{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r05-system-linear-equations-0047",
  "pilot_number": "R0014-07847",
  "number": 7847,
  "family_id": "qf-density16-system-classify-solve",
  "title": {
    "zh-Hant": "第 7847 題：兩列秩與完整解分類",
    "en": "Question 7847: Signed Determinant, Both Ranks, and System Class"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r05-system-linear-equations-0047",
    "batch_id": "batch-20261012-advance-20261016-r05-system-linear-equations-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-system-linear-equations"
    ],
    "title_zh": "兩列秩與完整解分類",
    "stem_zh": "R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 9], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。",
    "answer": {
      "kind": "expression",
      "value": "(1, 2, 2, 唯一解, [7/1, 2/1])",
      "explanation_zh": "有限結構核對=(1, 2, 2, 唯一解, [7/1, 2/1])。用有理Gaussian消去獨立計兩個秩；不一致答無解，兩個係數pivot才讀唯一解。有限精確核對不是一般形式證明。候選須同時滿足兩式，零縮放不當可逆列操作。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "兩列秩與完整解分類"
      ]
    },
    "variant": {
      "family_id": "qf-density16-system-classify-solve",
      "seed": 202610165047,
      "parameters": {
        "rows": [
          [
            1,
            1,
            9
          ],
          [
            0,
            1,
            2
          ]
        ],
        "domain": "R2_linear_system_integer_slice",
        "operation": "system-classify-solve",
        "semantic_template": "qf-density16-system-classify-solve/{\"rows\":[[1,1,9],[0,1,2]]}"
      }
    },
    "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-r05-system-linear-equations-0047",
    "batch_id": "batch-20261012-advance-20261016-r05-system-linear-equations-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "5a91f6b9a4c8e9f232f1c94c9f4dccc49aaa002ffd652bf701492330039962e7",
      "family_id": "qf-density16-system-classify-solve",
      "seed": 202610165047,
      "semantic_template": "qf-density16-system-classify-solve/{\"rows\":[[1,1,9],[0,1,2]]}"
    },
    "fields": {
      "title": "Signed Determinant, Both Ranks, and System Class",
      "stem": "Over R, rows=[[1, 1, 9], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency.",
      "answer": {
        "kind": "expression",
        "value": "(1, 2, 2, unique-solution, [7/1, 2/1])",
        "explanation": "Compare coefficient and augmented ranks using exact rational arithmetic: unequal ranks mean no solution, equal rank below 2 means infinitely many solutions, and rank 2 yields the ordered unique rational solution. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, 2, 2, unique-solution, [7/1, 2/1])."
      },
      "skills": [
        "Signed Determinant, Both Ranks, and System Class"
      ],
      "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": "R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 1, 9], [0, 1, 2]]。求 (有號D, 係數秩, 增廣秩, 無解/唯一解/無限多解, 唯一時有序有理解[x,y]或不適用/不唯一)。",
        "en": "Over R, rows=[[1, 1, 9], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. Give [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique] as an ordered tuple. A zero determinant alone does not decide inconsistency."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "rows=[[1, 1, 9], [0, 1, 2]]",
      "after": "each row [a,b,c] means a*x+b*y=c over R",
      "relation": "implies",
      "rule_id": "two_equations_real_system_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩式須同時成立，右側常數不是第三個未知數。退化零係數列也保留，不能先假設行列式非零。",
        "en": "Both equations must hold simultaneously.The right-hand constant is not a third unknown.Retain degenerate zero-coefficient rows rather than presuming a nonzero determinant."
      },
      "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": "before ae := mul(1, 1)",
      "after": "before ae = 1",
      "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 before ae，不以小數近似替代。",
          "en": "Recompute before ae with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "before ae"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "before bd := mul(1, 0)",
      "after": "before bd = 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 before bd，不以小數近似替代。",
          "en": "Recompute before bd with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "before bd"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "before D := sub(1, 0)",
      "after": "before D = 1",
      "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 before D，不以小數近似替代。",
          "en": "Recompute before D with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "before D"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "before af := mul(1, 2)",
      "after": "before af = 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 before af，不以小數近似替代。",
          "en": "Recompute before af with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "before af"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "before cd := mul(9, 0)",
      "after": "before cd = 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 before cd，不以小數近似替代。",
          "en": "Recompute before cd with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "0"
        ],
        "output": "0",
        "label": "before cd"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "before minor_x_rhs := sub(2, 0)",
      "after": "before minor_x_rhs = 2",
      "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 before minor_x_rhs，不以小數近似替代。",
          "en": "Recompute before minor_x_rhs with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "before minor_x_rhs"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "before bf := mul(1, 2)",
      "after": "before bf = 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 before bf，不以小數近似替代。",
          "en": "Recompute before bf with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "before bf"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "before ce := mul(9, 1)",
      "after": "before ce = 9",
      "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 before ce，不以小數近似替代。",
          "en": "Recompute before ce with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "1"
        ],
        "output": "9",
        "label": "before ce"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "before minor_y_rhs := sub(2, 9)",
      "after": "before minor_y_rhs = -7",
      "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 before minor_y_rhs，不以小數近似替代。",
          "en": "Recompute before minor_y_rhs with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "9"
        ],
        "output": "-7",
        "label": "before minor_y_rhs"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "phase=before;rows=[[1, 1, 9], [0, 1, 2]];2x2 minors=[1, 2, -7]",
      "after": "rankA=2;rankAb=2",
      "relation": "implies",
      "rule_id": "system_rank_from_complete_minors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "係數矩陣的非零2×2子式給秩2；否則查是否有非零係數給秩1。增廣矩陣有三個2×2子式，全部為0才再查非零格，不能漏右側常數。",
        "en": "A nonzero coefficient 2×2 minor gives rank2;otherwise a nonzero coefficient gives rank1.The augmented matrix has three 2×2 minors;if all vanish,check its entries,including the right-hand constants."
      },
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "before x numerator := sub(0, -7)",
      "after": "before x numerator = 7",
      "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 before x numerator，不以小數近似替代。",
          "en": "Recompute before x numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "-7"
        ],
        "output": "7",
        "label": "before x numerator"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "before x solution := div(7, 1)",
      "after": "before x solution = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分母已另核非零，作精確有理數除法。",
        "en": "The denominator has separately been checked nonzero;divide exactly as a rational number."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 before x solution，不以小數近似替代。",
          "en": "Recompute before x solution with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "7",
          "1"
        ],
        "output": "7",
        "label": "before x solution"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "before y solution := div(2, 1)",
      "after": "before y solution = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分母已另核非零，作精確有理數除法。",
        "en": "The denominator has separately been checked nonzero;divide exactly as a rational number."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 before y solution，不以小數近似替代。",
          "en": "Recompute before y solution with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "before y solution"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "before solution row0 ax := mul(1, 7)",
      "after": "before solution row0 ax = 7",
      "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 before solution row0 ax，不以小數近似替代。",
          "en": "Recompute before solution row0 ax with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "7"
        ],
        "output": "7",
        "label": "before solution row0 ax"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "before solution row0 by := mul(1, 2)",
      "after": "before solution row0 by = 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 before solution row0 by，不以小數近似替代。",
          "en": "Recompute before solution row0 by with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "before solution row0 by"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "before solution row0 lhs := sum(7, 2)",
      "after": "before solution row0 lhs = 9",
      "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 before solution row0 lhs，不以小數近似替代。",
          "en": "Recompute before solution row0 lhs with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "7",
          "2"
        ],
        "output": "9",
        "label": "before solution row0 lhs"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "before solution row0 residual := sub(9, 9)",
      "after": "before solution row0 residual = 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 before solution row0 residual，不以小數近似替代。",
          "en": "Recompute before solution row0 residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "9",
          "9"
        ],
        "output": "0",
        "label": "before solution row0 residual"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "before solution row1 ax := mul(0, 7)",
      "after": "before solution row1 ax = 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 before solution row1 ax，不以小數近似替代。",
          "en": "Recompute before solution row1 ax with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "7"
        ],
        "output": "0",
        "label": "before solution row1 ax"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "before solution row1 by := mul(1, 2)",
      "after": "before solution row1 by = 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 before solution row1 by，不以小數近似替代。",
          "en": "Recompute before solution row1 by with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "before solution row1 by"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "before solution row1 lhs := sum(0, 2)",
      "after": "before solution row1 lhs = 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 before solution row1 lhs，不以小數近似替代。",
          "en": "Recompute before solution row1 lhs with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "before solution row1 lhs"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "before solution row1 residual := sub(2, 2)",
      "after": "before solution row1 residual = 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 before solution row1 residual，不以小數近似替代。",
          "en": "Recompute before solution row1 residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "2"
        ],
        "output": "0",
        "label": "before solution row1 residual"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "phase=before;D=1;D*x=7;D*y=2",
      "after": "solution=[7/1, 2/1];row residuals=[0, 0]",
      "relation": "implies",
      "rule_id": "system_unique_solution_substitution",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非零D允許Cramer消去後除法：D·x=ce−bf，D·y=af−cd。再把有理解代回兩個原列，兩殘差都0；非零D也保證唯一。",
        "en": "Nonzero D permits division after Cramer elimination:D·x=ce−bf and D·y=af−cd.Substitute the rational pair into both original rows;both residuals are zero.Nonzero D also ensures uniqueness."
      },
      "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": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "phase=before;rankA=2;rankAb=2;unknowns=2",
      "after": "class=唯一解;solution/status=[7/1, 2/1]",
      "relation": "implies",
      "rule_id": "system_consistency_and_free_variables",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩秩不等表示有矛盾列，無解；相等且秩小於2有自由變數，在R上有無限多解；秩2才給唯一解。D=0時不做除以D。",
        "en": "Unequal ranks mean an inconsistent row and no solution.Equal rank below2 leaves a free variable and infinitely many real solutions.Only rank2 gives a unique solution.Never divide by D when D=0."
      },
      "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,
      2,
      "唯一解",
      [
        "7/1",
        "2/1"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(1, 2, 2, 唯一解, [7/1, 2/1])",
    "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-20261012-advance-20261016-r05-system-linear-equations-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261016-r05-system-linear-equations-foundations/questions.jsonl",
      "batch_sha256": "600d1fcb1634af8d3233900a81715f255bfc3a3d4fff09369711f40dea2bf385",
      "original_line": 47,
      "original_line_sha256": "a5d0bc5f439fafde5e8b5d5485223acc9fdc579b541760728bda5aa4d1e138e2",
      "question_compact_sha256": "5a91f6b9a4c8e9f232f1c94c9f4dccc49aaa002ffd652bf701492330039962e7",
      "english_row_compact_sha256": "1207e9fea2a7a8db792532cffc607bb4312336203cde1725b515da27efb73333",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261016-r05-system-linear-equations-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261016-r05-system-linear-equations-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-system-linear-equations": {
        "sha256": "62dc0d59fbb66d23a8ef47c7b2725a56713d925c7f2c8ef0a380c46552972403",
        "english_sha256": "7fc6d87498d9c2a7943b1715c26c7d3e2baf79d0f54d4beb9019b628da0010c4",
        "original_path": "public/data/mko/system-linear-equations.json",
        "english_path": "public/data/i18n/editions/mko-system-linear-equations.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d7860245c70062705fa4858c13821db6ff286678dec63f22464364f5d24cfcc1",
    "original_candidate_sha256": "fb792272caafa947caefedbe55307f643229c1224f2d84d1ae67c1e3f91eb090",
    "original_candidate_line_sha256": "fb792272caafa947caefedbe55307f643229c1224f2d84d1ae67c1e3f91eb090"
  },
  "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": 24,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "7084764ccbb66d5f71e08644edfd99ee99d18f01467391995adec0dcaa0e2a55",
      "arithmetic_steps": 20
    },
    "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."
  }
}