{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-linear-map-image-01",
  "mko_id": "mko-linear-map-image",
  "title": {
    "zh-Hant": "用一致性求完整像空間",
    "en": "The full image from consistency"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-linear-map-image-01",
    "source": {
      "mko_id": "mko-linear-map-image",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "9901554d389b316f90187d99a94cc8c17a03fc69cde338962d359b8d5e95eb8d",
      "english_sha256": "b431ff922c8e97a289c8ba4ddab5f8dc659cce34249fd397c52e03398f0f34c2"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "im T=span{(1,2)}；(3,6)在像內，(0,1)不在像內。",
      "en": "im T=span{(1,2)}; (3,6) lies in the image and (0,1) does not."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "T:R²→R²，A=[[1,2],[2,4]]，c₁=(1,2)，c₂=(2,4)。",
          "en": "T:R2 to R2, A=[[1,2],[2,4]], c1=(1,2), c2=(2,4)."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "A\\binom{x}{y}",
        "after_tex": "x\\binom{1}{2}+y\\binom{2}{4}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "矩陣輸出為columns的任意實線性組合，因此像包含於column span。",
          "en": "Every output is a real column combination, giving inclusion in the column span."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "係數x、y可取全部實數，不能只看columns本身。",
            "en": "x,y range over all reals, not merely the columns themselves."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "z=xc_1+yc_2",
        "after_tex": "z=T(x,y)\\Rightarrow\\operatorname{span}\\{c_1,c_2\\}\\subseteq\\operatorname{im}T",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任何column線性組合都由同一係數作輸入得到，證反向包含。",
          "en": "Any column combination is obtained by using its coefficients as input, proving reverse inclusion."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "兩包含合起來才得完整像，而非下界。",
            "en": "Both inclusions give the full image, not merely a lower bound."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S01",
          "S02"
        ],
        "before_tex": "c_2=2c_1",
        "after_tex": "T(x,y)=(x+2y)c_1",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩columns成比例，把任意輸出縮成單一參數s=x+2y。",
          "en": "The columns are proportional, reducing every output to parameter s=x+2y."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "s可任意實數，因可選x=s,y=0。",
            "en": "s can be any real because x=s,y=0 is available."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03"
        ],
        "before_tex": "s\\in\\mathbb R",
        "after_tex": "T(s,0)=(s,2s),\\quad\\operatorname{im}T=\\{(s,2s):s\\in\\mathbb R\\}",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "明示全稱前像證每個此形向量都可到達。",
          "en": "A universal preimage shows every vector of this form is reached."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "像只是R²內直線，不等於宣告codomain R²。",
            "en": "The image is a line inside R2, not the entire declared codomain."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S01",
          "S04"
        ],
        "before_tex": "A\\binom{x}{y}=\\binom{u}{v}",
        "after_tex": "x+2y=u,\\quad 2x+4y=v\\iff v=2u",
        "relation": "equivalent",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二方程減兩倍第一方程得必要條件；若v=2u則x=u,y=0給解，亦充分。",
          "en": "Subtracting twice equation one gives necessity; if v=2u, x=u,y=0 gives sufficiency."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "S01",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "兩方向均證，並非有限搜尋未命中的推斷。",
            "en": "Both directions are proved without a bounded no-hit inference."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05"
        ],
        "before_tex": "(u,v)=(3,6)",
        "after_tex": "6=2\\cdot3,\\quad T(3,0)=(3,6)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "一致性成立且有精確前像(3,0)，認證membership。",
          "en": "Consistency holds with exact preimage (3,0), certifying membership."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "前像非唯一；加任意t(−2,1)仍輸出(3,6)。",
            "en": "The preimage is not unique; adding t(-2,1) preserves output (3,6)."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S05"
        ],
        "before_tex": "(u,v)=(0,1)",
        "after_tex": "1\\ne2\\cdot0\\Rightarrow(0,1)\\notin\\operatorname{im}T",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全實數一致性失敗，故真非membership，不僅是某盒內無解。",
          "en": "Global consistency fails, proving nonmembership rather than just absence in a box."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "矛盾1=0排除所有實輸入。",
            "en": "The contradiction 1=0 excludes all real inputs."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S04",
          "S05"
        ],
        "before_tex": "|x|,|y|\\le1,\\quad x,y\\in\\mathbb Z",
        "after_tex": "-3\\le x+2y\\le3,\\quad(100,200)=T(100,0)\\in\\operatorname{im}T",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "小盒漏掉(100,200)，但它仍有全域前像，示範搜尋界線。",
          "en": "The small box misses (100,200), yet it has a global preimage, illustrating the search boundary."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "bounded無解不能代替S05的符號判準。",
            "en": "Bounded nonexistence cannot replace S05's symbolic criterion."
          }
        },
        "leaf_basis": "calc"
      }
    ],
    "result": {
      "zh-Hant": "im T=span{(1,2)}；(3,6)在像內，(0,1)不在像內。",
      "en": "im T=span{(1,2)}; (3,6) lies in the image and (0,1) does not."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "codomain R²等於image。",
          "en": "The codomain R2 equals the image."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "(0,1)不滿足v=2u。",
          "en": "(0,1) fails v=2u."
        }
      },
      {
        "claim": {
          "zh-Hant": "兩個columns便是全部像元素。",
          "en": "The two columns are all image elements."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "任意實倍數s(1,2)都是輸出，例如(3,6)。",
          "en": "Every real multiple s(1,2) is an output, such as (3,6)."
        }
      },
      {
        "claim": {
          "zh-Hant": "小盒找不到(100,200)表示非membership。",
          "en": "Missing (100,200) in a small box proves nonmembership."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "(100,0)是明示全域前像。",
          "en": "(100,0) is an explicit global preimage."
        }
      }
    ],
    "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": "本輪用精確 2×2 矩陣與 symbolic column-span certificate。membership 以 Ax=y 的精確一致性判定；bounded fiber 只提供 witness 或 bounded nonexistence，不可證全 R² 非 membership。像（image，亦常稱 range）不得與宣告陪域（codomain）或有限抽樣的輸出清單混同。",
          "en": "This slice uses exact 2-by-2 matrices and symbolic column-span certificates. Membership is exact real consistency of A*x=y. A bounded fiber search supplies witnesses or bounded nonexistence only, not global nonmembership in the real image. The image must not be conflated with the entire declared codomain or a sampled output list."
        },
        "source": {
          "mko_id": "mko-linear-map-image",
          "sha256": "9901554d389b316f90187d99a94cc8c17a03fc69cde338962d359b8d5e95eb8d",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "member",
        "lhs": "2*3",
        "rhs": "6",
        "step": "S06"
      },
      {
        "id": "nonmember",
        "lhs": "1",
        "rhs": "2*0",
        "step": "S07",
        "relation": "!="
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "精確2×2線性系統的全實數像，不以有限輸入清單當range。",
      "en": "The entire real image of an exact 2-by-2 system, not a finite sampled output list."
    },
    "md_symbols": [
      [
        "c_1,c_2",
        "矩陣columns",
        "matrix columns"
      ],
      [
        "u,v",
        "任意目標座標",
        "arbitrary target coordinates"
      ]
    ],
    "md_error_controls": [
      {
        "title": "像與陪域 / Image and codomain",
        "zh": "以Ax=y一致性分類，不由T:R²→R²猜滿射。",
        "en": "Use Ax=y consistency rather than guessing surjectivity from T:R2 to R2."
      },
      {
        "title": "盒界線 / Box boundary",
        "zh": "有限搜尋否定只適用已宣告輸入盒。",
        "en": "A finite search's negative result applies only to its declared input box."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "b01b892b391c388da6d5e1fa97da0a688e710e364c061e305952d03b4ff25546",
    "original_source_sha256": "841d6adb1aa0065b3f001c0945ad4d8276511c210c6df28756fc85df02fed2aa",
    "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-P015"
    ],
    "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."
  }
}