{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-vector-subspace-01",
  "mko_id": "mko-vector-subspace",
  "title": {
    "zh-Hant": "以全稱封閉認證過原點直線",
    "en": "Universal closure of a line through zero"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-vector-subspace-01",
    "source": {
      "mko_id": "mko-vector-subspace",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "a92727827e6ab175bea8b4da7021fe6b063af40e68f09beb9ad7437cd72c1abc",
      "english_sha256": "23093cbd91680008b49cf03eba212164f763c1dccecc9b30a56bccbe278919ca"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc",
      "subspace_def"
    ],
    "goal": {
      "zh-Hant": "U={(x,2x):x∈R}是R²子空間，仿射平移U+(0,1)不是。",
      "en": "U={(x,2x):x in R} is a subspace of R2, whereas U+(0,1) is not."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V=R²，U={(x,y):y−2x=0}，採通常向量加法與所有實純量。",
          "en": "V=R2, U={(x,y):y-2x=0}, with usual vector addition and all real scalars."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "(x,y)\\in U",
        "after_tex": "y=2x\\iff(x,y)=x(1,2)",
        "relation": "equivalent",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "解齊次方程得到完整參數化，不是只找到若干點。",
          "en": "Solving the homogeneous equation gives a complete parameterization, not just several points."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "反代x(1,2)給2x−2x=0，兩方向成立。",
            "en": "Substitution gives 2x-2x=0, verifying both directions."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "0\\in\\mathbb R",
        "after_tex": "(0,0)=0(1,2)\\in U",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "取參數0證含零向量，因而非空。",
          "en": "Parameter zero gives the zero vector, proving nonemptiness."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "含零只是必要條件，還未證封閉。",
            "en": "Containing zero is necessary but does not yet prove closure."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "u=a(1,2),\\ v=b(1,2),\\quad a,b\\in\\mathbb R",
        "after_tex": "u+v=(a+b)(1,2)\\in U",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任意兩元素之和仍為同一生成向量的實倍數。",
          "en": "The sum of any two elements remains a real multiple of the same generator."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "a、b全稱量化，不能以兩個數值例子替代。",
            "en": "a and b are arbitrary; two numerical examples would not replace the proof."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "u=a(1,2),\\quad\\lambda\\in\\mathbb R",
        "after_tex": "\\lambda u=(\\lambda a)(1,2)\\in U",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任意實純量乘法保持參數化，包含非整數與負純量。",
          "en": "Every real scalar preserves the parameterization, including negative and noninteger scalars."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "λ無任何有界或整數限制，只核λ=2不夠。",
            "en": "Lambda is neither bounded nor restricted to integers; checking only 2 is insufficient."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S02",
          "S03",
          "S04"
        ],
        "before_tex": "0\\in U,\\quad U+U\\subseteq U,\\quad\\mathbb R U\\subseteq U",
        "after_tex": "U\\le\\mathbb R^2",
        "relation": "implies",
        "rule": "subspace_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三項責任均以全稱代數完成，故U為子空間；其餘向量法則由R²限制。",
          "en": "All three obligations have universal algebraic proofs, so U is a subspace; other vector laws restrict from R2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S03",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "證書固定通常運算；不同運算不能沿用。",
            "en": "The certificate uses the usual operations and cannot be reused unchanged for different ones."
          }
        },
        "leaf_basis": "subspace_def"
      },
      {
        "id": "S06",
        "uses": [
          "G1"
        ],
        "before_tex": "Z=\\{(0,0)\\},\\quad V=\\mathbb R^2",
        "after_tex": "Z+Z=Z,\\ \\lambda Z\\subseteq Z;\\quad V+V\\subseteq V,\\ \\lambda V\\subseteq V",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "零空間只有零，閉性直接成立；全空間因實數和積仍實也封閉。",
          "en": "The zero space is closed directly; the whole space is closed because sums and products of reals are real."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "兩者皆含零，但這裡也另核加法與全部純量。",
            "en": "Both contain zero, and addition and all scalars are checked separately."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S02"
        ],
        "before_tex": "A=\\{(x,y):y=2x+1\\}",
        "after_tex": "(0,0)\\notin A",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "仿射平移直線不含零，立刻否證子空間。",
          "en": "The translated affine line omits zero, immediately refuting subspace status."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "(0,1)在A但0·(0,1)不在A，也給直接純量封閉反例。",
            "en": "(0,1) lies in A but zero times it does not, also directly refuting scalar closure."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S03"
        ],
        "before_tex": "B=\\{(x,0):x\\in\\mathbb R\\}\\cup\\{(0,y):y\\in\\mathbb R\\}",
        "after_tex": "(1,0),(0,1)\\in B,\\quad(1,1)\\notin B",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩座標軸聯集雖含零且各軸封閉，聯集不對加法封閉。",
          "en": "The union of coordinate axes contains zero and each axis is closed, but their union fails additive closure."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "一個明示反例足以否證全稱封閉；有限正例不足以證明。",
            "en": "One explicit counterexample refutes universal closure; finitely many positive samples cannot prove it."
          }
        },
        "leaf_basis": "calc"
      }
    ],
    "result": {
      "zh-Hant": "U={(x,2x):x∈R}是R²子空間，仿射平移U+(0,1)不是。",
      "en": "U={(x,2x):x in R} is a subspace of R2, whereas U+(0,1) is not."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "y=2x+1的直線也是子空間。",
          "en": "The line y=2x+1 is a subspace."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "不含(0,0)。",
          "en": "It omits (0,0)."
        }
      },
      {
        "claim": {
          "zh-Hant": "含零且對整數倍封閉就夠。",
          "en": "Containing zero and closure under integer multiples suffice."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "Z²對加法及整數倍封閉，但(1/2)(1,0)不在Z²。",
          "en": "Z2 is closed under addition and integer multiples, but (1/2)(1,0) is outside it."
        }
      },
      {
        "claim": {
          "zh-Hant": "兩個子空間的聯集必是子空間。",
          "en": "A union of two subspaces is a subspace."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "兩座標軸的聯集不含(1,0)+(0,1)=(1,1)。",
          "en": "The union of axes omits (1,0)+(0,1)=(1,1)."
        }
      }
    ],
    "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": "本輪只處理R²中的零空間、全空間、過原點直線與明示有限反例。正例使用齊次線性方程或span的全稱代數證書；有限抽樣通過不足以證子空間。",
          "en": "This slice handles the zero space, the whole space, lines through the origin and explicit finite counterexamples in R². Positive cases use universal algebraic certificates from homogeneous linear equations or spans. Finitely many successful samples do not prove a subspace."
        },
        "source": {
          "mko_id": "mko-vector-subspace",
          "sha256": "a92727827e6ab175bea8b4da7021fe6b063af40e68f09beb9ad7437cd72c1abc",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "subspace_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實向量空間V的子集U若含零且對加法及每個實純量封閉，則在限制運算下為子空間；其餘向量法則由V繼承。",
          "en": "A subset U of a real vector space V is a subspace under restricted operations if it contains zero and is closed under addition and every real scalar; the remaining vector laws are inherited from V."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "R²中齊次直線、零空間、全空間及明示反例；不用有限抽樣證正例。",
      "en": "A homogeneous line, zero space, whole R2 and explicit counterexamples; no sampled positive proof."
    },
    "md_symbols": [
      [
        "U",
        "齊次直線",
        "homogeneous line"
      ],
      [
        "\\lambda",
        "任意實純量",
        "arbitrary real scalar"
      ]
    ],
    "md_error_controls": [
      {
        "title": "全稱純量 / All real scalars",
        "zh": "必須對每個λ∈R證封閉，不能只核整數。",
        "en": "Closure must hold for every real lambda, not merely integers."
      },
      {
        "title": "齊次性 / Homogeneity",
        "zh": "y−2x=0含零；常數換成1便是不同仿射集合。",
        "en": "y-2x=0 contains zero; replacing zero by 1 gives a different affine set."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "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."
  }
}