{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261010-r01-polynomial-0059",
  "pilot_number": "R0007-04284",
  "number": 4284,
  "family_id": "qf-density10-poly-two-point-values",
  "title": {
    "zh-Hant": "第 4284 題：同一多項式的有序雙點值",
    "en": "Question 4284: Ordered Polynomial Values at Two Points"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261010-r01-polynomial-0059",
    "batch_id": "batch-20261008-advance-20261010-r01-polynomial-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-polynomial"
    ],
    "title_zh": "同一多項式的有序雙點值",
    "stem_zh": "升冪係數 f=[-4, 2, -1, 2]，a=-2、b=1。依序求 (f(a), f(b)) 的整數值。",
    "answer": {
      "kind": "expression",
      "value": "(-28, -1)",
      "explanation_zh": "有序雙點值=(-28, -1)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "同一多項式的有序雙點值"
      ]
    },
    "variant": {
      "family_id": "qf-density10-poly-two-point-values",
      "seed": 202610101059,
      "parameters": {
        "f": [
          -4,
          2,
          -1,
          2
        ],
        "a": -2,
        "b": 1,
        "representation": "ascending_powers",
        "operation": "density10_poly_two_point_values",
        "domain": "integer_polynomial_slice",
        "semantic_template": "qf-density10-poly-two-point-values/[[-4,2,-1,2],-2,1]"
      }
    },
    "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": "2026-10-08",
      "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-0059",
    "batch_id": "batch-20261008-advance-20261010-r01-polynomial-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8247e266f285707009fb4adecf022ea1374f1754c4eed9c6adad0e0c455382cf",
      "family_id": "qf-density10-poly-two-point-values",
      "seed": 202610101059,
      "semantic_template": "qf-density10-poly-two-point-values/[[-4,2,-1,2],-2,1]"
    },
    "fields": {
      "title": "Ordered Polynomial Values at Two Points",
      "stem": "Integer polynomials have ascending coefficients f=[-4, 2, -1, 2], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.",
      "answer": {
        "kind": "expression",
        "value": "(-28, -1)",
        "explanation": "Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-28, -1)."
      },
      "skills": [
        "Ordered Polynomial Values at Two Points"
      ],
      "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=[-4, 2, -1, 2]，a=-2、b=1。依序求 (f(a), f(b)) 的整數值。",
        "en": "Integer polynomials have ascending coefficients f=[-4, 2, -1, 2], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "f ascending coefficients=[-4, 2, -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": "[-4, 2, -1, 2]",
      "after": "Horner at a=-2",
      "relation": "implies",
      "rule_id": "polynomial_horner_evaluation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從最高非零次向常數項做Horner，不能把升冪清單直接反解為降冪。",
        "en": "Use Horner from the highest nonzero degree down to the constant;do not reinterpret ascending coefficients as descending."
      },
      "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": "a multiply before degree2 := mul(2, -2)",
      "after": "a multiply before degree2 = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一指定輸入。",
        "en": "Multiply by the same specified input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a multiply before degree2，不以小數近似替代。",
          "en": "Recompute a multiply before degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-2"
        ],
        "output": "-4",
        "label": "a multiply before degree2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "a add coefficient2 := sum(-4, -1)",
      "after": "a add coefficient2 = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的係數。",
        "en": "Then add the coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a add coefficient2，不以小數近似替代。",
          "en": "Recompute a add coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-4",
          "-1"
        ],
        "output": "-5",
        "label": "a add coefficient2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "a multiply before degree1 := mul(-5, -2)",
      "after": "a multiply before degree1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一指定輸入。",
        "en": "Multiply by the same specified input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a multiply before degree1，不以小數近似替代。",
          "en": "Recompute a multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-5",
          "-2"
        ],
        "output": "10",
        "label": "a multiply before degree1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "a add coefficient1 := sum(10, 2)",
      "after": "a add coefficient1 = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的係數。",
        "en": "Then add the coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a add coefficient1，不以小數近似替代。",
          "en": "Recompute a add coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "2"
        ],
        "output": "12",
        "label": "a add coefficient1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "a multiply before degree0 := mul(12, -2)",
      "after": "a multiply before degree0 = -24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一指定輸入。",
        "en": "Multiply by the same specified input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a multiply before degree0，不以小數近似替代。",
          "en": "Recompute a multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "-2"
        ],
        "output": "-24",
        "label": "a multiply before degree0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "a add coefficient0 := sum(-24, -4)",
      "after": "a add coefficient0 = -28",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的係數。",
        "en": "Then add the coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a add coefficient0，不以小數近似替代。",
          "en": "Recompute a add coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-24",
          "-4"
        ],
        "output": "-28",
        "label": "a add coefficient0"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "[-4, 2, -1, 2]",
      "after": "Horner at b=1",
      "relation": "implies",
      "rule_id": "polynomial_horner_evaluation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從最高非零次向常數項做Horner，不能把升冪清單直接反解為降冪。",
        "en": "Use Horner from the highest nonzero degree down to the constant;do not reinterpret ascending coefficients as descending."
      },
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "b multiply before degree2 := mul(2, 1)",
      "after": "b multiply before degree2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一指定輸入。",
        "en": "Multiply by the same specified input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 b multiply before degree2，不以小數近似替代。",
          "en": "Recompute b multiply before degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "b multiply before degree2"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "b add coefficient2 := sum(2, -1)",
      "after": "b add coefficient2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的係數。",
        "en": "Then add the coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 b add coefficient2，不以小數近似替代。",
          "en": "Recompute b add coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "-1"
        ],
        "output": "1",
        "label": "b add coefficient2"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "b multiply before degree1 := mul(1, 1)",
      "after": "b multiply before degree1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一指定輸入。",
        "en": "Multiply by the same specified input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 b multiply before degree1，不以小數近似替代。",
          "en": "Recompute b multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "b multiply before degree1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "b add coefficient1 := sum(1, 2)",
      "after": "b add coefficient1 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的係數。",
        "en": "Then add the coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 b add coefficient1，不以小數近似替代。",
          "en": "Recompute b add coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "2"
        ],
        "output": "3",
        "label": "b add coefficient1"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "b multiply before degree0 := mul(3, 1)",
      "after": "b multiply before degree0 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一指定輸入。",
        "en": "Multiply by the same specified input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 b multiply before degree0，不以小數近似替代。",
          "en": "Recompute b multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "1"
        ],
        "output": "3",
        "label": "b multiply before degree0"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "b add coefficient0 := sum(3, -4)",
      "after": "b add coefficient0 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的係數。",
        "en": "Then add the coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 b add coefficient0，不以小數近似替代。",
          "en": "Recompute b add coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "-4"
        ],
        "output": "-1",
        "label": "b add coefficient0"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "f(a)=-28;f(b)=-1",
      "after": "retain a,b order;two equal values do not make f constant",
      "relation": "implies",
      "rule_id": "two_values_not_polynomial_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩點值相等不代表函數為常數，例如x²在−1與1相等；本題只回報指定兩點值。",
        "en": "Equal values at two points do not make a polynomial constant,as x² at−1 and1 shows;report only the specified evaluations."
      },
      "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": [
      "-28",
      "-1"
    ]
  },
  "conclusion": {
    "canonical_value": "(-28, -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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    "question_binding": {
      "batch_id": "batch-20261008-advance-20261010-r01-polynomial-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261010-r01-polynomial-foundations/questions.jsonl",
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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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    "selected_part_sha256": "de1fb611cf230f243ddd3334fbc53c9205cbe528b78346a10544972dede78b95",
    "original_candidate_sha256": "4c5069d8cac8d36a5b9ee58e0cf86979de489b1a1fabdc7ffa98f8c09a1c8456",
    "original_candidate_line_sha256": "4c5069d8cac8d36a5b9ee58e0cf86979de489b1a1fabdc7ffa98f8c09a1c8456"
  },
  "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": {
      "semantic_review_sha256": "faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a",
      "question_first_witness_sha256": "4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 16,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "be99da9c82faa830b909b6ecd464f514c1d9f9e8b22fe5c664ae1955c43bc8ad",
      "arithmetic_steps": 12
    },
    "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."
  }
}