{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261019-r06-asymptotic-resource-bound-0040",
  "pilot_number": "R0017-09390",
  "number": 9390,
  "family_id": "qf-ext19-asymptotic-polynomial-degree",
  "title": {
    "zh-Hant": "第 9390 題：漸近資源界",
    "en": "Question 9390: Nonnegative Polynomial Degrees Including Both Zero Polynomials"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261019-r06-asymptotic-resource-bound-0040",
    "batch_id": "batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-asymptotic-resource-bound"
    ],
    "title_zh": "漸近資源界",
    "stem_zh": "明示非負函數族。操作=asymptotic-polynomial-degree；資料={\"coeffs_f\":[14,3,0],\"coeffs_g\":[1,15,1,0]}。請依 [coeffs_f,coeffs_g,degree_f,degree_g,bigO,bigOmega,theta] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([14, 3, 0], [1, 15, 1, 0], 1, 2, 是, 否, 否)",
      "explanation_zh": "第二批擴展核對=([14, 3, 0], [1, 15, 1, 0], 1, 2, 是, 否, 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；全尾段量化；有限benchmark不證Big-O或Theta；成本模型必須固定。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "漸近資源界"
      ]
    },
    "variant": {
      "family_id": "qf-ext19-asymptotic-polynomial-degree",
      "seed": 202610196040,
      "parameters": {
        "coeffs_f": [
          14,
          3,
          0
        ],
        "coeffs_g": [
          1,
          15,
          1,
          0
        ],
        "domain": "Nat_nonnegative_asymptotics",
        "operation": "asymptotic-polynomial-degree",
        "semantic_template": "qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[14,3,0],\"coeffs_g\":[1,15,1,0]}"
      }
    },
    "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、一般程式證明、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-20261019-r06-asymptotic-resource-bound-0040",
    "batch_id": "batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "55a19507b24573b93efc8b1750505d309e046dc523bda25f671eb605e2d8e8cd",
      "family_id": "qf-ext19-asymptotic-polynomial-degree",
      "seed": 202610196040,
      "semantic_template": "qf-ext19-asymptotic-polynomial-degree/{\"coeffs_f\":[14,3,0],\"coeffs_g\":[1,15,1,0]}"
    },
    "fields": {
      "title": "Nonnegative Polynomial Degrees Including Both Zero Polynomials",
      "stem": "Fix the cost model. Coefficients in ascending powers are f=[14, 3, 0], g=[1, 15, 1, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros.",
      "answer": {
        "kind": "expression",
        "value": "([14, 3, 0], [1, 15, 1, 0], 1, 2, Yes, No, No)",
        "explanation": "Classify the declared polynomial family with the zero cases explicit; a finite benchmark is not the full-tail argument. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([14, 3, 0], [1, 15, 1, 0], 1, 2, Yes, No, No)."
      },
      "skills": [
        "Nonnegative Polynomial Degrees Including Both Zero Polynomials"
      ],
      "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": "明示非負函數族。操作=asymptotic-polynomial-degree；資料={\"coeffs_f\":[14,3,0],\"coeffs_g\":[1,15,1,0]}。請依 [coeffs_f,coeffs_g,degree_f,degree_g,bigO,bigOmega,theta] 欄序回答。",
        "en": "Fix the cost model. Coefficients in ascending powers are f=[14, 3, 0], g=[1, 15, 1, 0]; all are nonnegative. Give (original f coefficients, original g coefficients, degree f, degree g, O?, Ω?, Θ?). Use degree -1 for the zero polynomial; retain original vectors including trailing zeros."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "fixed cost model;explicit nonnegative functions on natural n",
      "after": "O,Omega and Theta quantify positive constants over a full tail",
      "relation": "implies",
      "rule_id": "full_tail_not_finite_benchmark_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先固定成本模型與函數族。O要求存在C>0與N，使所有n≥N都滿足上界；Ω為對應正下界，Θ兩者同時。有限量測只描述有限窗口。",
        "en": "Fix the cost model and function family first.O requires C>0 and N with an upper bound for every n≥N;Omega is the corresponding positive lower bound,and Theta requires both.Finite measurements describe only a finite window."
      },
      "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": "f coefficient sum := sum(14, 3, 0)",
      "after": "f coefficient sum = 17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步原始操作數及指定次序精確計算，不用近似值或替換候選。",
        "en": "Calculate exactly in the specified source-operand order without approximation or substituting a different candidate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f coefficient sum，不以小數近似替代。",
          "en": "Recompute f coefficient sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "14",
          "3",
          "0"
        ],
        "output": "17",
        "label": "f coefficient sum"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "polynomial=f;original ascending coefficients=[14, 3, 0]",
      "after": "nonzero indices=[0, 1];degree=1;coefficient sum=17;leading coefficient/status=3",
      "relation": "implies",
      "rule_id": "nonnegative_polynomial_degree_and_bounds",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原係數向量含尾端0；最高非零索引為次數，零多項式另記−1。非零且係數非負時，n≥1有leading*n^degree≤P(n)≤sum(coeff)*n^degree，故無高次消去。",
        "en": "Retain the original coefficient vector including trailing zeros.The highest nonzero index is the degree;the zero polynomial has degree−1.For a nonzero polynomial with nonnegative coefficients and n≥1,leading*n^degree≤P(n)≤sum(coeff)*n^degree,so no leading cancellation occurs."
      },
      "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": "g coefficient sum := sum(1, 15, 1, 0)",
      "after": "g coefficient sum = 17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依本步原始操作數及指定次序精確計算，不用近似值或替換候選。",
        "en": "Calculate exactly in the specified source-operand order without approximation or substituting a different candidate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g coefficient sum，不以小數近似替代。",
          "en": "Recompute g coefficient sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "15",
          "1",
          "0"
        ],
        "output": "17",
        "label": "g coefficient sum"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "polynomial=g;original ascending coefficients=[1, 15, 1, 0]",
      "after": "nonzero indices=[0, 1, 2];degree=2;coefficient sum=17;leading coefficient/status=1",
      "relation": "implies",
      "rule_id": "nonnegative_polynomial_degree_and_bounds",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原係數向量含尾端0；最高非零索引為次數，零多項式另記−1。非零且係數非負時，n≥1有leading*n^degree≤P(n)≤sum(coeff)*n^degree，故無高次消去。",
        "en": "Retain the original coefficient vector including trailing zeros.The highest nonzero index is the degree;the zero polynomial has degree−1.For a nonzero polynomial with nonnegative coefficients and n≥1,leading*n^degree≤P(n)≤sum(coeff)*n^degree,so no leading cancellation occurs."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "direction=f versus g;degrees=(1,2);sums=(17,17);leading=(3,1)",
      "after": "Big-O=是;argument=C=17,N=1;upper numerator <= 17*n^1 <= C*1*n^2 <= C*denominator",
      "relation": "implies",
      "rule_id": "polynomial_quantified_upper_bound_case",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一分零多項式與非零次數情況。成功給正C與完整尾界；失敗對任意所提C,N都指定可更大的整數n，不以有限反例或數值樣本代替量化。Ω的判斷以交換分子、分母的O等價表達，再倒取正常數。",
        "en": "Separate zero and nonzero-degree cases.A success supplies positive C and a full-tail bound;a failure gives a larger integer n for every proposed C,N,not just a finite sample.Omega is obtained by swapping numerator/denominator in the equivalent O condition and reciprocating a positive constant."
      },
      "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": "direction=g versus f (Omega);degrees=(2,1);sums=(17,17);leading=(1,3)",
      "after": "Big-O=否;argument=for any C>0,N choose integer n>=max(N,1) and n>C*17/1;positive exponent gap=1 forces numerator>C*denominator",
      "relation": "implies",
      "rule_id": "polynomial_quantified_upper_bound_case",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一分零多項式與非零次數情況。成功給正C與完整尾界；失敗對任意所提C,N都指定可更大的整數n，不以有限反例或數值樣本代替量化。Ω的判斷以交換分子、分母的O等價表達，再倒取正常數。",
        "en": "Separate zero and nonzero-degree cases.A success supplies positive C and a full-tail bound;a failure gives a larger integer n for every proposed C,N,not just a finite sample.Omega is obtained by swapping numerator/denominator in the equivalent O condition and reciprocating a positive constant."
      },
      "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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "O(f,g)=是;O(g,f)=否",
      "after": "O=是;Omega=否;Theta=否",
      "relation": "implies",
      "rule_id": "zero_polynomial_tail_class_separation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "零對零時上下界0≤C*0都成立；零對非零只屬上界類，非零對零只屬下界類。非零多項式再比較次數，沒有把−1當一般非零常數次數。",
        "en": "For zero versus zero both bounds0≤C*0 hold.Zero versus nonzero gives only the upper class;nonzero versus zero gives only the lower class.Compare degrees only in nonzero cases;degree−1 is not a nonzero constant degree."
      },
      "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": [
      [
        14,
        3,
        0
      ],
      [
        1,
        15,
        1,
        0
      ],
      1,
      2,
      true,
      false,
      false
    ]
  },
  "conclusion": {
    "canonical_value": "([14, 3, 0], [1, 15, 1, 0], 1, 2, 是, 否, 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."
    }
  },
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    "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."
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}