{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261023-r01-complex-modulus-conjugate-0079",
  "pilot_number": "R0019-10329",
  "number": 10329,
  "family_id": "qf-mod23-complexmod-conjugation-product",
  "title": {
    "zh-Hant": "第 10329 題：複數模與共軛",
    "en": "Question 10329: Conjugating a Product versus Multiplying the Conjugates"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261023-r01-complex-modulus-conjugate-0079",
    "batch_id": "batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-complex-modulus-conjugate"
    ],
    "title_zh": "複數模與共軛",
    "stem_zh": "複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-9,0],\"right\":[1,-3]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([-9, 0], [1, -3], [-9, 27], [-9, -27], [-9, -27], 是)",
      "explanation_zh": "現代橋梁核對=([-9, 0], [1, -3], [-9, 27], [-9, -27], [-9, -27], 是)；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；模平方不偷換成未化簡根式；共軛依明示代數運算。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "複數模與共軛"
      ]
    },
    "variant": {
      "family_id": "qf-mod23-complexmod-conjugation-product",
      "seed": 202610231079,
      "parameters": {
        "left": [
          -9,
          0
        ],
        "right": [
          1,
          -3
        ],
        "domain": "Integer_pair_complex_invariants",
        "operation": "complexmod-conjugation-product",
        "semantic_template": "qf-mod23-complexmod-conjugation-product/{\"left\":[-9,0],\"right\":[1,-3]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261023-r01-complex-modulus-conjugate-0079",
    "batch_id": "batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1309707512251387a9fae1f11b5247a3f85a7fb90fb09a660f65645cd6963503",
      "family_id": "qf-mod23-complexmod-conjugation-product",
      "seed": 202610231079,
      "semantic_template": "qf-mod23-complexmod-conjugation-product/{\"left\":[-9,0],\"right\":[1,-3]}"
    },
    "fields": {
      "title": "Conjugating a Product versus Multiplying the Conjugates",
      "stem": "Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-9, 0], right=[1, -3]. Give (left,right,product,conjugate of product,product of conjugates,equal?).",
      "answer": {
        "kind": "expression",
        "value": "([-9, 0], [1, -3], [-9, 27], [-9, -27], [-9, -27], Yes)",
        "explanation": "Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 0], [1, -3], [-9, 27], [-9, -27], [-9, -27], Yes)."
      },
      "skills": [
        "Conjugating a Product versus Multiplying the Conjugates"
      ],
      "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": "複數以整數有序對表示。操作=complexmod-conjugation-product；資料={\"left\":[-9,0],\"right\":[1,-3]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。",
        "en": "Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-9, 0], right=[1, -3]. Give (left,right,product,conjugate of product,product of conjugates,equal?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared complex ordered pair=[-9, 0]",
      "after": "exact integer coefficients in the algebra with i^2=-1",
      "relation": "implies",
      "rule_id": "exact_complex_pair_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只依明示複數代數運算，不當成實向量逐項乘法；零複數也合法，未要求非零。",
        "en": "Use the declared complex algebra,not componentwise real-vector multiplication.The zero complex number is allowed."
      },
      "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": "original product real-real := mul(-9, 1)",
      "after": "original product real-real = -9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original product real-real，不以小數近似替代。",
          "en": "Recompute original product real-real with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-9",
          "1"
        ],
        "output": "-9",
        "label": "original product real-real"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "original product imaginary-imaginary := mul(0, -3)",
      "after": "original product imaginary-imaginary = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original product imaginary-imaginary，不以小數近似替代。",
          "en": "Recompute original product imaginary-imaginary with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-3"
        ],
        "output": "0",
        "label": "original product imaginary-imaginary"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "original product real coefficient := sub(-9, 0)",
      "after": "original product real coefficient = -9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original product real coefficient，不以小數近似替代。",
          "en": "Recompute original product real coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-9",
          "0"
        ],
        "output": "-9",
        "label": "original product real coefficient"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "original product real-imaginary := mul(-9, -3)",
      "after": "original product real-imaginary = 27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original product real-imaginary，不以小數近似替代。",
          "en": "Recompute original product real-imaginary with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-9",
          "-3"
        ],
        "output": "27",
        "label": "original product real-imaginary"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "original product imaginary-real := mul(0, 1)",
      "after": "original product imaginary-real = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original product imaginary-real，不以小數近似替代。",
          "en": "Recompute original product imaginary-real with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "original product imaginary-real"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "original product imaginary coefficient := sum(27, 0)",
      "after": "original product imaginary coefficient = 27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original product imaginary coefficient，不以小數近似替代。",
          "en": "Recompute original product imaginary coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "27",
          "0"
        ],
        "output": "27",
        "label": "original product imaginary coefficient"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "original product:left=[-9, 0];right=[1, -3];i^2=-1",
      "after": "complex product=[-9, 27]",
      "relation": "implies",
      "rule_id": "complex_product_ordered_component_expansion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "展開(a+bi)(c+di)=(ac−bd)+(ad+bc)i，四個乘積各保留其來源位置與符號。",
        "en": "Expand(a+bi)(c+di)=(ac−bd)+(ad+bc)i,retaining each of the four products' source positions and signs."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "original product negate imaginary coefficient := sub(0, 27)",
      "after": "original product negate imaginary coefficient = -27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original product negate imaginary coefficient，不以小數近似替代。",
          "en": "Recompute original product negate imaginary coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "27"
        ],
        "output": "-27",
        "label": "original product negate imaginary coefficient"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "original product:original ordered pair=[-9, 27]",
      "after": "conjugate pair=[-9, -27]",
      "relation": "implies",
      "rule_id": "complex_conjugate_real_fixed_imaginary_negated",
      "conditions": [],
      "explanation": {
        "zh-Hant": "(a,b)表示a+bi；共軛保留實部，僅把虛係數b改成−b，不交換座標。",
        "en": "The pair(a,b) denotes a+bi.Conjugation fixes the real part and negates only b;it does not exchange coordinates."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "left value negate imaginary coefficient := sub(0, 0)",
      "after": "left value negate imaginary coefficient = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left value negate imaginary coefficient，不以小數近似替代。",
          "en": "Recompute left value negate imaginary coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "left value negate imaginary coefficient"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "left value:original ordered pair=[-9, 0]",
      "after": "conjugate pair=[-9, 0]",
      "relation": "implies",
      "rule_id": "complex_conjugate_real_fixed_imaginary_negated",
      "conditions": [],
      "explanation": {
        "zh-Hant": "(a,b)表示a+bi；共軛保留實部，僅把虛係數b改成−b，不交換座標。",
        "en": "The pair(a,b) denotes a+bi.Conjugation fixes the real part and negates only b;it does not exchange coordinates."
      },
      "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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "right value negate imaginary coefficient := sub(0, -3)",
      "after": "right value negate imaginary coefficient = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right value negate imaginary coefficient，不以小數近似替代。",
          "en": "Recompute right value negate imaginary coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "-3"
        ],
        "output": "3",
        "label": "right value negate imaginary coefficient"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "right value:original ordered pair=[1, -3]",
      "after": "conjugate pair=[1, 3]",
      "relation": "implies",
      "rule_id": "complex_conjugate_real_fixed_imaginary_negated",
      "conditions": [],
      "explanation": {
        "zh-Hant": "(a,b)表示a+bi；共軛保留實部，僅把虛係數b改成−b，不交換座標。",
        "en": "The pair(a,b) denotes a+bi.Conjugation fixes the real part and negates only b;it does not exchange coordinates."
      },
      "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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "product of conjugates real-real := mul(-9, 1)",
      "after": "product of conjugates real-real = -9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 product of conjugates real-real，不以小數近似替代。",
          "en": "Recompute product of conjugates real-real with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-9",
          "1"
        ],
        "output": "-9",
        "label": "product of conjugates real-real"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "product of conjugates imaginary-imaginary := mul(0, 3)",
      "after": "product of conjugates imaginary-imaginary = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 product of conjugates imaginary-imaginary，不以小數近似替代。",
          "en": "Recompute product of conjugates imaginary-imaginary with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "3"
        ],
        "output": "0",
        "label": "product of conjugates imaginary-imaginary"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "product of conjugates real coefficient := sub(-9, 0)",
      "after": "product of conjugates real coefficient = -9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 product of conjugates real coefficient，不以小數近似替代。",
          "en": "Recompute product of conjugates real coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-9",
          "0"
        ],
        "output": "-9",
        "label": "product of conjugates real coefficient"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "product of conjugates real-imaginary := mul(-9, 3)",
      "after": "product of conjugates real-imaginary = -27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 product of conjugates real-imaginary，不以小數近似替代。",
          "en": "Recompute product of conjugates real-imaginary with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-9",
          "3"
        ],
        "output": "-27",
        "label": "product of conjugates real-imaginary"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "product of conjugates imaginary-real := mul(0, 1)",
      "after": "product of conjugates imaginary-real = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 product of conjugates imaginary-real，不以小數近似替代。",
          "en": "Recompute product of conjugates imaginary-real with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "product of conjugates imaginary-real"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "product of conjugates imaginary coefficient := sum(-27, 0)",
      "after": "product of conjugates imaginary coefficient = -27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 product of conjugates imaginary coefficient，不以小數近似替代。",
          "en": "Recompute product of conjugates imaginary coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-27",
          "0"
        ],
        "output": "-27",
        "label": "product of conjugates imaginary coefficient"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "product of conjugates:left=[-9, 0];right=[1, 3];i^2=-1",
      "after": "complex product=[-9, -27]",
      "relation": "implies",
      "rule_id": "complex_product_ordered_component_expansion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "展開(a+bi)(c+di)=(ac−bd)+(ad+bc)i，四個乘積各保留其來源位置與符號。",
        "en": "Expand(a+bi)(c+di)=(ac−bd)+(ad+bc)i,retaining each of the four products' source positions and signs."
      },
      "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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "conjugate of computed product=[-9, -27];computed product of conjugates=[-9, -27]",
      "after": "equal=是",
      "relation": "implies",
      "rule_id": "conjugation_product_two_routes",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩路各自完整展開，保留原left/right次序與每個中間對；這些精確代值不宣稱新增一般形式證據。",
        "en": "Expand both routes completely,retaining original left/right roles and every intermediate pair.These exact instances are not new general formal Evidence."
      },
      "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": [
      [
        -9,
        0
      ],
      [
        1,
        -3
      ],
      [
        -9,
        27
      ],
      [
        -9,
        -27
      ],
      [
        -9,
        -27
      ],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([-9, 0], [1, -3], [-9, 27], [-9, -27], [-9, -27], 是)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations/questions.jsonl",
      "batch_sha256": "47ceca5a202830b45dbe720a14de817fd991d8fa3c57819dddfc0697eca0414e",
      "original_line": 79,
      "original_line_sha256": "3ecd53f8a1e4b1710fc54a1a64102868f4f9b274b9adab88e90671d555f02ac5",
      "question_compact_sha256": "1309707512251387a9fae1f11b5247a3f85a7fb90fb09a660f65645cd6963503",
      "english_row_compact_sha256": "d392b779b1fd0764ed12580acdd94aa093ef31c8e742a7776cec72c57cf6010c",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-complex-modulus-conjugate": {
        "sha256": "c4c1f275b4ae1708bd308a1ea5ce57ee90a10921251139a1eb880ffd2101842d",
        "english_sha256": "4d553f442775fa75aa08d2cdf7a109593034fd6b47602575bbec0612a1970cb4",
        "original_path": "public/data/mko/complex-modulus-conjugate.json",
        "english_path": "public/data/i18n/editions/mko-complex-modulus-conjugate.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "7359cb2346bdb6bf38b7ca4a5fd5b5a99c3007a711f34346754d57b77b2dfcc8",
    "original_candidate_sha256": "ec8c8771fa390b4da34944ce914b256ff7a46a9d078586a770b143c0cacba895",
    "original_candidate_line_sha256": "ec8c8771fa390b4da34944ce914b256ff7a46a9d078586a770b143c0cacba895"
  },
  "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-p20",
    "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": "eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5",
      "question_first_witness_sha256": "3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419"
    },
    "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": "89babf386b33dfef14125b147885ec530cfea2b4efa29d61e2927f86d1ec0b2b",
      "arithmetic_steps": 15
    },
    "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."
  }
}