{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261023-r02-linear-combination-span-0065",
  "pilot_number": "R0020-10815",
  "number": 10815,
  "family_id": "qf-mod23-span-r2-coordinate-solve",
  "title": {
    "zh-Hant": "第 10815 題：有限線性組合",
    "en": "Question 10815: Unique Real Coordinates in Two Independent Plane Generators"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261023-r02-linear-combination-span-0065",
    "batch_id": "batch-20261013-advance-20261023-r02-linear-combination-span-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-combination-span"
    ],
    "title_zh": "有限線性組合",
    "stem_zh": "固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[2,-3]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([1, 1], [1, -1], [2, -3], -2, -1/2, 5/2, [2, -3])",
      "explanation_zh": "現代橋梁核對=([1, 1], [1, -1], [2, -3], -2, -1/2, 5/2, [2, -3])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；span量化全部有限組合；不包含無限級數。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有限線性組合"
      ]
    },
    "variant": {
      "family_id": "qf-mod23-span-r2-coordinate-solve",
      "seed": 202610232065,
      "parameters": {
        "v1": [
          1,
          1
        ],
        "v2": [
          1,
          -1
        ],
        "target": [
          2,
          -3
        ],
        "domain": "Finite_coordinate_linear_combinations",
        "operation": "span-r2-coordinate-solve",
        "semantic_template": "qf-mod23-span-r2-coordinate-solve/{\"target\":[2,-3],\"v1\":[1,1],\"v2\":[1,-1]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261023-r02-linear-combination-span-0065",
    "batch_id": "batch-20261013-advance-20261023-r02-linear-combination-span-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "6edbbff72eaef7b403236b1aaf78b69f6242a9d69b27ffc60e791ade5d2222cc",
      "family_id": "qf-mod23-span-r2-coordinate-solve",
      "seed": 202610232065,
      "semantic_template": "qf-mod23-span-r2-coordinate-solve/{\"target\":[2,-3],\"v1\":[1,1],\"v2\":[1,-1]}"
    },
    "fields": {
      "title": "Unique Real Coordinates in Two Independent Plane Generators",
      "stem": "Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[2, -3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "([1, 1], [1, -1], [2, -3], -2, -1/2, 5/2, [2, -3])",
        "explanation": "Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [2, -3], -2, -1/2, 5/2, [2, -3])."
      },
      "skills": [
        "Unique Real Coordinates in Two Independent Plane Generators"
      ],
      "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": "固定維度與明示有限生成族。操作=span-r2-coordinate-solve；資料={\"v1\":[1,1],\"v2\":[1,-1],\"target\":[2,-3]}。請依 [v1,v2,target,det,a,b,reconstructed] 欄序回答。",
        "en": "Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[2, -3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "fixed real coordinate space;span uses all finite real linear combinations",
      "after": "given coefficient witness and existence of some coefficients are separate",
      "relation": "implies",
      "rule_id": "span_quantifier_and_witness_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "span量化所有有限實係數組合；本題的整數／有理資料是明示切片，一個指定見證失敗不等於所有係數都失敗。",
        "en": "Span quantifies all finite real linear combinations.These integer/rational instances are declared slices;a failed designated witness does not show every coefficient choice fails."
      },
      "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": "column determinant first product := mul(1, -1)",
      "after": "column determinant first product = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 column determinant first product，不以小數近似替代。",
          "en": "Recompute column determinant first product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "-1",
        "label": "column determinant first product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "column determinant second product := mul(1, 1)",
      "after": "column determinant second product = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 column determinant second product，不以小數近似替代。",
          "en": "Recompute column determinant second product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "column determinant second product"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "signed column determinant := sub(-1, 1)",
      "after": "signed column determinant = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 signed column determinant，不以小數近似替代。",
          "en": "Recompute signed column determinant with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-2",
        "label": "signed column determinant"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "first column=[1, 1];second column=[1, -1];target=[2, -3]",
      "after": "signed determinant=-2;independent generators=是",
      "relation": "implies",
      "rule_id": "R2_column_independence_before_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩生成元按原次序作矩陣直欄；非零有號行列式才可唯一解座標，不取絕對值或交換角色。",
        "en": "Use the generators as columns in their original order.A nonzero signed determinant permits unique coordinates;do not take its absolute value or exchange roles."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "first coordinate numerator first product := mul(2, -1)",
      "after": "first coordinate numerator first product = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first coordinate numerator first product，不以小數近似替代。",
          "en": "Recompute first coordinate numerator first product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-1"
        ],
        "output": "-2",
        "label": "first coordinate numerator first product"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "first coordinate numerator second product := mul(1, -3)",
      "after": "first coordinate numerator second product = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first coordinate numerator second product，不以小數近似替代。",
          "en": "Recompute first coordinate numerator second product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-3"
        ],
        "output": "-3",
        "label": "first coordinate numerator second product"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "first coordinate numerator := sub(-2, -3)",
      "after": "first coordinate numerator = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first coordinate numerator，不以小數近似替代。",
          "en": "Recompute first coordinate numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-2",
          "-3"
        ],
        "output": "1",
        "label": "first coordinate numerator"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "second coordinate numerator first product := mul(1, -3)",
      "after": "second coordinate numerator first product = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second coordinate numerator first product，不以小數近似替代。",
          "en": "Recompute second coordinate numerator first product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-3"
        ],
        "output": "-3",
        "label": "second coordinate numerator first product"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "second coordinate numerator second product := mul(2, 1)",
      "after": "second coordinate numerator second product = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second coordinate numerator second product，不以小數近似替代。",
          "en": "Recompute second coordinate numerator second product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "second coordinate numerator second product"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "second coordinate numerator := sub(-3, 2)",
      "after": "second coordinate numerator = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second coordinate numerator，不以小數近似替代。",
          "en": "Recompute second coordinate numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-3",
          "2"
        ],
        "output": "-5",
        "label": "second coordinate numerator"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "coefficient of first generator := div(1, -2)",
      "after": "coefficient of first generator = -1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 coefficient of first generator，不以小數近似替代。",
          "en": "Recompute coefficient of first generator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "-2"
        ],
        "output": "-1/2",
        "label": "coefficient of first generator"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "coefficient of second generator := div(-5, -2)",
      "after": "coefficient of second generator = 5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 coefficient of second generator，不以小數近似替代。",
          "en": "Recompute coefficient of second generator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-5",
          "-2"
        ],
        "output": "5/2",
        "label": "coefficient of second generator"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "R2 reconstructed coordinate0 first contribution := mul(-1/2, 1)",
      "after": "R2 reconstructed coordinate0 first contribution = -1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 R2 reconstructed coordinate0 first contribution，不以小數近似替代。",
          "en": "Recompute R2 reconstructed coordinate0 first contribution with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1/2",
          "1"
        ],
        "output": "-1/2",
        "label": "R2 reconstructed coordinate0 first contribution"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "R2 reconstructed coordinate0 second contribution := mul(5/2, 1)",
      "after": "R2 reconstructed coordinate0 second contribution = 5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 R2 reconstructed coordinate0 second contribution，不以小數近似替代。",
          "en": "Recompute R2 reconstructed coordinate0 second contribution with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5/2",
          "1"
        ],
        "output": "5/2",
        "label": "R2 reconstructed coordinate0 second contribution"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "R2 reconstructed coordinate0 sum := sum(-1/2, 5/2)",
      "after": "R2 reconstructed coordinate0 sum = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 R2 reconstructed coordinate0 sum，不以小數近似替代。",
          "en": "Recompute R2 reconstructed coordinate0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1/2",
          "5/2"
        ],
        "output": "2",
        "label": "R2 reconstructed coordinate0 sum"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "R2 reconstructed coordinate1 first contribution := mul(-1/2, 1)",
      "after": "R2 reconstructed coordinate1 first contribution = -1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 R2 reconstructed coordinate1 first contribution，不以小數近似替代。",
          "en": "Recompute R2 reconstructed coordinate1 first contribution with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1/2",
          "1"
        ],
        "output": "-1/2",
        "label": "R2 reconstructed coordinate1 first contribution"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "R2 reconstructed coordinate1 second contribution := mul(5/2, -1)",
      "after": "R2 reconstructed coordinate1 second contribution = -5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 R2 reconstructed coordinate1 second contribution，不以小數近似替代。",
          "en": "Recompute R2 reconstructed coordinate1 second contribution with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5/2",
          "-1"
        ],
        "output": "-5/2",
        "label": "R2 reconstructed coordinate1 second contribution"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "R2 reconstructed coordinate1 sum := sum(-1/2, -5/2)",
      "after": "R2 reconstructed coordinate1 sum = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 R2 reconstructed coordinate1 sum，不以小數近似替代。",
          "en": "Recompute R2 reconstructed coordinate1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1/2",
          "-5/2"
        ],
        "output": "-3",
        "label": "R2 reconstructed coordinate1 sum"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "first-generator coefficient=-1/2;second-generator coefficient=5/2",
      "after": "reconstructed target=[2, -3]",
      "relation": "implies",
      "rule_id": "R2_exact_coordinate_back_substitution",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用兩個求得係數代回原欄向量，兩個座標都等於目標才完成；分母規約不改生成元順序。",
        "en": "Substitute both coefficients back into the original columns and recover both target coordinates.Denominator normalization does not change generator order."
      },
      "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": [
      [
        1,
        1
      ],
      [
        1,
        -1
      ],
      [
        2,
        -3
      ],
      -2,
      "-1/2",
      "5/2",
      [
        "2",
        "-3"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([1, 1], [1, -1], [2, -3], -2, -1/2, 5/2, [2, -3])",
    "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-20261013-advance-20261023-r02-linear-combination-span-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261023-r02-linear-combination-span-foundations/questions.jsonl",
      "batch_sha256": "da062918e67f92a171ca3b576c4e8fd07cc820916a8047a29a22a0da6e0df6d4",
      "original_line": 65,
      "original_line_sha256": "f7b7d9a19253da2e37b788506d9ac739f9a2617ddebd38d032f7a6441082d3a4",
      "question_compact_sha256": "6edbbff72eaef7b403236b1aaf78b69f6242a9d69b27ffc60e791ade5d2222cc",
      "english_row_compact_sha256": "15a601b8436c87172858388fd5649c5978576cdde982cd240baae21f85033fae",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261023-r02-linear-combination-span-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261023-r02-linear-combination-span-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-linear-combination-span": {
        "sha256": "6f4f8ff504708ab9492740d70255cb2d85c406d8eaa891eef387935b3d46dad3",
        "english_sha256": "3d16797b83376915e08d16c68eee0e87ae7d9b28ca5d5ba76a9e2466ce8195b3",
        "original_path": "public/data/mko/linear-combination-span.json",
        "english_path": "public/data/i18n/editions/mko-linear-combination-span.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "bd67681fffd9e29972c322d497cd945a668a794dd25e0c8cd26f1a6ce22e4dfb",
    "original_candidate_sha256": "38a42cd115babf1a71a6154ffac095373e0a52e882beb0ee566b6346ce6fa1ca",
    "original_candidate_line_sha256": "38a42cd115babf1a71a6154ffac095373e0a52e882beb0ee566b6346ce6fa1ca"
  },
  "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-20261026-006-p21",
    "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": "b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f",
      "question_first_witness_sha256": "3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4"
    },
    "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": "2493b668f1b6153c6d07385c944b3f3e4bfbc8995e4474c2622e8892f1dd1712",
      "arithmetic_steps": 17
    },
    "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."
  }
}