{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r05-system-linear-equations-0090",
  "pilot_number": "R0014-07890",
  "number": 7890,
  "family_id": "qf-density16-system-row-scaling-solution",
  "title": {
    "zh-Hant": "第 7890 題：第一列縮放的操作資格",
    "en": "Question 7890: Row Scaling: Classification versus Operation Reversibility"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r05-system-linear-equations-0090",
    "batch_id": "batch-20261012-advance-20261016-r05-system-linear-equations-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-system-linear-equations"
    ],
    "title_zh": "第一列縮放的操作資格",
    "stem_zh": "R上兩元兩式；rows每列[a,b,c]代表a*x+b*y=c。rows=[[1, 0, 2], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。",
    "answer": {
      "kind": "expression",
      "value": "([[1, 0, 2], [0, 1, 2]], [1, 2, 2, 唯一解, [2/1, 2/1]], [1, 2, 2, 唯一解, [2/1, 2/1]], 是, 非零縮放才是可逆列操作)",
      "explanation_zh": "有限結構核對=([[1, 0, 2], [0, 1, 2]], [1, 2, 2, 唯一解, [2/1, 2/1]], [1, 2, 2, 唯一解, [2/1, 2/1]], 是, 非零縮放才是可逆列操作)。連同右側常數縮放，再以Gaussian核前後；零縮放可能在冗餘系統保留解集但操作仍不可逆。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": 2,
      "skills": [
        "第一列縮放的操作資格"
      ]
    },
    "variant": {
      "family_id": "qf-density16-system-row-scaling-solution",
      "seed": 202610165090,
      "parameters": {
        "rows": [
          [
            1,
            0,
            2
          ],
          [
            0,
            1,
            2
          ]
        ],
        "t": 1,
        "domain": "R2_linear_system_integer_slice",
        "operation": "system-row-scaling-solution",
        "semantic_template": "qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,2],[0,1,2]],\"t\":1}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261016-r05-system-linear-equations-0090",
    "batch_id": "batch-20261012-advance-20261016-r05-system-linear-equations-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "fd64a1e405d16556cae5d035059dbd59b7fb587679419d1d28b8f93787af9ca8",
      "family_id": "qf-density16-system-row-scaling-solution",
      "seed": 202610165090,
      "semantic_template": "qf-density16-system-row-scaling-solution/{\"rows\":[[1,0,2],[0,1,2]],\"t\":1}"
    },
    "fields": {
      "title": "Row Scaling: Classification versus Operation Reversibility",
      "stem": "Over R, rows=[[1, 0, 2], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal.",
      "answer": {
        "kind": "expression",
        "value": "([[1, 0, 2], [0, 1, 2]], [1, 2, 2, unique-solution, [2/1, 2/1]], [1, 2, 2, unique-solution, [2/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)",
        "explanation": "Recompute both ranks/classes after scaling the whole row; zero scaling can preserve a redundant system’s solution set yet remains a noninvertible row operation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0, 2], [0, 1, 2]], [1, 2, 2, unique-solution, [2/1, 2/1]], [1, 2, 2, unique-solution, [2/1, 2/1]], Yes, Only nonzero scaling is an invertible row operation)."
      },
      "skills": [
        "Row Scaling: Classification versus Operation Reversibility"
      ],
      "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, 0, 2], [0, 1, 2]]、t=1。只把row1乘t。求 (新rows, 前分類列[D,rankA,rankAb,class,solution或marker], 後分類列, t≠0可逆操作資格, 固定說明字串「非零縮放才是可逆列操作」)。第4欄不是解集是否恰巧等價。",
        "en": "Over R, rows=[[1, 0, 2], [0, 1, 2]], where each [a,b,c] means a*x+b*y=c. t=1; multiply only row1 by t, including its right-side constant. Give (new rows, original classification record [signed D, rankA, rankAb, no-solution/unique-solution/infinitely-many-solutions, ordered rational solution [x,y] or not-applicable/not-unique], new classification record, is t≠0 an invertible operation?, fixed text “Only nonzero scaling is an invertible row operation”). Field 4 is not whether the solution sets happen to be equal."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "rows=[[1, 0, 2], [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(0, 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": [
          "0",
          "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(2, 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": [
          "2",
          "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(0, 2)",
      "after": "before bf = 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 bf，不以小數近似替代。",
          "en": "Recompute before bf with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "before bf"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "before ce := mul(2, 1)",
      "after": "before ce = 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 ce，不以小數近似替代。",
          "en": "Recompute before ce with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "before ce"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "before minor_y_rhs := sub(0, 2)",
      "after": "before minor_y_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_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": [
          "0",
          "2"
        ],
        "output": "-2",
        "label": "before minor_y_rhs"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "phase=before;rows=[[1, 0, 2], [0, 1, 2]];2x2 minors=[1, 2, -2]",
      "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, -2)",
      "after": "before x numerator = 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 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",
          "-2"
        ],
        "output": "2",
        "label": "before x numerator"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "before x solution := div(2, 1)",
      "after": "before x 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 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": [
          "2",
          "1"
        ],
        "output": "2",
        "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, 2)",
      "after": "before solution row0 ax = 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 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",
          "2"
        ],
        "output": "2",
        "label": "before solution row0 ax"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "before solution row0 by := mul(0, 2)",
      "after": "before solution row0 by = 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 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": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "before solution row0 by"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "before solution row0 lhs := sum(2, 0)",
      "after": "before solution row0 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 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": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "before solution row0 lhs"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "before solution row0 residual := sub(2, 2)",
      "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": [
          "2",
          "2"
        ],
        "output": "0",
        "label": "before solution row0 residual"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "before solution row1 ax := mul(0, 2)",
      "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",
          "2"
        ],
        "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=2;D*y=2",
      "after": "solution=[2/1, 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=[2/1, 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
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "scaled row1[0] := mul(1, 1)",
      "after": "scaled row1[0] = 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 scaled row1[0]，不以小數近似替代。",
          "en": "Recompute scaled row1[0] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "scaled row1[0]"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "scaled row1[1] := mul(1, 0)",
      "after": "scaled row1[1] = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用本位置明示的純量與分量相乘，不改成內積或逐項矩陣乘法。",
        "en": "Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scaled row1[1]，不以小數近似替代。",
          "en": "Recompute scaled row1[1] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "scaled row1[1]"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "scaled row1[2] := mul(1, 2)",
      "after": "scaled row1[2] = 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 scaled row1[2]，不以小數近似替代。",
          "en": "Recompute scaled row1[2] with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "scaled row1[2]"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "operation=system-row-scaling-solution;t=1;old rows=[[1, 0, 2], [0, 1, 2]]",
      "after": "new rows=[[1, 0, 2], [0, 1, 2]];invertible=是;inverse=row1 -> row1/(1)",
      "relation": "implies",
      "rule_id": "row_operation_inverse_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整列操作包括右側常數。加任意t倍另一列的逆操作是加−t倍；縮放只有t≠0才可除回。t=0可能恰好保留冗餘系統的解集，但操作本身仍不可逆。",
        "en": "A row operation includes the right-hand constant.Adding any t times the other row is undone by adding−t times it.Scaling can be divided back only for t≠0.At t=0 a redundant system may coincidentally keep its solution set,but the operation is still not invertible."
      },
      "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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "after ae := mul(1, 1)",
      "after": "after 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 after ae，不以小數近似替代。",
          "en": "Recompute after ae with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "after ae"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "after bd := mul(0, 0)",
      "after": "after 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 after bd，不以小數近似替代。",
          "en": "Recompute after bd with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "after bd"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "after D := sub(1, 0)",
      "after": "after 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 after D，不以小數近似替代。",
          "en": "Recompute after D with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "after D"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "after af := mul(1, 2)",
      "after": "after 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 after af，不以小數近似替代。",
          "en": "Recompute after af with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "after af"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "after cd := mul(2, 0)",
      "after": "after 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 after cd，不以小數近似替代。",
          "en": "Recompute after cd with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "after cd"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "after minor_x_rhs := sub(2, 0)",
      "after": "after 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 after minor_x_rhs，不以小數近似替代。",
          "en": "Recompute after minor_x_rhs with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "after minor_x_rhs"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "after bf := mul(0, 2)",
      "after": "after bf = 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 after bf，不以小數近似替代。",
          "en": "Recompute after bf with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "after bf"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "after ce := mul(2, 1)",
      "after": "after ce = 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 after ce，不以小數近似替代。",
          "en": "Recompute after ce with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "after ce"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "after minor_y_rhs := sub(0, 2)",
      "after": "after minor_y_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 after minor_y_rhs，不以小數近似替代。",
          "en": "Recompute after minor_y_rhs with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "2"
        ],
        "output": "-2",
        "label": "after minor_y_rhs"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "phase=after;rows=[[1, 0, 2], [0, 1, 2]];2x2 minors=[1, 2, -2]",
      "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": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "after x numerator := sub(0, -2)",
      "after": "after x numerator = 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 after x numerator，不以小數近似替代。",
          "en": "Recompute after x numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "2",
        "label": "after x numerator"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "after x solution := div(2, 1)",
      "after": "after x 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 after x solution，不以小數近似替代。",
          "en": "Recompute after x solution with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "after x solution"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "after y solution := div(2, 1)",
      "after": "after 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 after y solution，不以小數近似替代。",
          "en": "Recompute after y solution with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "after y solution"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "after solution row0 ax := mul(1, 2)",
      "after": "after solution row0 ax = 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 after solution row0 ax，不以小數近似替代。",
          "en": "Recompute after solution row0 ax with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "after solution row0 ax"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "after solution row0 by := mul(0, 2)",
      "after": "after solution row0 by = 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 after solution row0 by，不以小數近似替代。",
          "en": "Recompute after solution row0 by with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "after solution row0 by"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "after solution row0 lhs := sum(2, 0)",
      "after": "after solution row0 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 after solution row0 lhs，不以小數近似替代。",
          "en": "Recompute after solution row0 lhs with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "after solution row0 lhs"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "after solution row0 residual := sub(2, 2)",
      "after": "after 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 after solution row0 residual，不以小數近似替代。",
          "en": "Recompute after solution row0 residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "2"
        ],
        "output": "0",
        "label": "after solution row0 residual"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "after solution row1 ax := mul(0, 2)",
      "after": "after 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 after solution row1 ax，不以小數近似替代。",
          "en": "Recompute after solution row1 ax with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "after solution row1 ax"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "after solution row1 by := mul(1, 2)",
      "after": "after 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 after solution row1 by，不以小數近似替代。",
          "en": "Recompute after solution row1 by with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "after solution row1 by"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "after solution row1 lhs := sum(0, 2)",
      "after": "after 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 after solution row1 lhs，不以小數近似替代。",
          "en": "Recompute after solution row1 lhs with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "after solution row1 lhs"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "after solution row1 residual := sub(2, 2)",
      "after": "after 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 after solution row1 residual，不以小數近似替代。",
          "en": "Recompute after solution row1 residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "2"
        ],
        "output": "0",
        "label": "after solution row1 residual"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "phase=after;D=1;D*x=2;D*y=2",
      "after": "solution=[2/1, 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": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "phase=after;rankA=2;rankAb=2;unknowns=2",
      "after": "class=唯一解;solution/status=[2/1, 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",
          "0",
          "2"
        ],
        [
          0,
          1,
          2
        ]
      ],
      [
        "1",
        2,
        2,
        "唯一解",
        [
          "2/1",
          "2/1"
        ]
      ],
      [
        "1",
        2,
        2,
        "唯一解",
        [
          "2/1",
          "2/1"
        ]
      ],
      "是",
      "非零縮放才是可逆列操作"
    ]
  },
  "conclusion": {
    "canonical_value": "([[1, 0, 2], [0, 1, 2]], [1, 2, 2, 唯一解, [2/1, 2/1]], [1, 2, 2, 唯一解, [2/1, 2/1]], 是, 非零縮放才是可逆列操作)",
    "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": 90,
      "original_line_sha256": "ac8d01be65a2d5e18b00b7f6648975b5f39a81ac7ddea52679b9ca2c8830e2b9",
      "question_compact_sha256": "fd64a1e405d16556cae5d035059dbd59b7fb587679419d1d28b8f93787af9ca8",
      "english_row_compact_sha256": "fce0e84a60dbff1b58c27de88d83c16025f0035b54d18a6a49150d8fa6defb50",
      "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": "c6c4ce74297886859a3db0d861fb38a279b36c6a55620ef79ad30ec5eea8bdf0",
    "original_candidate_line_sha256": "c6c4ce74297886859a3db0d861fb38a279b36c6a55620ef79ad30ec5eea8bdf0"
  },
  "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": 51,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "9ad8e2ad9535bb12eaf37d91c69a51239564fc9b0aeab6a8042e21db82ad78f7",
      "arithmetic_steps": 43
    },
    "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."
  }
}