{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261023-r01-complex-modulus-conjugate-0030",
  "pilot_number": "R0019-10355",
  "number": 10355,
  "family_id": "qf-mod23-complexmod-modulus-square",
  "title": {
    "zh-Hant": "第 10355 題：複數模與共軛",
    "en": "Question 10355: Exact Modulus Squared without an Unrequested Root"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261023-r01-complex-modulus-conjugate-0030",
    "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-modulus-square；資料={\"value\":[-8,1]}。請依 [value,modulus_square] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([-8, 1], 65)",
      "explanation_zh": "現代橋梁核對=([-8, 1], 65)；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-modulus-square",
      "seed": 202610231030,
      "parameters": {
        "value": [
          -8,
          1
        ],
        "domain": "Integer_pair_complex_invariants",
        "operation": "complexmod-modulus-square",
        "semantic_template": "qf-mod23-complexmod-modulus-square/{\"value\":[-8,1]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261023-r01-complex-modulus-conjugate-0030",
    "batch_id": "batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
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      "family_id": "qf-mod23-complexmod-modulus-square",
      "seed": 202610231030,
      "semantic_template": "qf-mod23-complexmod-modulus-square/{\"value\":[-8,1]}"
    },
    "fields": {
      "title": "Exact Modulus Squared without an Unrequested Root",
      "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. value=[-8, 1]. Give (value,modulus squared).",
      "answer": {
        "kind": "expression",
        "value": "([-8, 1], 65)",
        "explanation": "Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 1], 65)."
      },
      "skills": [
        "Exact Modulus Squared without an Unrequested Root"
      ],
      "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-modulus-square；資料={\"value\":[-8,1]}。請依 [value,modulus_square] 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. value=[-8, 1]. Give (value,modulus squared)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared complex ordered pair=[-8, 1]",
      "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": "real coefficient squared := mul(-8, -8)",
      "after": "real coefficient squared = 64",
      "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 real coefficient squared，不以小數近似替代。",
          "en": "Recompute real coefficient squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-8",
          "-8"
        ],
        "output": "64",
        "label": "real coefficient squared"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "imaginary coefficient squared := mul(1, 1)",
      "after": "imaginary coefficient squared = 1",
      "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 imaginary coefficient squared，不以小數近似替代。",
          "en": "Recompute imaginary coefficient squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "imaginary coefficient squared"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "exact squared modulus := sum(64, 1)",
      "after": "exact squared modulus = 65",
      "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 exact squared modulus，不以小數近似替代。",
          "en": "Recompute exact squared modulus with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "64",
          "1"
        ],
        "output": "65",
        "label": "exact squared modulus"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "value=[-8, 1];real square=64;imaginary square=1",
      "after": "modulus squared=65",
      "relation": "implies",
      "rule_id": "modulus_squared_not_unrequested_square_root",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本題要|z|²=a²+b²，不是|z|；不擅自改成根式、近似值或負平方根。",
        "en": "The requested value is|z|²=a²+b²,not|z|.Do not substitute a radical,approximation or negative square root."
      },
      "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": [
      [
        -8,
        1
      ],
      65
    ]
  },
  "conclusion": {
    "canonical_value": "([-8, 1], 65)",
    "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."
  },
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      "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."
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}