{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261010-r01-polynomial-0022",
  "pilot_number": "R0007-04222",
  "number": 4222,
  "family_id": "qf-density10-poly-add-coefficients",
  "title": {
    "zh-Hant": "第 4222 題：多項式和的完整係數",
    "en": "Question 4222: Complete Polynomial Sum Coefficients"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261010-r01-polynomial-0022",
    "batch_id": "batch-20261008-advance-20261010-r01-polynomial-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-polynomial"
    ],
    "title_zh": "多項式和的完整係數",
    "stem_zh": "升冪係數 f=[9, 22, -1, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。",
    "answer": {
      "kind": "expression",
      "value": "[10, 20, 1, 1]",
      "explanation_zh": "和的升冪係數=[10, 20, 1, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "多項式和的完整係數"
      ]
    },
    "variant": {
      "family_id": "qf-density10-poly-add-coefficients",
      "seed": 202610101022,
      "parameters": {
        "f": [
          9,
          22,
          -1,
          2
        ],
        "g": [
          1,
          -2,
          2,
          -1
        ],
        "representation": "ascending_powers",
        "operation": "density10_poly_add_coefficients",
        "domain": "integer_polynomial_slice",
        "semantic_template": "qf-density10-poly-add-coefficients/[[1,-2,2,-1],[9,22,-1,2]]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261010-r01-polynomial-0022",
    "batch_id": "batch-20261008-advance-20261010-r01-polynomial-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "fa8858a590e3fdb62ffa9f40a40dcd6389a7e8cb97732b04a93ea99c2977abc8",
      "family_id": "qf-density10-poly-add-coefficients",
      "seed": 202610101022,
      "semantic_template": "qf-density10-poly-add-coefficients/[[1,-2,2,-1],[9,22,-1,2]]"
    },
    "fields": {
      "title": "Complete Polynomial Sum Coefficients",
      "stem": "Integer polynomials have ascending coefficients f=[9, 22, -1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.",
      "answer": {
        "kind": "expression",
        "value": "[10, 20, 1, 1]",
        "explanation": "Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [10, 20, 1, 1]."
      },
      "skills": [
        "Complete Polynomial Sum Coefficients"
      ],
      "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": "升冪係數 f=[9, 22, -1, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。",
        "en": "Integer polynomials have ascending coefficients f=[9, 22, -1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "f ascending coefficients=[9, 22, -1, 2]",
      "after": "coefficient k multiplies x^k;keep all required degrees",
      "relation": "implies",
      "rule_id": "ascending_polynomial_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "係數按升冪排列，常數在索引0；完整係數運算不是由單點相等推論恆等，原始係數清單保持不變。",
        "en": "Coefficients are in ascending powers with the constant at index0.Full coefficient operations are not identities inferred from one-point agreement;retain the original coefficient list."
      },
      "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": "sum coefficient0 := sum(9, 1)",
      "after": "sum coefficient0 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合併同次冪係數，缺項以0補齊。",
        "en": "Combine coefficients of the same power,padding a missing term with0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum coefficient0，不以小數近似替代。",
          "en": "Recompute sum coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "1"
        ],
        "output": "10",
        "label": "sum coefficient0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "sum coefficient1 := sum(22, -2)",
      "after": "sum coefficient1 = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合併同次冪係數，缺項以0補齊。",
        "en": "Combine coefficients of the same power,padding a missing term with0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum coefficient1，不以小數近似替代。",
          "en": "Recompute sum coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "22",
          "-2"
        ],
        "output": "20",
        "label": "sum coefficient1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "sum coefficient2 := sum(-1, 2)",
      "after": "sum coefficient2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合併同次冪係數，缺項以0補齊。",
        "en": "Combine coefficients of the same power,padding a missing term with0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum coefficient2，不以小數近似替代。",
          "en": "Recompute sum coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "2"
        ],
        "output": "1",
        "label": "sum coefficient2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "sum coefficient3 := sum(2, -1)",
      "after": "sum coefficient3 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合併同次冪係數，缺項以0補齊。",
        "en": "Combine coefficients of the same power,padding a missing term with0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum coefficient3，不以小數近似替代。",
          "en": "Recompute sum coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "-1"
        ],
        "output": "1",
        "label": "sum coefficient3"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "[10, 20, 1, 1]",
      "after": "canonical ascending coefficient list",
      "relation": "implies",
      "rule_id": "polynomial_sum_coefficients",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐項相加保持多項式恆等；末端零可移除，零多項式仍列[0]。",
        "en": "Coefficientwise addition preserves the polynomial identity;remove trailing zeros but retain[0] for the zero polynomial."
      },
      "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": [
      "10",
      "20",
      "1",
      "1"
    ]
  },
  "conclusion": {
    "canonical_value": "[10, 20, 1, 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": {
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      "batch_sha256": "a405b95ddca6b0d773ef4f12f900f94620ea7ec880a98ae7ebae6688f5e23ad6",
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      "question_compact_sha256": "fa8858a590e3fdb62ffa9f40a40dcd6389a7e8cb97732b04a93ea99c2977abc8",
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      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261010-r01-polynomial-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261010-r01-polynomial-foundations.en.jsonl"
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    "mko_bindings": {
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        "original_path": "public/data/mko/polynomial.json",
        "english_path": "public/data/i18n/editions/mko-polynomial.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "de1fb611cf230f243ddd3334fbc53c9205cbe528b78346a10544972dede78b95",
    "original_candidate_sha256": "f1078a12d89eb9f45f6868fb884e747eee16e8a0277b9880a3683d403b7d953a",
    "original_candidate_line_sha256": "f1078a12d89eb9f45f6868fb884e747eee16e8a0277b9880a3683d403b7d953a"
  },
  "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-20261026-004-p8",
    "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": {
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      "question_first_witness_sha256": "4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
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    "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."
  }
}