{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-direct-sum-01",
  "mko_id": "mko-direct-sum",
  "title": {
    "zh-Hant": "R³中唯一分解與覆蓋",
    "en": "Unique decomposition and coverage in R3"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-direct-sum-01",
    "source": {
      "mko_id": "mko-direct-sum",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "3eedbed2a646d34aa3bf6b0c5eb4e3482fcd8c816d40ca7c5652a6bf7224b9e8",
      "english_sha256": "1fe3402da619d87b341bcf2c97aad834c0b1ea87391581c56651a5dc46bc3779"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "R³=span{e₁}⊕span{e₂,e₃}，唯一性與覆蓋分別成立。",
      "en": "R3=span{e1} direct-sum span{e2,e3}, with uniqueness and coverage separately verified."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V=R³；e₁=(1,0,0),e₂=(0,1,0),e₃=(0,0,1)；U=span{e₁}，W=span{e₂,e₃}。",
          "en": "V=R3; e1=(1,0,0),e2=(0,1,0),e3=(0,0,1); U=span{e1}, W=span{e2,e3}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "U=\\operatorname{span}\\{e_1\\},\\quad W=\\operatorname{span}\\{e_2,e_3\\}",
        "after_tex": "U=\\{(a,0,0):a\\in\\mathbb R\\},\\quad W=\\{(0,b,c):b,c\\in\\mathbb R\\}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整參數化兩子空間；各生成族由座標比較證獨立。",
          "en": "Both subspaces are fully parameterized; each generating family is independent by coordinate comparison."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "W不可漏e₃，否則覆蓋將失敗。",
            "en": "Omitting e3 from W would invalidate coverage."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "v\\in U\\cap W",
        "after_tex": "v=(a,0,0)=(0,b,c)\\Rightarrow a=b=c=0",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "比較全部座標，交集只有零向量。",
          "en": "Comparing every coordinate makes the intersection the zero subspace."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "零確在兩者中，所以交集不是空集。",
            "en": "Zero belongs to both, so the intersection is not empty."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "(x,y,z)\\in V",
        "after_tex": "(x,y,z)=(x,0,0)+(0,y,z)\\in U+W",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任意來源向量有兩個合法summands，故U+W=V。",
          "en": "Every ambient vector has two valid summands, so U+W=V."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "零交集與覆蓋是分開責任，此步才證後者。",
            "en": "Zero intersection and coverage are separate; this step establishes coverage."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S01",
          "S02"
        ],
        "before_tex": "u+w=u'+w',\\quad u,u'\\in U,\\quad w,w'\\in W",
        "after_tex": "u-u'=w'-w\\in U\\cap W=\\{0\\}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩種分解的差在交集中，故差為零。",
          "en": "The difference of two decompositions lies in the intersection and therefore vanishes."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "子空間對差封閉才保證左右屬各自子空間。",
            "en": "Closure under differences is needed to place each side in its respective subspace."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S03",
          "S04"
        ],
        "before_tex": "u-u'=0,\\quad w'-w=0",
        "after_tex": "u=u',\\quad w=w';\\qquad V=U\\oplus W",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "唯一性與覆蓋合併，認證母空間內直和。",
          "en": "Uniqueness combines with coverage to certify the internal direct sum of V."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "只由交集零只能保證U+W中的唯一性，不能省S03。",
            "en": "Zero intersection alone gives uniqueness within U+W; S03 cannot be omitted."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S01",
          "S05"
        ],
        "before_tex": "[e_1\\ e_2\\ e_3]=I_3",
        "after_tex": "\\operatorname{rank}[e_1\\ e_2\\ e_3]=3,\\quad1+2=3",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整聯合基底矩陣為I₃，三pivot再核獨立與覆蓋。",
          "en": "The full concatenated basis matrix is I3; its three pivots recheck independence and coverage."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "此rank綁定同一U、W，不是任意維度相加。",
            "en": "The rank certificate concerns the same U,W, not arbitrary dimensions."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1"
        ],
        "before_tex": "L_1=\\mathbb R(1,0),\\ L_2=\\mathbb R(0,1),\\ L_3=\\mathbb R(1,1)",
        "after_tex": "L_i\\cap L_j=\\{0\\}\\ (i\\ne j),\\quad(1,0)+(0,1)-(1,1)=0",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "R²三條直線兩兩零交集，但存在整體非平凡關係，因此不是三項直和。",
          "en": "Three R2 lines have pairwise zero intersections yet a nontrivial global relation, so the three-way sum is not direct."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "三個非零summands合為零，直接否證唯一性。",
            "en": "Three nonzero summands add to zero, directly refuting uniqueness."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S03",
          "S04",
          "S05"
        ],
        "before_tex": "U\\times W=\\{(u,w):u\\in U,w\\in W\\}",
        "after_tex": "\\Phi:U\\times W\\to V,\\quad\\Phi(u,w)=u+w\\text{ bijective}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "外和元素是有序對；本例覆蓋給滿射、唯一性給單射。",
          "en": "External-sum elements are ordered pairs; coverage gives surjectivity and uniqueness gives injectivity here."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "雙射不使有序對字面等於母空間向量。",
            "en": "A bijection does not make ordered pairs literally equal to ambient vectors."
          }
        },
        "leaf_basis": "calc"
      }
    ],
    "result": {
      "zh-Hant": "R³=span{e₁}⊕span{e₂,e₃}，唯一性與覆蓋分別成立。",
      "en": "R3=span{e1} direct-sum span{e2,e3}, with uniqueness and coverage separately verified."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "R³兩座標軸零交集就覆蓋R³。",
          "en": "Two zero-intersection axes cover R3."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "span{e₁}+span{e₂}漏e₃。",
          "en": "span{e1}+span{e2} misses e3."
        }
      },
      {
        "claim": {
          "zh-Hant": "三項兩兩零交集即直和。",
          "en": "Pairwise zero intersections suffice for three summands."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "上述Lᵢ有非平凡三項關係。",
          "en": "The displayed lines have a nontrivial three-term relation."
        }
      },
      {
        "claim": {
          "zh-Hant": "外和元素就是u+w。",
          "en": "External-sum elements are literally u+w."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "外載體是(u,w)，Φ才送至u+w。",
          "en": "The external carrier contains (u,w); Phi sends it to u+w."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "calc",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "假設通常實數域與座標向量運算、矩陣乘法、線性獨立及生成的定義；所用消去均為可逆初等列運算。",
          "en": "Assume the real field, usual coordinate-vector arithmetic, matrix multiplication, and definitions of independence and span; elimination uses invertible elementary row operations."
        }
      },
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "在有限維實向量空間以完整基底與exact rank核對二項內直和。U交W為零只保證U+W內的唯一性；V=U直和W還需U+W=V。三項以上不得以兩兩交集為零代替整體唯一性，外直和則使用乘積型新載體。",
          "en": "Use complete bases and exact ranks in a finite-dimensional real ambient space. U∩W={0} ensures uniqueness in U+W, while V=U⊕W also needs U+W=V. For three or more subspaces pairwise zero intersections do not replace global uniqueness; an external direct sum uses a new product-type carrier."
        },
        "source": {
          "mko_id": "mko-direct-sum",
          "sha256": "3eedbed2a646d34aa3bf6b0c5eb4e3482fcd8c816d40ca7c5652a6bf7224b9e8",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "dims",
        "lhs": "1+2",
        "rhs": "3",
        "step": "S06"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限維二項內直和，另列多項及外載體界線。",
      "en": "A finite-dimensional two-summand internal sum, with many-summand and external-carrier boundaries."
    },
    "md_symbols": [
      [
        "U,W",
        "內部子空間",
        "internal subspaces"
      ],
      [
        "\\Phi",
        "外和到內和的加法映射",
        "addition map from external to internal sum"
      ]
    ],
    "md_error_controls": [
      {
        "title": "覆蓋另核 / Separate coverage",
        "zh": "零交集給唯一性，U+W=V給覆蓋，兩者均要。",
        "en": "Zero intersection gives uniqueness; U+W=V gives coverage. Both are needed."
      },
      {
        "title": "多項獨立 / Global independence",
        "zh": "多項直和核全部基底聯集整體獨立。",
        "en": "A many-summand direct sum requires global independence of all basis columns."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "original_source_sha256": "e587d57a8edbf7ea1be841fcab7e765b95d2765b0c3bcff302efa7213d0cb9b6",
    "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": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
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    "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."
  }
}