{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-r03-factorization-0012",
  "pilot_number": "R0019-10062",
  "number": 10062,
  "family_id": "qf-factorization-product",
  "title": {
    "zh-Hant": "第 10062 題：因子乘積的完整係數",
    "en": "Question 10062: Complete Coefficients of a Factor Product"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-r03-factorization-0012",
    "batch_id": "batch-20261003-r03-factorization-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-factorization"
    ],
    "title_zh": "因子乘積的完整係數",
    "stem_zh": "在 Q[x] 中，係數向量由低次至高次排列。因子 [-1, 1] 與 [13, 1] 相乘後，規範係數向量為何？",
    "answer": {
      "kind": "expression",
      "value": "[-13, 12, 1]",
      "explanation_zh": "逐項相乘並按指數和合併係數，乘積為 [-13, 12, 1]。比較的是完整係數，不是有限代入值。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "因子乘積的完整係數"
      ]
    },
    "variant": {
      "family_id": "qf-factorization-product",
      "seed": 202610033012,
      "parameters": {
        "left": [
          -1,
          1
        ],
        "right": [
          13,
          1
        ],
        "operation": "product",
        "semantic_template": "qf-factorization-product/{\"left\":[\"-1\",\"1\"],\"operation\":\"product\",\"right\":[\"13\",\"1\"]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication"
      ],
      "warning_zh": "僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-03",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261003-r03-factorization-0012",
    "batch_id": "batch-20261003-r03-factorization-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8bc952999e08e31f1eda744794bc92dd1705d314094ecc435c2b4b15ff2cb1df",
      "family_id": "qf-factorization-product",
      "seed": 202610033012,
      "semantic_template": "qf-factorization-product/{\"left\":[\"-1\",\"1\"],\"operation\":\"product\",\"right\":[\"13\",\"1\"]}"
    },
    "fields": {
      "title": "Complete Coefficients of a Factor Product",
      "stem": "In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-1, 1] and [13, 1] and return the normalized complete coefficient vector.",
      "answer": {
        "kind": "expression",
        "value": "[-13, 12, 1]",
        "explanation": "Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-13, 12, 1]; the identity is determined by all coefficients rather than sampled values."
      },
      "skills": [
        "Complete Coefficients of a Factor Product"
      ],
      "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": "在 Q[x] 中，係數向量由低次至高次排列。因子 [-1, 1] 與 [13, 1] 相乘後，規範係數向量為何？",
        "en": "In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-1, 1] and [13, 1] and return the normalized complete coefficient vector."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "Q[x];ascending coefficients;input vectors may have trailing zeros",
      "after": "equality uses every normalized coefficient,not numerical sampling",
      "relation": "implies",
      "rule_id": "factorization_coefficient_identity_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原因子順序與原向量，規範只刪高次尾零；不要依semantic身份排序交換原輸入，也不以單點相等代替多項式相等。",
        "en": "Preserve the original factor order and vectors.Normalization removes only trailing high-power zeros;do not reorder source factors by semantic identity or infer polynomial equality from one point."
      },
      "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": "raw left=[-1, 1];raw right=[13, 1]",
      "after": "normalized left=[-1, 1];normalized right=[13, 1]",
      "relation": "implies",
      "rule_id": "factor_vectors_trailing_zero_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只刪末端零，不刪中間零，也不交換兩個向量；空向量與全零向量都代表零。",
        "en": "Remove trailing zeros only,not internal zeros or the original vector order.Empty and all-zero vectors both represent zero."
      },
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "power k=0;left=[-1, 1];right=[13, 1]",
      "after": "all index pairs i+j=k=[[0, 0]]",
      "relation": "implies",
      "rule_id": "product_coefficient_complete_index_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪向量的索引就是次數；只收i+j=k且兩索引各自合法的全部項，不把乘法當逐項相乘。",
        "en": "Ascending-vector indices are powers.Collect every valid pair with i+j=k;polynomial multiplication is not entrywise multiplication."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "coefficient0 term0,0 := mul(-1, 13)",
      "after": "coefficient0 term0,0 = -13",
      "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 coefficient0 term0,0，不以小數近似替代。",
          "en": "Recompute coefficient0 term0,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "13"
        ],
        "output": "-13",
        "label": "coefficient0 term0,0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "coefficient0 complete sum := sum(-13)",
      "after": "coefficient0 complete sum = -13",
      "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 coefficient0 complete sum，不以小數近似替代。",
          "en": "Recompute coefficient0 complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-13"
        ],
        "output": "-13",
        "label": "coefficient0 complete sum"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "power k=1;left=[-1, 1];right=[13, 1]",
      "after": "all index pairs i+j=k=[[0, 1], [1, 0]]",
      "relation": "implies",
      "rule_id": "product_coefficient_complete_index_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪向量的索引就是次數；只收i+j=k且兩索引各自合法的全部項，不把乘法當逐項相乘。",
        "en": "Ascending-vector indices are powers.Collect every valid pair with i+j=k;polynomial multiplication is not entrywise multiplication."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "coefficient1 term0,1 := mul(-1, 1)",
      "after": "coefficient1 term0,1 = -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 coefficient1 term0,1，不以小數近似替代。",
          "en": "Recompute coefficient1 term0,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "coefficient1 term0,1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "coefficient1 term1,0 := mul(1, 13)",
      "after": "coefficient1 term1,0 = 13",
      "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 coefficient1 term1,0，不以小數近似替代。",
          "en": "Recompute coefficient1 term1,0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "13"
        ],
        "output": "13",
        "label": "coefficient1 term1,0"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "coefficient1 complete sum := sum(-1, 13)",
      "after": "coefficient1 complete sum = 12",
      "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 coefficient1 complete sum，不以小數近似替代。",
          "en": "Recompute coefficient1 complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "13"
        ],
        "output": "12",
        "label": "coefficient1 complete sum"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "power k=2;left=[-1, 1];right=[13, 1]",
      "after": "all index pairs i+j=k=[[1, 1]]",
      "relation": "implies",
      "rule_id": "product_coefficient_complete_index_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪向量的索引就是次數；只收i+j=k且兩索引各自合法的全部項，不把乘法當逐項相乘。",
        "en": "Ascending-vector indices are powers.Collect every valid pair with i+j=k;polynomial multiplication is not entrywise multiplication."
      },
      "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": "coefficient2 term1,1 := mul(1, 1)",
      "after": "coefficient2 term1,1 = 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 coefficient2 term1,1，不以小數近似替代。",
          "en": "Recompute coefficient2 term1,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "coefficient2 term1,1"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "coefficient2 complete sum := sum(1)",
      "after": "coefficient2 complete sum = 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 coefficient2 complete sum，不以小數近似替代。",
          "en": "Recompute coefficient2 complete sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "coefficient2 complete sum"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "all collected product coefficients=[-13, 12, 1]",
      "after": "normalized complete product=[-13, 12, 1]",
      "relation": "implies",
      "rule_id": "complete_product_not_sampled_values",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全部有限係數已求出；去掉高次尾零，零多項式輸出[0]。判的是係數恆等，不是幾個點的數值吻合。",
        "en": "All finitely many coefficients were obtained.Remove trailing high-power zeros and display zero as[0].This is coefficient identity,not agreement at a few sample points."
      },
      "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": [
      "-13",
      "12",
      "1"
    ]
  },
  "conclusion": {
    "canonical_value": "[-13, 12, 1]",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261003-r03-factorization-foundations",
      "batch_path": "public/data/questions/batches/batch-20261003-r03-factorization-foundations/questions.jsonl",
      "batch_sha256": "219ab610272e69af0c134f62a7796121c871cf30cb8a070384b366251b557e3f",
      "original_line": 12,
      "original_line_sha256": "461d731f1db43d2ed7acca158c0c87120fe04266b3ed1a706207c9a346393791",
      "question_compact_sha256": "8bc952999e08e31f1eda744794bc92dd1705d314094ecc435c2b4b15ff2cb1df",
      "english_row_compact_sha256": "1380cfc89adc4f2e9d4c07b1c3735e324e86f2a4bbf3383cec35dbc1fc3d6c76",
      "english_manifest_path": "public/data/i18n/editions/batch-20261003-r03-factorization-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261003-r03-factorization-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-factorization": {
        "sha256": "375bd5e9c171dd33297e9e1954a6c9c2b48d97cebbbc4dd93b44c36108f04015",
        "english_sha256": "a24152abff47aa4af453747775e20a1c3ccccf16231b2c4d63aa1d68f06febf7",
        "original_path": "public/data/mko/factorization.json",
        "english_path": "public/data/i18n/editions/mko-factorization.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "7359cb2346bdb6bf38b7ca4a5fd5b5a99c3007a711f34346754d57b77b2dfcc8",
    "original_candidate_sha256": "9804f0c1773a139bb0f957cee9b723566e10d5f04a9b927305ad83dd51506423",
    "original_candidate_line_sha256": "9804f0c1773a139bb0f957cee9b723566e10d5f04a9b927305ad83dd51506423"
  },
  "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": 13,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "cfe008a9dab1a9c35c646b2318a6911efdd69a7974b4eaa62b8bef4f34f68d70",
      "arithmetic_steps": 7
    },
    "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."
  }
}