{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261020-r02-ordinary-differential-equation-0046",
  "pilot_number": "R0002-01921",
  "number": 1921,
  "family_id": "qf-app20-ode-first-order-residual",
  "title": {
    "zh-Hant": "第 1921 題：多項式 ODE 候選核對",
    "en": "Question 1921: One-Point Residual for a First-Order Relation"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261020-r02-ordinary-differential-equation-0046",
    "batch_id": "batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-ordinary-differential-equation"
    ],
    "title_zh": "多項式 ODE 候選核對",
    "stem_zh": "整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[8,1,2],\"p\":3,\"q\":20,\"x\":3}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([8, 1, 2], 3, 29, 13, 3, 20, -94, 否)",
      "explanation_zh": "應用基礎核對=([8, 1, 2], 3, 29, 13, 3, 20, -94, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；殘差核對不證解存在唯一；只核對明示點或初值。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "多項式 ODE 候選核對"
      ]
    },
    "variant": {
      "family_id": "qf-app20-ode-first-order-residual",
      "seed": 202610202046,
      "parameters": {
        "coefficients": [
          8,
          1,
          2
        ],
        "p": 3,
        "q": 20,
        "x": 3,
        "domain": "Polynomial_candidate_ODE",
        "operation": "ode-first-order-residual",
        "semantic_template": "qf-app20-ode-first-order-residual/{\"coefficients\":[8,1,2],\"p\":3,\"q\":20,\"x\":3}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、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-20261020-r02-ordinary-differential-equation-0046",
    "batch_id": "batch-20261013-advance-20261020-r02-ordinary-differential-equation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1e234b806e4ac12bda982d63d56b170db381b51c4c56940ef7a083db41066b05",
      "family_id": "qf-app20-ode-first-order-residual",
      "seed": 202610202046,
      "semantic_template": "qf-app20-ode-first-order-residual/{\"coefficients\":[8,1,2],\"p\":3,\"q\":20,\"x\":3}"
    },
    "fields": {
      "title": "One-Point Residual for a First-Order Relation",
      "stem": "The integer-coefficient polynomial candidate has ascending-power coefficients=[8, 1, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=3, check y'=p*y+q with p=3, q=20. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?).",
      "answer": {
        "kind": "expression",
        "value": "([8, 1, 2], 3, 29, 13, 3, 20, -94, No)",
        "explanation": "Differentiate and substitute exactly at this point. A zero point residual does not prove the candidate solves the ODE on an interval. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([8, 1, 2], 3, 29, 13, 3, 20, -94, No)."
      },
      "skills": [
        "One-Point Residual for a First-Order Relation"
      ],
      "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": "整係數多項式候選。操作=ode-first-order-residual；資料={\"coefficients\":[8,1,2],\"p\":3,\"q\":20,\"x\":3}。請依 [coefficients,x,y,y_prime,p,q,residual,is_solution_at_x] 欄序回答。",
        "en": "The integer-coefficient polynomial candidate has ascending-power coefficients=[8, 1, 2]. Exact checks concern only the specified point or initial conditions, not existence, uniqueness or a solution on an entire interval. At the specified point x=3, check y'=p*y+q with p=3, q=20. Give (coefficients, x, y(x), y'(x), p, q, y'−p*y−q, zero residual?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "candidate y ascending coefficients=[8, 1, 2]",
      "after": "audit point x=3",
      "relation": "implies",
      "rule_id": "polynomial_candidate_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原係數清單完整保留，包括零最高次項；只檢查題目指定的多項式候選與位置。",
        "en": "Retain the original coefficient list,including top zero terms,and audit only the specified polynomial candidate and 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": "y′ coefficient at degree0 := mul(1, 1)",
      "after": "y′ coefficient at degree0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用原次數乘係數，次數再降1；常數項微分為0。",
        "en": "Multiply by the original degree and lower it by1;the constant term differentiates to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′ coefficient at degree0，不以小數近似替代。",
          "en": "Recompute y′ coefficient at degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "y′ coefficient at degree0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "y′ coefficient at degree1 := mul(2, 2)",
      "after": "y′ coefficient at degree1 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用原次數乘係數，次數再降1；常數項微分為0。",
        "en": "Multiply by the original degree and lower it by1;the constant term differentiates to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′ coefficient at degree1，不以小數近似替代。",
          "en": "Recompute y′ coefficient at degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "y′ coefficient at degree1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "y ascending coefficients=[8, 1, 2]",
      "after": "evaluate at x=3",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original 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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "y Horner multiply before degree1 := mul(2, 3)",
      "after": "y Horner multiply before degree1 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute y Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3"
        ],
        "output": "6",
        "label": "y Horner multiply before degree1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "y Horner add degree1 := sum(6, 1)",
      "after": "y Horner add degree1 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y Horner add degree1，不以小數近似替代。",
          "en": "Recompute y Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "1"
        ],
        "output": "7",
        "label": "y Horner add degree1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "y Horner multiply before degree0 := mul(7, 3)",
      "after": "y Horner multiply before degree0 = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute y Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "3"
        ],
        "output": "21",
        "label": "y Horner multiply before degree0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "y Horner add degree0 := sum(21, 8)",
      "after": "y Horner add degree0 = 29",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y Horner add degree0，不以小數近似替代。",
          "en": "Recompute y Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "8"
        ],
        "output": "29",
        "label": "y Horner add degree0"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "y′ ascending coefficients=[1, 4]",
      "after": "evaluate at x=3",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original 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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "y′ Horner multiply before degree0 := mul(4, 3)",
      "after": "y′ Horner multiply before degree0 = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′ Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute y′ Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "3"
        ],
        "output": "12",
        "label": "y′ Horner multiply before degree0"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "y′ Horner add degree0 := sum(12, 1)",
      "after": "y′ Horner add degree0 = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′ Horner add degree0，不以小數近似替代。",
          "en": "Recompute y′ Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "1"
        ],
        "output": "13",
        "label": "y′ Horner add degree0"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "p·y := mul(3, 29)",
      "after": "p·y = 87",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先計算右邊的一次倍數項。",
        "en": "Compute the scaled y term on the right first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 p·y，不以小數近似替代。",
          "en": "Recompute p·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "29"
        ],
        "output": "87",
        "label": "p·y"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "y′−p·y := sub(13, 87)",
      "after": "y′−p·y = -74",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "殘差採左減右的固定方向。",
        "en": "Use the fixed left-minus-right residual convention."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′−p·y，不以小數近似替代。",
          "en": "Recompute y′−p·y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "13",
          "87"
        ],
        "output": "-74",
        "label": "y′−p·y"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "y′−p·y−q := sub(-74, 20)",
      "after": "y′−p·y−q = -94",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再減掉常數右側項q。",
        "en": "Then subtract the constant right-side term q."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 y′−p·y−q，不以小數近似替代。",
          "en": "Recompute y′−p·y−q with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-74",
          "20"
        ],
        "output": "-94",
        "label": "y′−p·y−q"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "residual at x=3 is -94",
      "after": "equation holds at this point: 否",
      "relation": "implies",
      "rule_id": "point_residual_verdict",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只證在這一點是否成立；一點殘差為0不等於全實數上的微分方程解。",
        "en": "This decides the equation at this point only;a zero residual at one point is not a solution on all reals."
      },
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "global solution would require the residual identity on the declared domain",
      "after": "no automatic existence,uniqueness or global-solution claim",
      "relation": "implies",
      "rule_id": "ODE_scope_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "方程階數由方程的非零最高導數係數決定，不由候選多項式次數決定；本題不套一般存在唯一性定理。",
        "en": "Equation order depends on the nonzero leading derivative coefficient,not the candidate polynomial's degree;no general existence-uniqueness theorem is invoked."
      },
      "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,
        2
      ],
      3,
      "29",
      "13",
      3,
      20,
      "-94",
      false
    ],
    "first_coefficients": [
      "1",
      "4"
    ],
    "second_coefficients": null
  },
  "conclusion": {
    "canonical_value": "([8, 1, 2], 3, 29, 13, 3, 20, -94, 否)",
    "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-20261013-advance-20261020-r02-ordinary-differential-equation-foundations",
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