{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261024-r01-linear-independence-basis-0060",
  "pilot_number": "R0021-11235",
  "number": 11235,
  "family_id": "qf-mod24-linbasis-coordinate-witness",
  "title": {
    "zh-Hant": "第 11235 題：線性獨立與基底",
    "en": "Question 11235: Unique Coordinates Only after Qualifying an Ordered Plane Basis"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261024-r01-linear-independence-basis-0060",
    "batch_id": "batch-20261013-advance-20261024-r01-linear-independence-basis-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-independence-basis"
    ],
    "title_zh": "線性獨立與基底",
    "stem_zh": "明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,2],[0,1]],\"target\":[-3,-1]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[1, 2], [0, 1]], [-3, -1], 1, -3, 5, [-3, -1])",
      "explanation_zh": "依賴結構核對=([[1, 2], [0, 1]], [-3, -1], 1, -3, 5, [-3, -1])；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；基底必須同時線性獨立並生成宣告空間；行列式與座標唯一性只用於方形R2基底。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "線性獨立與基底"
      ]
    },
    "variant": {
      "family_id": "qf-mod24-linbasis-coordinate-witness",
      "seed": 202610241060,
      "parameters": {
        "basis": [
          [
            1,
            2
          ],
          [
            0,
            1
          ]
        ],
        "target": [
          -3,
          -1
        ],
        "domain": "Exact_finite_coordinate_basis",
        "operation": "linbasis-coordinate-witness",
        "semantic_template": "qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,2],[0,1]],\"target\":[-3,-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長期性、PDE適定性、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-20261024-r01-linear-independence-basis-0060",
    "batch_id": "batch-20261013-advance-20261024-r01-linear-independence-basis-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "fcfcd8df5847c2846e1eaa6ebd1c75802da4e5a5286d194f5bcca98e43dbec4b",
      "family_id": "qf-mod24-linbasis-coordinate-witness",
      "seed": 202610241060,
      "semantic_template": "qf-mod24-linbasis-coordinate-witness/{\"basis\":[[1,2],[0,1]],\"target\":[-3,-1]}"
    },
    "fields": {
      "title": "Unique Coordinates Only after Qualifying an Ordered Plane Basis",
      "stem": "Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 2], [0, 1]], target=[-3, -1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.",
      "answer": {
        "kind": "expression",
        "value": "([[1, 2], [0, 1]], [-3, -1], 1, -3, 5, [-3, -1])",
        "explanation": "Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [-3, -1], 1, -3, 5, [-3, -1])."
      },
      "skills": [
        "Unique Coordinates Only after Qualifying an Ordered Plane Basis"
      ],
      "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": "明示有限座標向量族。操作=linbasis-coordinate-witness；資料={\"basis\":[[1,2],[0,1]],\"target\":[-3,-1]}。請依 [basis,target,det,first_coordinate,second_coordinate,reconstructed] 欄序回答。",
        "en": "Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 2], [0, 1]], target=[-3, -1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "basis determinant first product := mul(1, 1)",
      "after": "basis determinant first product = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 basis determinant first product，不以小數近似替代。",
          "en": "Recompute basis determinant first product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "basis determinant first product"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "basis determinant second product := mul(0, 2)",
      "after": "basis determinant second product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 basis determinant second product，不以小數近似替代。",
          "en": "Recompute basis determinant second product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "basis determinant second product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "basis determinant signed difference := sub(1, 0)",
      "after": "basis determinant signed difference = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 basis determinant signed difference，不以小數近似替代。",
          "en": "Recompute basis determinant signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "basis determinant signed difference"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ordered basis columns=[[1, 2], [0, 1]];target=[-3, -1];signed determinant=1",
      "after": "nonzero determinant;unique real coordinates;Cramer division permitted",
      "relation": "implies",
      "rule_id": "ordered_basis_qualification_before_coordinates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有號行列式非零才有唯一座標，按原基底次序解，不取絕對值或預先假設可逆。",
        "en": "A nonzero signed determinant gives unique coordinates.Solve in the original basis order,without taking an absolute value or assuming invertibility."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "first coordinate numerator first product := mul(-3, 1)",
      "after": "first coordinate numerator first product = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "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": [
          "-3",
          "1"
        ],
        "output": "-3",
        "label": "first coordinate numerator first product"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "first coordinate numerator second product := mul(0, -1)",
      "after": "first coordinate numerator second product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "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": [
          "0",
          "-1"
        ],
        "output": "0",
        "label": "first coordinate numerator second product"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "first coordinate numerator signed difference := sub(-3, 0)",
      "after": "first coordinate numerator signed difference = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first coordinate numerator signed difference，不以小數近似替代。",
          "en": "Recompute first coordinate numerator signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-3",
          "0"
        ],
        "output": "-3",
        "label": "first coordinate numerator signed difference"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "second coordinate numerator first product := mul(1, -1)",
      "after": "second coordinate numerator first product = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "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",
          "-1"
        ],
        "output": "-1",
        "label": "second coordinate numerator first product"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "second coordinate numerator second product := mul(-3, 2)",
      "after": "second coordinate numerator second product = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "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": [
          "-3",
          "2"
        ],
        "output": "-6",
        "label": "second coordinate numerator second product"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "second coordinate numerator signed difference := sub(-1, -6)",
      "after": "second coordinate numerator signed difference = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second coordinate numerator signed difference，不以小數近似替代。",
          "en": "Recompute second coordinate numerator signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "-6"
        ],
        "output": "5",
        "label": "second coordinate numerator signed difference"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "first basis coordinate := div(-3, 1)",
      "after": "first basis coordinate = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first basis coordinate，不以小數近似替代。",
          "en": "Recompute first basis coordinate with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-3",
          "1"
        ],
        "output": "-3",
        "label": "first basis coordinate"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "second basis coordinate := div(5, 1)",
      "after": "second basis coordinate = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second basis coordinate，不以小數近似替代。",
          "en": "Recompute second basis coordinate with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "5",
          "1"
        ],
        "output": "5",
        "label": "second basis coordinate"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "coordinate reconstruction:ordered vectors=[[1, 2], [0, 1]];aligned weights=[-3, 5]",
      "after": "dimension=2;preserve duplicates and order",
      "relation": "implies",
      "rule_id": "ordered_vector_weight_pairing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留向量順序、重複項與係數配對；同一座標加完所有向量的貢獻。",
        "en": "Preserve vector order,duplicates and coefficient pairing.Sum every vector's contribution within the same coordinate."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "coordinate reconstruction coordinate0 term0 := mul(-3, 1)",
      "after": "coordinate reconstruction coordinate0 term0 = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 coordinate reconstruction coordinate0 term0，不以小數近似替代。",
          "en": "Recompute coordinate reconstruction coordinate0 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "1"
        ],
        "output": "-3",
        "label": "coordinate reconstruction coordinate0 term0"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "coordinate reconstruction coordinate0 term1 := mul(5, 0)",
      "after": "coordinate reconstruction coordinate0 term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 coordinate reconstruction coordinate0 term1，不以小數近似替代。",
          "en": "Recompute coordinate reconstruction coordinate0 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "0"
        ],
        "output": "0",
        "label": "coordinate reconstruction coordinate0 term1"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "coordinate reconstruction coordinate0 sum := sum(-3, 0)",
      "after": "coordinate reconstruction coordinate0 sum = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 coordinate reconstruction coordinate0 sum，不以小數近似替代。",
          "en": "Recompute coordinate reconstruction coordinate0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-3",
          "0"
        ],
        "output": "-3",
        "label": "coordinate reconstruction coordinate0 sum"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "coordinate reconstruction coordinate1 term0 := mul(-3, 2)",
      "after": "coordinate reconstruction coordinate1 term0 = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 coordinate reconstruction coordinate1 term0，不以小數近似替代。",
          "en": "Recompute coordinate reconstruction coordinate1 term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-3",
          "2"
        ],
        "output": "-6",
        "label": "coordinate reconstruction coordinate1 term0"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "coordinate reconstruction coordinate1 term1 := mul(5, 1)",
      "after": "coordinate reconstruction coordinate1 term1 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 coordinate reconstruction coordinate1 term1，不以小數近似替代。",
          "en": "Recompute coordinate reconstruction coordinate1 term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "1"
        ],
        "output": "5",
        "label": "coordinate reconstruction coordinate1 term1"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "coordinate reconstruction coordinate1 sum := sum(-6, 5)",
      "after": "coordinate reconstruction coordinate1 sum = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 coordinate reconstruction coordinate1 sum，不以小數近似替代。",
          "en": "Recompute coordinate reconstruction coordinate1 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "5"
        ],
        "output": "-1",
        "label": "coordinate reconstruction coordinate1 sum"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "basis target residual0 := sub(-3, -3)",
      "after": "basis target residual0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 basis target residual0，不以小數近似替代。",
          "en": "Recompute basis target residual0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-3",
          "-3"
        ],
        "output": "0",
        "label": "basis target residual0"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "basis target residual1 := sub(-1, -1)",
      "after": "basis target residual1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步已指定的角色逐項精確運算；保留符號、分母與原始配對。",
        "en": "Compute exactly with this step's specified roles,retaining signs,denominators and original pairings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 basis target residual1，不以小數近似替代。",
          "en": "Recompute basis target residual1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "0",
        "label": "basis target residual1"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "coordinates=[-3, 5];reconstructed=[-3, -1];target=[-3, -1];residuals=[0, 0]",
      "after": "both original equations satisfied;uniqueness follows from qualified basis",
      "relation": "implies",
      "rule_id": "coordinate_reconstruction_and_uniqueness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "重組兩個原方程全通過給存在性；唯一性另由非零行列式，不由一次代值成功推得。",
        "en": "Reconstructing both original equations proves existence.Uniqueness follows separately from the nonzero determinant,not from one successful substitution."
      },
      "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,
          2
        ],
        [
          0,
          1
        ]
      ],
      [
        -3,
        -1
      ],
      1,
      "-3",
      "5",
      [
        "-3",
        "-1"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([[1, 2], [0, 1]], [-3, -1], 1, -3, 5, [-3, -1])",
    "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-20261024-r01-linear-independence-basis-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261024-r01-linear-independence-basis-foundations/questions.jsonl",
      "batch_sha256": "9c09bb587cfb555cea8fbb6e53978a87ed1a994c6c16dd9c77a76f0a37571632",
      "original_line": 60,
      "original_line_sha256": "8b71fa57aba35662e764c39ad037468b12ed0371b345fc56116f98ff218a3fbe",
      "question_compact_sha256": "fcfcd8df5847c2846e1eaa6ebd1c75802da4e5a5286d194f5bcca98e43dbec4b",
      "english_row_compact_sha256": "e8221e703e0ed6faea66077386fa6931c21f45504fc4f6ec226a90e5cb1cde21",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261024-r01-linear-independence-basis-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261024-r01-linear-independence-basis-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-linear-independence-basis": {
        "sha256": "78c45988a718f6a7ca4acf2aba00ea4df794c8f141cefe822bf72206fa2b2f62",
        "english_sha256": "48c43771b8f6734ece79626851e05a5dca9bb984732eb3f69cffe2c6e364a912",
        "original_path": "public/data/mko/linear-independence-basis.json",
        "english_path": "public/data/i18n/editions/mko-linear-independence-basis.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "18bbb89127e14ce3466c1a8730f1c3c6e22a8f59a9148433b2abc7f6ea274ee5",
    "original_candidate_sha256": "133950f5207c64ae093898acc8b5750e59c2d723ae1a2847bb4794feb7974dc7",
    "original_candidate_line_sha256": "133950f5207c64ae093898acc8b5750e59c2d723ae1a2847bb4794feb7974dc7"
  },
  "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-p22",
    "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": "47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258",
      "question_first_witness_sha256": "a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 22,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "9d8b834e5c40bef9f5b4295bcd21ff0e49dba32042a1f41c7ecfe6c6ea575686",
      "arithmetic_steps": 19
    },
    "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."
  }
}