{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r03-linear-transformation-0055",
  "pilot_number": "R0014-07580",
  "number": 7580,
  "family_id": "qf-density16-linear-scalar-values",
  "title": {
    "zh-Hant": "第 7580 題：矩陣映射的純量兩側值",
    "en": "Question 7580: Matrix-Model Scalar Routes and Defect"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r03-linear-transformation-0055",
    "batch_id": "batch-20261012-advance-20261016-r03-linear-transformation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-transformation"
    ],
    "title_zh": "矩陣映射的純量兩側值",
    "stem_zh": "R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[-8, 1]、c=0。求 (Mv, M(cv), cMv, 後兩側之差)。",
    "answer": {
      "kind": "expression",
      "value": "([-8, 2], [0, 0], [0, 0], [0, 0])",
      "explanation_zh": "有限結構核對=([-8, 2], [0, 0], [0, 0], [0, 0])。一條路先縮放輸入，一條路先映射；兩路以BigInt核分量。有限精確核對不是一般形式證明。有限測值不是任意函數線性證明，零映射可以線性。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "矩陣映射的純量兩側值"
      ]
    },
    "variant": {
      "family_id": "qf-density16-linear-scalar-values",
      "seed": 202610163055,
      "parameters": {
        "M": [
          [
            1,
            0
          ],
          [
            0,
            2
          ]
        ],
        "v": [
          -8,
          1
        ],
        "c": 0,
        "domain": "R2_column_integer_slice",
        "operation": "linear-scalar-values",
        "semantic_template": "qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[-8,1]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261016-r03-linear-transformation-0055",
    "batch_id": "batch-20261012-advance-20261016-r03-linear-transformation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "bc280294b4bb1c614703caba38ad51a2df3cee406011e5ce86eb2ef6bf8fdaeb",
      "family_id": "qf-density16-linear-scalar-values",
      "seed": 202610163055,
      "semantic_template": "qf-density16-linear-scalar-values/{\"M\":[[1,0],[0,2]],\"c\":0,\"v\":[-8,1]}"
    },
    "fields": {
      "title": "Matrix-Model Scalar Routes and Defect",
      "stem": "Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[-8, 1], c=0. Give (Mv, M(cv), cMv, difference of the last two sides).",
      "answer": {
        "kind": "expression",
        "value": "([-8, 2], [0, 0], [0, 0], [0, 0])",
        "explanation": "One route scales the input first and the other scales the image; compare exact components, including zero cases. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8, 2], [0, 0], [0, 0], [0, 0])."
      },
      "skills": [
        "Matrix-Model Scalar Routes and Defect"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "R²到R²列向量，明示2×2矩陣模型。M=[[1, 0], [0, 2]]、v=[-8, 1]、c=0。求 (Mv, M(cv), cMv, 後兩側之差)。",
        "en": "Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. M=[[1, 0], [0, 2]], v=[-8, 1], c=0. Give (Mv, M(cv), cMv, difference of the last two sides)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "M=[[1, 0], [0, 2]];v=[-8, 1]",
      "after": "declared R2 matrix model;finite values are not a test of an arbitrary function",
      "relation": "implies",
      "rule_id": "declared_linear_model_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "明示矩陣模型Mv在全R²上由分配律給出線性映射；本題列出的有限代值不是對任意函數的線性證明。零映射也線性，不要求非零或可逆。",
        "en": "The declared matrix model Mv gives a linear map on all R² by distributivity.The listed finite evaluations do not prove an arbitrary function linear.The zero map is linear;nonzero values or invertibility are not required."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Mv[0] term0 := mul(1, -8)",
      "after": "Mv[0] term0 = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv[0] term0，不以小數近似替代。",
          "en": "Recompute Mv[0] term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-8"
        ],
        "output": "-8",
        "label": "Mv[0] term0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Mv[0] term1 := mul(0, 1)",
      "after": "Mv[0] term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv[0] term1，不以小數近似替代。",
          "en": "Recompute Mv[0] term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "Mv[0] term1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Mv[0] row sum := sum(-8, 0)",
      "after": "Mv[0] row sum = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv[0] row sum，不以小數近似替代。",
          "en": "Recompute Mv[0] row sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-8",
          "0"
        ],
        "output": "-8",
        "label": "Mv[0] row sum"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Mv[1] term0 := mul(0, -8)",
      "after": "Mv[1] term0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv[1] term0，不以小數近似替代。",
          "en": "Recompute Mv[1] term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-8"
        ],
        "output": "0",
        "label": "Mv[1] term0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Mv[1] term1 := mul(2, 1)",
      "after": "Mv[1] term1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv[1] term1，不以小數近似替代。",
          "en": "Recompute Mv[1] term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "Mv[1] term1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Mv[1] row sum := sum(0, 2)",
      "after": "Mv[1] row sum = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mv[1] row sum，不以小數近似替代。",
          "en": "Recompute Mv[1] row sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "Mv[1] row sum"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "cv[0] := mul(0, -8)",
      "after": "cv[0] = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 cv[0]，不以小數近似替代。",
          "en": "Recompute cv[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-8"
        ],
        "output": "0",
        "label": "cv[0]"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "cv[1] := mul(0, 1)",
      "after": "cv[1] = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 cv[1]，不以小數近似替代。",
          "en": "Recompute cv[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "cv[1]"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "M(cv)[0] term0 := mul(1, 0)",
      "after": "M(cv)[0] term0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M(cv)[0] term0，不以小數近似替代。",
          "en": "Recompute M(cv)[0] term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "M(cv)[0] term0"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "M(cv)[0] term1 := mul(0, 0)",
      "after": "M(cv)[0] term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M(cv)[0] term1，不以小數近似替代。",
          "en": "Recompute M(cv)[0] term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "M(cv)[0] term1"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "M(cv)[0] row sum := sum(0, 0)",
      "after": "M(cv)[0] row sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M(cv)[0] row sum，不以小數近似替代。",
          "en": "Recompute M(cv)[0] row sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "M(cv)[0] row sum"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "M(cv)[1] term0 := mul(0, 0)",
      "after": "M(cv)[1] term0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M(cv)[1] term0，不以小數近似替代。",
          "en": "Recompute M(cv)[1] term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "M(cv)[1] term0"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "M(cv)[1] term1 := mul(2, 0)",
      "after": "M(cv)[1] term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M(cv)[1] term1，不以小數近似替代。",
          "en": "Recompute M(cv)[1] term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "M(cv)[1] term1"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "M(cv)[1] row sum := sum(0, 0)",
      "after": "M(cv)[1] row sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把已對齊的同一分量或同一位置相加，不跨分量配對。",
        "en": "Add the already aligned component or position,without mixing coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 M(cv)[1] row sum，不以小數近似替代。",
          "en": "Recompute M(cv)[1] row sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "M(cv)[1] row sum"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "cMv[0] := mul(0, -8)",
      "after": "cMv[0] = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 cMv[0]，不以小數近似替代。",
          "en": "Recompute cMv[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-8"
        ],
        "output": "0",
        "label": "cMv[0]"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "cMv[1] := mul(0, 2)",
      "after": "cMv[1] = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 cMv[1]，不以小數近似替代。",
          "en": "Recompute cMv[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "cMv[1]"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "linear scalar residual[0] := sub(0, 0)",
      "after": "linear scalar residual[0] = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按題目指定的前項減後項，保留殘差的正負。",
        "en": "Subtract the second operand from the first in the requested order,retaining the residual sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 linear scalar residual[0]，不以小數近似替代。",
          "en": "Recompute linear scalar residual[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "linear scalar residual[0]"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "linear scalar residual[1] := sub(0, 0)",
      "after": "linear scalar residual[1] = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按題目指定的前項減後項，保留殘差的正負。",
        "en": "Subtract the second operand from the first in the requested order,retaining the residual sign."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 linear scalar residual[1]，不以小數近似替代。",
          "en": "Recompute linear scalar residual[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "linear scalar residual[1]"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "c=0;v=[-8, 1]",
      "after": "left=[0, 0];right=[0, 0];residual=[0, 0]",
      "relation": "implies",
      "rule_id": "matrix_scalar_value_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "c為0或負值仍合法；先縮放輸入與先映射輸出都要完整代值，不把有理除法當先決條件。",
        "en": "A zero or negative c is allowed.Evaluate both scaling-before-input and scaling-after-output fully;no division is required."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        "-8",
        "2"
      ],
      [
        "0",
        "0"
      ],
      [
        "0",
        "0"
      ],
      [
        "0",
        "0"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([-8, 2], [0, 0], [0, 0], [0, 0])",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261012-advance-20261016-r03-linear-transformation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261016-r03-linear-transformation-foundations/questions.jsonl",
      "batch_sha256": "915d95dff60999b46e55ccc20b8ae66dc4b4ee601782d8d932e1810cf76686f1",
      "original_line": 55,
      "original_line_sha256": "6b681c998e4d3c3c6ef11b9f2e719f5daad5c53d1e221dbb1e80c0f7785c76fe",
      "question_compact_sha256": "bc280294b4bb1c614703caba38ad51a2df3cee406011e5ce86eb2ef6bf8fdaeb",
      "english_row_compact_sha256": "703f5309f7c4f7d979c544d715135be3a098df9c47fccc7e112fa3dd5fce9dbc",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261016-r03-linear-transformation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261016-r03-linear-transformation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-linear-transformation": {
        "sha256": "59ce4d8fac03871ea16eb62a76c9fae255470dc391ff7312ad4da120bafbf132",
        "english_sha256": "d95d3ac4155053ddc0d186089ae6ec2d7e4e20468aa14880d371ac260c3cd0a3",
        "original_path": "public/data/mko/linear-transformation.json",
        "english_path": "public/data/i18n/editions/mko-linear-transformation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d7860245c70062705fa4858c13821db6ff286678dec63f22464364f5d24cfcc1",
    "original_candidate_sha256": "e7f2bc4bdf5561b28f3b207a804b9cbaee16e1c8fa0e8492add4974cf198eb79",
    "original_candidate_line_sha256": "e7f2bc4bdf5561b28f3b207a804b9cbaee16e1c8fa0e8492add4974cf198eb79"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-005-p15",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77",
      "question_first_witness_sha256": "17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 20,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "e2a8fe90f63783ded670b8eda5748b8ce615c75f7f668194e7336021a0872bc9",
      "arithmetic_steps": 18
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}