{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261019-r06-asymptotic-resource-bound-0081",
  "pilot_number": "R0017-09306",
  "number": 9306,
  "family_id": "qf-ext19-asymptotic-benchmark-insufficiency",
  "title": {
    "zh-Hant": "第 9306 題：漸近資源界",
    "en": "Question 9306: Finite Sample Results versus Independently Known Tail Classes"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261019-r06-asymptotic-resource-bound-0081",
    "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-benchmark-insufficiency；資料={\"p\":5,\"q\":5,\"bound\":5}。請依 [p,q,bound,samples,sample_le,asymptotic_class,samples_sufficient,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(5, 5, 5, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243], [4, 1024, 1024], [5, 3125, 3125]], 是, [是, 是, 是], 否, 有限benchmark不足以證全尾段)",
      "explanation_zh": "第二批擴展核對=(5, 5, 5, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243], [4, 1024, 1024], [5, 3125, 3125]], 是, [是, 是, 是], 否, 有限benchmark不足以證全尾段)；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": 1,
      "skills": [
        "漸近資源界"
      ]
    },
    "variant": {
      "family_id": "qf-ext19-asymptotic-benchmark-insufficiency",
      "seed": 202610196081,
      "parameters": {
        "p": 5,
        "q": 5,
        "bound": 5,
        "domain": "Nat_nonnegative_asymptotics",
        "operation": "asymptotic-benchmark-insufficiency",
        "semantic_template": "qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":5,\"p\":5,\"q\":5}"
      }
    },
    "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-0081",
    "batch_id": "batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f854950938f2879e946fa3aa9c4263e9994f5a923a9c18598218f56d1abefb67",
      "family_id": "qf-ext19-asymptotic-benchmark-insufficiency",
      "seed": 202610196081,
      "semantic_template": "qf-ext19-asymptotic-benchmark-insufficiency/{\"bound\":5,\"p\":5,\"q\":5}"
    },
    "fields": {
      "title": "Finite Sample Results versus Independently Known Tail Classes",
      "stem": "Fix f(n)=n^5, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,5]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label).",
      "answer": {
        "kind": "expression",
        "value": "(5, 5, 5, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243], [4, 1024, 1024], [5, 3125, 3125]], Yes, [Yes, Yes, Yes], No, finite-benchmark-insufficient-for-a-full-tail-proof)",
        "explanation": "Use the explicit family for asymptotic classification. A short sample can agree despite the wrong eventual order, or fail at zero despite a valid eventual upper bound. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, 5, 5, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243], [4, 1024, 1024], [5, 3125, 3125]], Yes, [Yes, Yes, Yes], No, finite-benchmark-insufficient-for-a-full-tail-proof)."
      },
      "skills": [
        "Finite Sample Results versus Independently Known Tail Classes"
      ],
      "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-benchmark-insufficiency；資料={\"p\":5,\"q\":5,\"bound\":5}。請依 [p,q,bound,samples,sample_le,asymptotic_class,samples_sufficient,boundary] 欄序回答。",
        "en": "Fix f(n)=n^5, g(n)=n^5, with n natural and the algebraic convention 0^0=1. Sample the complete window [0,5]. Give (p, q, bound, every [n,f(n),g(n)] sample, all sampled f≤g?, independently derived [O?,Ω?,Θ?], do samples alone suffice?, fixed finite-benchmark-insufficiency label)."
      }
    }
  ],
  "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": "sample natural window=[0,5];exponents=(5,5)",
      "after": "algebraic polynomial convention0^0=1",
      "relation": "implies",
      "rule_id": "benchmark_zero_power_model_convention",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這個明示代數成本模型採n^0=1，含n=0；不把此局部約定宣稱為所有分析極限問題的0^0定義。",
        "en": "This explicit algebraic cost model uses n^0=1 including n=0.This local convention is not a universal definition for0^0 in all analytic limit problems."
      },
      "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": "(0)^5",
      "after": "benchmark n0 first power = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "5"
        ],
        "output": "0",
        "label": "benchmark n0 first power"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "(0)^5",
      "after": "benchmark n0 second power = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "5"
        ],
        "output": "0",
        "label": "benchmark n0 second power"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "sample n=0;f(n)=0;g(n)=0",
      "after": "sample f<=g=是",
      "relation": "implies",
      "rule_id": "finite_benchmark_each_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只比較這個原窗口位置的兩個精確值，不擅自延長樣本範圍。",
        "en": "Compare the two exact values at this original window position without silently extending the sample domain."
      },
      "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": "(1)^5",
      "after": "benchmark n1 first power = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "1",
          "5"
        ],
        "output": "1",
        "label": "benchmark n1 first power"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "(1)^5",
      "after": "benchmark n1 second power = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "1",
          "5"
        ],
        "output": "1",
        "label": "benchmark n1 second power"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "sample n=1;f(n)=1;g(n)=1",
      "after": "sample f<=g=是",
      "relation": "implies",
      "rule_id": "finite_benchmark_each_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只比較這個原窗口位置的兩個精確值，不擅自延長樣本範圍。",
        "en": "Compare the two exact values at this original window position without silently extending the sample domain."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "(2)^5",
      "after": "benchmark n2 first power = 32",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "2",
          "5"
        ],
        "output": "32",
        "label": "benchmark n2 first power"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "(2)^5",
      "after": "benchmark n2 second power = 32",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "2",
          "5"
        ],
        "output": "32",
        "label": "benchmark n2 second power"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "sample n=2;f(n)=32;g(n)=32",
      "after": "sample f<=g=是",
      "relation": "implies",
      "rule_id": "finite_benchmark_each_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只比較這個原窗口位置的兩個精確值，不擅自延長樣本範圍。",
        "en": "Compare the two exact values at this original window position without silently extending the sample domain."
      },
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "(3)^5",
      "after": "benchmark n3 first power = 243",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "3",
          "5"
        ],
        "output": "243",
        "label": "benchmark n3 first power"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "(3)^5",
      "after": "benchmark n3 second power = 243",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "3",
          "5"
        ],
        "output": "243",
        "label": "benchmark n3 second power"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "sample n=3;f(n)=243;g(n)=243",
      "after": "sample f<=g=是",
      "relation": "implies",
      "rule_id": "finite_benchmark_each_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只比較這個原窗口位置的兩個精確值，不擅自延長樣本範圍。",
        "en": "Compare the two exact values at this original window position without silently extending the sample domain."
      },
      "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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "(4)^5",
      "after": "benchmark n4 first power = 1024",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "4",
          "5"
        ],
        "output": "1024",
        "label": "benchmark n4 first power"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "(4)^5",
      "after": "benchmark n4 second power = 1024",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "4",
          "5"
        ],
        "output": "1024",
        "label": "benchmark n4 second power"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "sample n=4;f(n)=1024;g(n)=1024",
      "after": "sample f<=g=是",
      "relation": "implies",
      "rule_id": "finite_benchmark_each_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只比較這個原窗口位置的兩個精確值，不擅自延長樣本範圍。",
        "en": "Compare the two exact values at this original window position without silently extending the sample domain."
      },
      "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": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "(5)^5",
      "after": "benchmark n5 first power = 3125",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "5",
          "5"
        ],
        "output": "3125",
        "label": "benchmark n5 first power"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "(5)^5",
      "after": "benchmark n5 second power = 3125",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "5",
          "5"
        ],
        "output": "3125",
        "label": "benchmark n5 second power"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "sample n=5;f(n)=3125;g(n)=3125",
      "after": "sample f<=g=是",
      "relation": "implies",
      "rule_id": "finite_benchmark_each_sample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只比較這個原窗口位置的兩個精確值，不擅自延長樣本範圍。",
        "en": "Compare the two exact values at this original window position without silently extending the sample domain."
      },
      "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": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "phase=independent declared power model;positive coefficients=(1,1);exponents=(5,5);tail n>=1",
      "after": "O=是;Omega=是;Theta=是",
      "relation": "implies",
      "rule_id": "monomial_full_tail_exponent_proof",
      "conditions": [],
      "explanation": {
        "zh-Hant": "n≥1且p≤q時a*n^p≤a*n^q≤a*b*n^q，所以指定C=a、N=1有效。p≥q時n^q≤a*n^p，所以指定c=1/b、N=1有效。反向失敗時正次方比率無界：任意C,N都可取n≥max(N,1)且n>C*b/a，使n^(p−q)≥n而違反上界；下界失敗把f,g角色交換。不是由樣本比較指數。",
        "en": "For n≥1 and p≤q,a*n^p≤a*n^q≤a*b*n^q,so prescribed C=a,N=1 works.For p≥q,n^q≤a*n^p,so prescribed c=1/b,N=1 works.In a failing direction the positive-power ratio is unbounded:for any C,N choose n≥max(N,1) and n>C*b/a;then n^(p−q)≥n violates the upper bound.For a failed lower bound interchange f and g.This is not a sample-based exponent inference."
      },
      "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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "all finite sample comparisons pass=是;model-derived classes=[是, 是, 是]",
      "after": "samples alone sufficient=否;boundary=有限benchmark不足以證全尾段",
      "relation": "implies",
      "rule_id": "finite_benchmark_not_tail_proof",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分類來自上面明示冪函數的全尾段推理，不來自有限表。短窗口可全部通過而尾段次序相反，也可在0失敗但最終上界成立；兩種證據不要混為一個PASS。",
        "en": "The classes come from full-tail reasoning for the declared power model,not the finite table.A short window can pass with the opposite eventual order,or fail at0 despite a valid eventual upper bound.Do not merge these evidence roots into one PASS."
      },
      "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": [
      5,
      5,
      5,
      [
        [
          0,
          "0",
          "0"
        ],
        [
          1,
          "1",
          "1"
        ],
        [
          2,
          "32",
          "32"
        ],
        [
          3,
          "243",
          "243"
        ],
        [
          4,
          "1024",
          "1024"
        ],
        [
          5,
          "3125",
          "3125"
        ]
      ],
      true,
      [
        true,
        true,
        true
      ],
      false,
      "有限benchmark不足以證全尾段"
    ]
  },
  "conclusion": {
    "canonical_value": "(5, 5, 5, [[0, 0, 0], [1, 1, 1], [2, 32, 32], [3, 243, 243], [4, 1024, 1024], [5, 3125, 3125]], 是, [是, 是, 是], 否, 有限benchmark不足以證全尾段)",
    "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-20261019-r06-asymptotic-resource-bound-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations/questions.jsonl",
      "batch_sha256": "e327d392c461513525af13baeae11c1e83c474ede1181e0ac60e1c3f0e53f946",
      "original_line": 81,
      "original_line_sha256": "7548b83b1d0220cc14d44331f508b42205373ee71287b14da11fbe4128bc1d63",
      "question_compact_sha256": "f854950938f2879e946fa3aa9c4263e9994f5a923a9c18598218f56d1abefb67",
      "english_row_compact_sha256": "d100b2b4cfe06dc206db39e956560ab9bc763fa6786a2c1d590838c020fecb9b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261019-r06-asymptotic-resource-bound-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-asymptotic-resource-bound": {
        "sha256": "2b5dc69aebd1ca2d1e4c3c4bb10e9c0c889c653d311aebba7e59d53def35d42e",
        "english_sha256": "9efe714ae882b4c30cff9d151ada3de59893c982eddf24b68e74196cb3c2bc29",
        "original_path": "public/data/mko/asymptotic-resource-bound.json",
        "english_path": "public/data/i18n/editions/mko-asymptotic-resource-bound.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "20e59f30c0b6218e6d9614c16170d770e56a2f1e4ae5ce39d0a8a3db206afa61",
    "original_candidate_sha256": "a9351467c638ffe70b3361fc6fde07ea32038e27e668932d4e3464dc6b992182",
    "original_candidate_line_sha256": "a9351467c638ffe70b3361fc6fde07ea32038e27e668932d4e3464dc6b992182",
    "mko_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c",
      "previous_english_sha256": {
        "mko-asymptotic-resource-bound": "4ae951a94a3de8afff8e41d6edead7e6ee397c76b4ecae8cad617c6d6429361f"
      }
    }
  },
  "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-20261026-006-p18",
    "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": "abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8",
      "question_first_witness_sha256": "43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 22,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "8c3c9a9e3e71a892b3c05b99140911bbcd6312a6b660b6edf4c6a873522d54bb",
      "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."
  }
}