relation": "implies",
      "rule_id": "component_zero_length_reflexive_path",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點以長度0路徑到自己，孤立點也必須納入，不能因沒有邊而漏掉。",
        "en": "The start reaches itself by a length-zero path.Even an isolated vertex must be included rather than omitted for having no edges."
      },
      "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": "class representative0:layer=0;frontier=[0];all frontier neighbors=[2, 4, 5]",
      "after": "previously seen=[0];new vertices=[2, 4, 5]",
      "relation": "implies",
      "rule_id": "component_complete_frontier_expansion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每層遍歷該層所有頂點的所有鄰居，去重後只加入未訪者；不提前停止於看起來足夠的子圖。",
        "en": "At each layer inspect every neighbor of every frontier vertex,then add only unvisited distinct labels.Do not stop at a plausible-looking fragment."
      },
      "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": "class representative0:new vertex=2;earliest frontier parent=0;parent path=[0]",
      "after": "append actual parent edge;reachable path=[0, 2]",
      "relation": "implies",
      "rule_id": "component_each_new_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已有父路徑後，接上一條原圖邊得到新頂點路徑，故每個被加入的點真的可達。",
        "en": "Extend the established parent path by an original graph edge,so every added vertex really is reachable."
      },
      "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": "class representative0:new vertex=4;earliest frontier parent=0;parent path=[0]",
      "after": "append actual parent edge;reachable path=[0, 4]",
      "relation": "implies",
      "rule_id": "component_each_new_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已有父路徑後，接上一條原圖邊得到新頂點路徑，故每個被加入的點真的可達。",
        "en": "Extend the established parent path by an original graph edge,so every added vertex really is reachable."
      },
      "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": "class representative0:new vertex=5;earliest frontier parent=0;parent path=[0]",
      "after": "append actual parent edge;reachable path=[0, 5]",
      "relation": "implies",
      "rule_id": "component_each_new_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已有父路徑後，接上一條原圖邊得到新頂點路徑，故每個被加入的點真的可達。",
        "en": "Extend the established parent path by an original graph edge,so every added vertex really is reachable."
      },
      "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": "class representative0:layer=1;frontier=[2, 4, 5];all frontier neighbors=[0, 1, 2, 3, 4, 5]",
      "after": "previously seen=[0, 2, 4, 5];new vertices=[1, 3]",
      "relation": "implies",
      "rule_id": "component_complete_frontier_expansion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每層遍歷該層所有頂點的所有鄰居，去重後只加入未訪者；不提前停止於看起來足夠的子圖。",
        "en": "At each layer inspect every neighbor of every frontier vertex,then add only unvisited distinct labels.Do not stop at a plausible-looking fragment."
      },
      "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": "class representative0:new vertex=1;earliest frontier parent=4;parent path=[0, 4]",
      "after": "append actual parent edge;reachable path=[0, 4, 1]",
      "relation": "implies",
      "rule_id": "component_each_new_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已有父路徑後，接上一條原圖邊得到新頂點路徑，故每個被加入的點真的可達。",
        "en": "Extend the established parent path by an original graph edge,so every added vertex really is reachable."
      },
      "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": "class representative0:new vertex=3;earliest frontier parent=2;parent path=[0, 2]",
      "after": "append actual parent edge;reachable path=[0, 2, 3]",
      "relation": "implies",
      "rule_id": "component_each_new_vertex_path_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "已有父路徑後，接上一條原圖邊得到新頂點路徑，故每個被加入的點真的可達。",
        "en": "Extend the established parent path by an original graph edge,so every added vertex really is reachable."
      },
      "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": "class representative0:layer=2;frontier=[1, 3];all frontier neighbors=[1, 2, 3, 4]",
      "after": "previously seen=[0, 1, 2, 3, 4, 5];new vertices=[]",
      "relation": "implies",
      "rule_id": "component_complete_frontier_expansion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每層遍歷該層所有頂點的所有鄰居，去重後只加入未訪者；不提前停止於看起來足夠的子圖。",
        "en": "At each layer inspect every neighbor of every frontier vertex,then add only unvisited distinct labels.Do not stop at a plausible-looking fragment."
      },
      "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": "class representative0:final visited=[0, 1, 2, 3, 4, 5];all [vertex,neighbors,outside neighbors] rows=[[0, [2, 4, 5], []], [1, [3, 4], []], [2, [0, 3, 4, 5], []], [3, [1, 2, 4], []], [4, [0, 1, 2, 3], []], [5, [0, 2], []]]",
      "after": "no edge leaves visited;every visited vertex has a path;visited equals full reachable set",
      "relation": "implies",
      "rule_id": "component_reachable_and_closed_set_completeness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "路徑見證給已訪⊆可達；搜尋完後無邊離開已訪集合，任一路徑若到外面必有第一條離開邊，矛盾，因此可達⊆已訪。兩方向才是完整分量，不是遍歷順序。",
        "en": "Path witnesses give visited⊆reachable.At completion no edge leaves visited.If a path reached outside,it would have a first leaving edge,a contradiction,so reachable⊆visited.Both directions establish the whole component,not a traversal order."
      },
      "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": "least unvisited representative=0;derived component=[0, 1, 2, 3, 4, 5]",
      "after": "all visited labels=[0, 1, 2, 3, 4, 5]",
      "relation": "implies",
      "rule_id": "component_least_unvisited_class_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "類內標籤遞增，類間按最小未訪代表順序輸出；不以BFS發現順序充當canonical分量次序。",
        "en": "Sort labels within each class and order classes by the least unvisited representative.BFS discovery order is not the canonical component order."
      },
      "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": "class representative6:start vertex=6;complete adjacency=[[2, 4, 5], [3, 4], [0, 3, 4, 5], [1, 2, 4], [0, 1, 2, 3], [0, 2], []]",
      "after": "initial visited=[6];path to start=[6];edge length0",
      "relation": "implies",
      "rule_id": "component_zero_length_reflexive_path",
      "conditions": [],
      "explanation": {
        "zh-Hant": "起點以長度0路徑到自己，孤立點也必須納入，不能因沒有邊而漏掉。",
        "en": "The start reaches itself by a length-zero path.Even an isolated vertex must be included rather than omitted for having no edges."
      },
      "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": "class representative6:layer=0;frontier=[6];all frontier neighbors=[]",
      "after": "previously seen=[6];new vertices=[]",
      "relation": "implies",
      "rule_id": "component_complete_frontier_expansion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每層遍歷該層所有頂點的所有鄰居，去重後只加入未訪者；不提前停止於看起來足夠的子圖。",
        "en": "At each layer inspect every neighbor of every frontier vertex,then add only unvisited distinct labels.Do not stop at a plausible-looking fragment."
      },
      "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": "class representative6:final visited=[6];all [vertex,neighbors,outside neighbors] rows=[[6, [], []]]",
      "after": "no edge leaves visited;every visited vertex has a path;visited equals full reachable set",
      "relation": "implies",
      "rule_id": "component_reachable_and_closed_set_completeness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "路徑見證給已訪⊆可達；搜尋完後無邊離開已訪集合，任一路徑若到外面必有第一條離開邊，矛盾，因此可達⊆已訪。兩方向才是完整分量，不是遍歷順序。",
        "en": "Path witnesses give visited⊆reachable.At completion no edge leaves visited.If a path reached outside,it would have a first leaving edge,a contradiction,so reachable⊆visited.Both directions establish the whole component,not a traversal order."
      },
      "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": "least unvisited representative=6;derived component=[6]",
      "after": "all visited labels=[0, 1, 2, 3, 4, 5, 6]",
      "relation": "implies",
      "rule_id": "component_least_unvisited_class_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "類內標籤遞增，類間按最小未訪代表順序輸出；不以BFS發現順序充當canonical分量次序。",
        "en": "Sort labels within each class and order classes by the least unvisited representative.BFS discovery order is not the canonical component order."
      },
      "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": "all components=[[0, 1, 2, 3, 4, 5], [6]];full carrier=[0, 1, 2, 3, 4, 5, 6]",
      "after": "covers=是;disjoint=是;singleton components=[[6]]",
      "relation": "implies",
      "rule_id": "component_partition_full_carrier_not_edge_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整可達等價類覆蓋所有頂點且不相交；孤立點因零長度路徑各自成單點類，不只分割出現在邊中的標籤。",
        "en": "Complete reachability classes cover all vertices disjointly.Isolated vertices form singleton classes by zero-length paths;do not partition only edge-present labels."
      },
      "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": [
      [
        [
          0,
          1,
          2,
          3,
          4,
          5
        ],
        [
          6
        ]
      ],
      true,
      true,
      [
        [
          6
        ]
      ],
      "length_zero_paths_include_every_vertex"
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 1, 2, 3, 4, 5], [6]], 是, 是, [[6]], length_zero_paths_include_every_vertex)",
    "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-20261028-r07-connected-component-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261028-r07-connected-component-foundations/questions.jsonl",
      "batch_sha256": "99467f90a1d5917daeb151fa02072250527bf26705993bbd963c365a7184090c",
      "original_line": 81,
      "original_line_sha256": "27a153c3ef428f37a975f8a49ff29f67343716718576a84e2740df546a36b168",
      "question_compact_sha256": "f8bb787b608f4e61f46e4bf0bf4469e430a84085c0679ea32e677d19d6ec229e",
      "english_row_compact_sha256": "50140f601f18c327d1b82a0cd93832c7805fefbb58dbe327167ac3aeb56ecf9b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261028-r07-connected-component-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261028-r07-connected-component-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-connected-component": {
        "sha256": "f921926c2c42fafc8d64364eb7bd06488901e081b0117ba17fc4265610e54eab",
        "english_sha256": "285fe3c5fee3788c2fe30437f7518390875655888e5723faf06e1126fca40995",
        "original_path": "public/data/mko/connected-component.json",
        "english_path": "public/data/i18n/editions/mko-connected-component.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "e9c0ae2888b9f0f177728712cf7eb267a13a2a12f233a4c64ea50cb104f58bb0",
    "original_candidate_sha256": "d38407c56d04cb8702f88b2c990f893cc52869d888377158bdf7bf40155e740f",
    "original_candidate_line_sha256": "d38407c56d04cb8702f88b2c990f893cc52869d888377158bdf7bf40155e740f"
  },
  "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-20261026-007-p27",
    "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": "f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798",
      "question_first_witness_sha256": "cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 17,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "31a2ebf6a9d7ec7b4bbc81cc54c882d7e05c8263c4952f37191b447b00fff574",
      "arithmetic_steps": 0
    },
    "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."
  }
}