{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-r02-polynomial-0024",
  "pilot_number": "R0067-34299",
  "number": 34299,
  "family_id": "qf-polynomial-exact-evaluation",
  "title": {
    "zh-Hant": "第 34299 題：多項式精確代入",
    "en": "Question 34299: Exact Polynomial Evaluation"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-r02-polynomial-0024",
    "batch_id": "batch-20261003-r02-polynomial-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-polynomial"
    ],
    "title_zh": "多項式精確代入",
    "stem_zh": "本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [11, -1, 1]，求 p(11) 的精確值。",
    "answer": {
      "kind": "numeric",
      "value": "121",
      "explanation_zh": "p(11)=121。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "多項式精確代入"
      ]
    },
    "variant": {
      "family_id": "qf-polynomial-exact-evaluation",
      "seed": 202610032024,
      "parameters": {
        "coefficients": [
          11,
          -1,
          1
        ],
        "x": 11,
        "operation": "evaluate",
        "semantic_template": "qf-polynomial-exact-evaluation/{\"coefficients\":[\"11\",\"-1\",\"1\"],\"operation\":\"evaluate\",\"x\":11}"
      }
    },
    "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-r02-polynomial-0024",
    "batch_id": "batch-20261003-r02-polynomial-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2630b5b71e8f8bcbad648ee2bc65c277cb7d156c53c80d843c0680c029138439",
      "family_id": "qf-polynomial-exact-evaluation",
      "seed": 202610032024,
      "semantic_template": "qf-polynomial-exact-evaluation/{\"coefficients\":[\"11\",\"-1\",\"1\"],\"operation\":\"evaluate\",\"x\":11}"
    },
    "fields": {
      "title": "Exact Polynomial Evaluation",
      "stem": "In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [11, -1, 1]. Compute p(11) exactly.",
      "answer": {
        "kind": "numeric",
        "value": "121",
        "explanation": "Multiply each coefficient by the corresponding natural-number power and add: p(11)=121. This is evaluation at one input and does not establish a general polynomial identity."
      },
      "skills": [
        "Exact Polynomial Evaluation"
      ],
      "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": "本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [11, -1, 1]，求 p(11) 的精確值。",
        "en": "In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [11, -1, 1]. Compute p(11) exactly."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-polynomial-exact-evaluation, [coefficients, operation, x]]",
      "after": "derived = [evaluate, no-invented-domain]",
      "relation": "implies",
      "rule_id": "R67_SOURCE_specific_keys_operation_domain_and_numeric_bank",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本bank核keys與operation/domain：legacy為各表underscore，MOD50用local hyphen；BASE參數較寬不加未知欄限制。Number.isInteger與SAFE兩種來源界分開，原JSON數字表示仍完整保存。",
        "en": "Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。",
          "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [polynomial ORIGINAL, [11, -1, 1]]",
      "after": "derived = [[11, -1, 1], [11, -1, 1]]",
      "relation": "implies",
      "rule_id": "Polynomial_RAW_ALL_SAFE_coefficients_trim_ONLY_trailing_zero",
      "conditions": [],
      "explanation": {
        "zh-Hant": "Q[x]的係數按低次到高次，raw長0..32、每格先驗SAFE，包括尾端0；只刪trailing zeros，不刪內部0。零多項式內部空向量，顯示[0]，不可改原source向量。",
        "en": "In Q[x],coefficients run from low to high degree.Type EVERY RAW SAFE cell of a0..32-entry list,including trailing zeros.Trim ONLY trailing zeros,not internal zeros.The zero polynomial has empty internal support but displays[0];keep the ORIGINAL source vector unchanged."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。",
          "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Horner ORIGINAL prefix times x at degree2 := mul(0, 11)",
      "after": "Horner ORIGINAL prefix times x at degree2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。",
        "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Horner ORIGINAL prefix times x at degree2，不以小數近似替代。",
          "en": "Recompute Horner ORIGINAL prefix times x at degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "11"
        ],
        "output": "0",
        "label": "Horner ORIGINAL prefix times x at degree2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Horner ORIGINAL add coefficient at degree2 := sum(0, 1)",
      "after": "Horner ORIGINAL add coefficient at degree2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。",
        "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Horner ORIGINAL add coefficient at degree2，不以小數近似替代。",
          "en": "Recompute Horner ORIGINAL add coefficient at degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "Horner ORIGINAL add coefficient at degree2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Horner ORIGINAL prefix times x at degree1 := mul(1, 11)",
      "after": "Horner ORIGINAL prefix times x at degree1 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。",
        "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Horner ORIGINAL prefix times x at degree1，不以小數近似替代。",
          "en": "Recompute Horner ORIGINAL prefix times x at degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "11"
        ],
        "output": "11",
        "label": "Horner ORIGINAL prefix times x at degree1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Horner ORIGINAL add coefficient at degree1 := sum(11, -1)",
      "after": "Horner ORIGINAL add coefficient at degree1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。",
        "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Horner ORIGINAL add coefficient at degree1，不以小數近似替代。",
          "en": "Recompute Horner ORIGINAL add coefficient at degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "11",
          "-1"
        ],
        "output": "10",
        "label": "Horner ORIGINAL add coefficient at degree1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Horner ORIGINAL prefix times x at degree0 := mul(10, 11)",
      "after": "Horner ORIGINAL prefix times x at degree0 = 110",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。",
        "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Horner ORIGINAL prefix times x at degree0，不以小數近似替代。",
          "en": "Recompute Horner ORIGINAL prefix times x at degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "11"
        ],
        "output": "110",
        "label": "Horner ORIGINAL prefix times x at degree0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Horner ORIGINAL add coefficient at degree0 := sum(110, 11)",
      "after": "Horner ORIGINAL add coefficient at degree0 = 121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原操作數與原次序精確計算；legacy衍生整數不重套raw SAFE界，MOD50有理數則另保naive pre-reduction來源界。",
        "en": "Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Horner ORIGINAL add coefficient at degree0，不以小數近似替代。",
          "en": "Recompute Horner ORIGINAL add coefficient at degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "110",
          "11"
        ],
        "output": "121",
        "label": "Horner ORIGINAL add coefficient at degree0"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "source roles = [[11, -1, 1], 11]",
      "after": "derived = [121]",
      "relation": "implies",
      "rule_id": "Polynomial_HORNER_complete_exact_evaluation_NOT_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由最高冪往下反覆乘x再加本係數，涵蓋全部係數；zero polynomial空loop值0。這只求本x的值，不以兩式單點相等認證恆等式，也不限制衍生整數大小。",
        "en": "Starting at the highest power,repeatedly multiply by x and add THIS coefficient,covering ALL coefficients.The zero polynomial’s empty loop gives0.This evaluates THIS input only;one-point agreement is not a polynomial identity.Derived integers remain unbounded."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。",
          "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "121"
  },
  "conclusion": {
    "canonical_value": "121",
    "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-r02-polynomial-foundations",
      "batch_path": "public/data/questions/batches/batch-20261003-r02-polynomial-foundations/questions.jsonl",
      "batch_sha256": "2294545062c111a50b54219e05944a8e143ded85eb92974a7ac019620d363675",
      "original_line": 24,
      "original_line_sha256": "ab76ab89f38502a50e6422145e9b55dbcecdd6061f75c4609bdaea4ad60273a9",
      "question_compact_sha256": "2630b5b71e8f8bcbad648ee2bc65c277cb7d156c53c80d843c0680c029138439",
      "english_row_compact_sha256": "e0fed83a02256436c7fc79c819137e5e181d47a5c9cda4c43c75bf8916410199",
      "english_manifest_path": "public/data/i18n/editions/batch-20261003-r02-polynomial-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261003-r02-polynomial-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-polynomial": {
        "sha256": "0a0d76b12363d10d9ecbe8b45d7bebb1136637db1c41e6308481c675c8db9e6c",
        "english_sha256": "4c5e7d9539a5612d401321e99d2e0f4667c558a65abb5f412bf6cb3518f9fbc8",
        "original_path": "public/data/mko/polynomial.json",
        "english_path": "public/data/i18n/editions/mko-polynomial.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6bb156cec7bdb6baf2497762702fc01f94fd1db55a5926bc575b12ff9237a4de",
    "original_candidate_sha256": "6453a063334d2718d929a59ad76d21429e35f06f0fd0205e63452143bb65d5b5",
    "original_candidate_line_sha256": "6453a063334d2718d929a59ad76d21429e35f06f0fd0205e63452143bb65d5b5"
  },
  "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-20261026-012-p68",
    "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": "c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c",
      "question_first_witness_sha256": "12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "384ad339670ba8b3789c02f6ea45c128a06377a8aacdf009a2d2570b574950b9",
      "arithmetic_steps": 6
    },
    "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."
  }
}