{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-quotient-vector-space-01",
  "mko_id": "mko-quotient-vector-space",
  "title": {
    "zh-Hant": "對角線的商空間",
    "en": "Quotient by the diagonal line"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-quotient-vector-space-01",
    "source": {
      "mko_id": "mko-quotient-vector-space",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "0f381f20543e1db106973cf749847f066bf4cb237412005cfb871209df548ed0",
      "english_sha256": "2591a61d4bb585ffcd836dbd49d9f2f74f4f3731c5c0b78d89bfbeb40b2729cd"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "R²/span{(1,1)}的基底為[(0,1)]，維度1；零coset是整條對角線。",
      "en": "R2/span{(1,1)} has basis [(0,1)] and dimension 1; its zero coset is the whole diagonal."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V=R²，W={t(1,1):t∈R}；[v]=v+W，q(x,y)=y−x。",
          "en": "V=R2, W={t(1,1):t in R}; [v]=v+W and q(x,y)=y-x."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "W=\\{(t,t):t\\in\\mathbb R\\}",
        "after_tex": "0\\in W,\\quad(s,s)+(t,t)=(s+t,s+t),\\quad\\lambda(t,t)=(\\lambda t,\\lambda t)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全稱封閉認證W子空間，非零(1,1)為完整一項基底。",
          "en": "Universal closure qualifies W as a subspace, with complete one-vector basis (1,1)."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "先認證子空間才能建立向量商。",
            "en": "Subspace qualification precedes the vector quotient."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "v=(x,y),\\quad u=(a,b)",
        "after_tex": "v-u\\in W\\iff y-b=x-a\\iff q(v)=q(u)",
        "relation": "equivalent",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩代表同類恰當q相同，cosets是固定y−x的整條線。",
          "en": "Two vectors represent the same coset exactly when their q-values agree; cosets are whole lines with a fixed value of y-x."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "雙向等價涵蓋全部實座標，不是有限membership測試。",
            "en": "The equivalence covers all real coordinates, not finite membership tests."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S01",
          "S02"
        ],
        "before_tex": "v'=v+w_1,\\quad u'=u+w_2,\\quad w_1,w_2\\in W",
        "after_tex": "(v'+u')-(v+u)=w_1+w_2\\in W",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任意代表替換使和只差W元素，商加法良定。",
          "en": "Any representative replacement changes the sum by W, proving quotient addition well-defined."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "使用W加法封閉，非只測特定代表。",
            "en": "This uses additive closure of W, not preferred-representative tests."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S01",
          "S02"
        ],
        "before_tex": "v'=v+w,\\quad w\\in W,\\quad\\lambda\\in\\mathbb R",
        "after_tex": "\\lambda v'-\\lambda v=\\lambda w\\in W",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任意實純量仍保持同類，商純量乘法良定。",
          "en": "Every real scalar preserves the class, making scalar multiplication well-defined."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "包含λ=0與負純量，無額外限制。",
            "en": "Zero and negative scalars are included without extra restrictions."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S01",
          "S02"
        ],
        "before_tex": "[0]=0+W",
        "after_tex": "[0]=W,\\quad[(1,1)]=[(0,0)]",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "零商元素是整個W，不是單元素集合。",
          "en": "The quotient zero is all of W, not a singleton."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "兩個向量代表不相等但差在W，故cosets相等。",
            "en": "The representative vectors differ, but their difference is in W, so their cosets agree."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S02",
          "S04"
        ],
        "before_tex": "v=(x,y),\\quad b=(0,1)",
        "after_tex": "v-(y-x)b=(x,x)\\in W\\Rightarrow[v]=(y-x)[b]",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個coset都是[b]的實倍數，完整生成。",
          "en": "Every coset is a real multiple of [b], giving full spanning."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "使用已證良定的商純量乘法，不是數值除法。",
            "en": "This uses the well-defined quotient scalar operation, not numerical division."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S01",
          "S04",
          "S05"
        ],
        "before_tex": "\\lambda[b]=[0]",
        "after_tex": "(0,\\lambda)\\in W\\iff\\lambda=0",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對角向量第一座標零則第二也零，因此[b]獨立且非零。",
          "en": "A diagonal vector with first coordinate zero has second coordinate zero, proving [b] independent and nonzero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "非零向量不保證非零coset，此處另核b不在W。",
            "en": "A nonzero vector need not have a nonzero coset; b outside W is checked here."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S01",
          "S06",
          "S07"
        ],
        "before_tex": "\\{[b]\\}\\text{ independent and spanning}",
        "after_tex": "\\dim(V/W)=1=2-1=\\dim V-\\dim W",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "以三個完整基底核此例維度差公式：V兩項，W一項，商一項。",
          "en": "Three complete bases verify the dimension difference here: two for V, one for W, one for the quotient."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S06",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "這是實例證書，不把它當抽象一般定理證明。",
            "en": "This is an instance certificate, not a general abstract proof."
          }
        },
        "leaf_basis": "mko_definition"
      }
    ],
    "result": {
      "zh-Hant": "R²/span{(1,1)}的基底為[(0,1)]，維度1；零coset是整條對角線。",
      "en": "R2/span{(1,1)} has basis [(0,1)] and dimension 1; its zero coset is the whole diagonal."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "零coset只有(0,0)。",
          "en": "The zero coset contains only (0,0)."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "它是W，含全部(t,t)。",
          "en": "It is W, containing all (t,t)."
        }
      },
      {
        "claim": {
          "zh-Hant": "[(1,0)]與[(2,1)]不同。",
          "en": "[(1,0)] differs from [(2,1)]."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "差為(1,1)在W中。",
          "en": "Their difference (1,1) is in W."
        }
      },
      {
        "claim": {
          "zh-Hant": "每個非零向量代表非零coset。",
          "en": "Every nonzero vector represents a nonzero coset."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "(1,1)非零但屬W。",
          "en": "(1,1) is nonzero but lies in 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": "v與u代表同一coset恰當v-u在W；quotient元素是cosets而非唯一representatives。核[v]+[u]=[v+u]與lambda[v]=[lambda v]的代表元獨立性。有限維時dim(V/W)=dimV-dimW，零coset是W。",
          "en": "Vectors v,u name the same coset exactly when v−u lies in W. Quotient elements are cosets, not unique representatives. Check representative independence of [v]+[u]=[v+u] and scalar*[v]=[scalar*v]. In finite dimension dim(V/W)=dim(V)−dim(W), and the zero coset is W."
        },
        "source": {
          "mko_id": "mko-quotient-vector-space",
          "sha256": "0f381f20543e1db106973cf749847f066bf4cb237412005cfb871209df548ed0",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "difference",
        "lhs": "2-1",
        "rhs": "1",
        "step": "S08"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限維實商空間的完整等價與良定性；不談拓撲商。",
      "en": "Full equivalence and representative-independent operations for a finite-dimensional real quotient vector space; no topological quotient is considered."
    },
    "md_symbols": [
      [
        "[v]",
        "完整coset",
        "whole coset"
      ],
      [
        "q",
        "coset標籤y−x",
        "coset label y-x"
      ],
      [
        "b",
        "(0,1)基底代表",
        "basis representative (0,1)"
      ]
    ],
    "md_error_controls": [
      {
        "title": "零型別 / Zero type",
        "zh": "零商元素是W本身，非唯一代表。",
        "en": "The quotient zero is W itself, not a unique representative."
      },
      {
        "title": "雙良定性 / Both well-defined operations",
        "zh": "加法用W封閉，純量用λW⊆W，都須全稱。",
        "en": "Addition and scalar multiplication each require universal closure of W."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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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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}