{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-spanning-tree-01",
  "mko_id": "mko-spanning-tree",
  "title": {
    "zh-Hant": "方形父圖內的spanning tree",
    "en": "A spanning tree inside a square parent graph"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-spanning-tree-01",
    "source": {
      "mko_id": "mko-spanning-tree",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "f93b1e5d271c80d9a571d19d5ddba8bceb178dce9bc9c8e6f3b1b0279a0d3833",
      "english_sha256": "32cad5f8907e94e691500db0fe35e01257bbe42816f694ffb01b2277a26d725d"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "核全頂點、選邊存在、connected及acyclic。",
      "en": "Audit all vertices, edge membership, connectivity and acyclicity."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V={0,1,2,3}，父圖邊e0={0,1}、e1={1,2}、e2={2,3}、e3={0,3}；選T={e0,e1,e2}且保留全部V。",
          "en": "V={0,1,2,3}; parent edges are e0={0,1},e1={1,2},e2={2,3},e3={0,3}; select T={e0,e1,e2} while retaining all V."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "T=\\{e_0,e_1,e_2\\}",
        "after_tex": "T\\subseteq E,\\quad V_T=V",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "選邊全在父圖，頂點集不只取某個component。",
          "en": "Every selected edge belongs to the parent and the vertex set is all V."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "spanning要求全V。",
            "en": "Spanning requires all V."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "0\\to1\\to2\\to3\\text{ using }T",
        "after_tex": "\\text{every vertex is reachable from }0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "明列0到1、2、3的path前綴。",
          "en": "The path prefixes explicitly reach1,2,3 from 0."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "無向path可反向，故任兩點可經0相連。",
            "en": "Undirected paths can reverse, connecting any two vertices via0."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "\\text{all vertices reachable from }0",
        "after_tex": "(V,T)\\text{ connected}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "可達性見證完整覆蓋四點。",
          "en": "Reachability witnesses cover all four vertices."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "父圖也包含T，因此父圖connected。",
            "en": "The parent also contains T and is connected."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "(V,T):\\quad\\deg_T(0)=1",
        "after_tex": "0\\text{ cannot lie on a cycle}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "cycle上的頂點至少有兩條cycle邊，但0只有一邊。",
          "en": "A cycle vertex needs two incident cycle edges, whereas0 has only one edge."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "此為針對T的degree，不是父圖degree。",
            "en": "Use T-degrees, not parent degrees."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "\\text{delete leaf }0\\text{ and }e_0",
        "after_tex": "1-2-3\\text{ remains; delete leaf }1\\text{ then edge }\\{2,3\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "依序刪葉，不可能刪掉任何cycle；最後沒有邊。",
          "en": "Delete leaves successively, removing no possible cycle, until no edges remain."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "若原有cycle，刪葉後會保留，與末態矛盾。",
            "en": "Any original cycle would survive leaf deletion, contradicting the final state."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "(V,T)\\text{ connected and acyclic}",
        "after_tex": "T\\text{ is a spanning tree},\\quad|T|=3=|V|-1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "樹定義與全V條件完成證書；邊數另核一致。",
          "en": "The tree definition and all-V condition certify spanning; the edge count cross-checks it."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不是只數三邊就稱樹。",
            "en": "The tree claim does not rest merely on counting three edges."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "H:\\ V=\\{0,1,2,3\\},\\ E_H=\\{01,12,02\\}",
        "after_tex": "|E_H|=3=|V|-1,\\quad H\\text{ disconnected and cyclic}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三角形加孤立3是n-1邊不足的完整反例。",
          "en": "A triangle plus isolated 3 completely refutes the n-1-only criterion."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "反例graph與原父圖分開。",
            "en": "The counterexample graph is separate from the original parent."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "V=\\{v\\},\\quad T=\\varnothing",
        "after_tex": "\\text{connected length-zero paths and no cycles}\\implies\\text{spanning tree}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "保留單頂點邊界，零條邊等於n-1。",
          "en": "Retain the one-vertex case, with zero edges equal to n-1."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "沒有weights，故沒有minimum weight比較。",
            "en": "No weights were given, so no minimum-weight comparison is made."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "T為父圖spanning tree；三邊數目只作交叉核對。",
      "en": "T is a spanning tree of the parent; the three-edge count is only a cross-check."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "n-1邊必是tree。",
          "en": "Every n-1-edge graph is a tree."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "三角形加孤立點有三邊但不connected。",
          "en": "A triangle plus an isolate has three edges but is disconnected."
        }
      },
      {
        "claim": {
          "zh-Hant": "單點零邊不是tree。",
          "en": "A singleton with no edges is not a tree."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "它connected且acyclic。",
          "en": "It is connected and acyclic."
        }
      },
      {
        "claim": {
          "zh-Hant": "T是minimum-weight tree。",
          "en": "T is minimum-weight."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "沒有weights或比較證書，無此結論資格。",
          "en": "No weights or comparison certificate were supplied."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "完整有限connected simple parent graph與選邊子圖。核選邊存在、全頂點、connected/acyclic；connected+n−1 edges或acyclic+n−1 edges可作exact certificate，n−1 alone不足。保留single vertex；不作weighted/MST claim。",
          "en": "Use a fully specified finite connected simple parent graph and a subgraph formed from selected parent edges. Check edge membership, all parent vertices, connectivity and acyclicity. Connectivity plus n−1 edges, or acyclicity plus n−1 edges, certifies a tree; n−1 edges alone do not. Retain the one-vertex case and make no weighted or minimum-spanning-tree claim."
        },
        "source": {
          "mko_id": "mko-spanning-tree",
          "sha256": "f93b1e5d271c80d9a571d19d5ddba8bceb178dce9bc9c8e6f3b1b0279a0d3833",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限無向簡單圖的邊為無loop、無重複unordered pairs；圖性質須對完整明示V、E逐一檢查。 connected表示任兩頂點有path；acyclic表示無cycle；tree同時connected及acyclic；spanning tree保留父圖全部頂點。",
          "en": "A finite simple undirected graph has loop-free duplicate-free unordered edges; graph properties are audited on all declared V and E. Connected means every vertex pair has a path; acyclic means no cycle; a tree is connected and acyclic; a spanning tree retains every parent vertex."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "4-1",
        "rhs": "3"
      },
      {
        "id": "N02",
        "lhs": "1-1",
        "rhs": "0"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "unweighted模型，不作MST宣稱。",
      "en": "An unweighted model with no MST claim."
    },
    "md_symbols": [
      [
        "V,E",
        "完整頂點與邊集合",
        "complete vertex and edge sets"
      ],
      [
        "T",
        "父圖選邊集合",
        "selected parent edges"
      ]
    ],
    "md_error_controls": [
      {
        "title": "兩種degree / Two degrees",
        "zh": "刪葉論證使用選邊子圖degree。",
        "en": "Leaf deletion uses selected-subgraph degrees."
      },
      {
        "title": "n-1界線 / n-1 boundary",
        "zh": "先證connected與acyclic，不能反過來只數邊。",
        "en": "Prove connectedness and acyclicity rather than relying on edge count alone."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "da0088130c6f6dd6a4f6f6e08bb0139a4ba1ab80dea27a8a789279270b89cb0f",
    "original_source_sha256": "d11ac46e36437f1a939508dc2799a977a5bbb303e3c22bcb76ede0878f29768b",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [
      "mko_definition"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-P027",
      "BR142-S075",
      "BR142-S076"
    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}