{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0065",
  "pilot_number": "R0041-21440",
  "number": 21440,
  "family_id": "qf-mod40-bw-oscillatory-tail",
  "title": {
    "zh-Hant": "第 21440 題：實數列Bolzano－Weierstrass定理",
    "en": "Question 21440: Parity Subsequences Need a Strict All-Later Epsilon Witness"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0065",
    "batch_id": "batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-bolzano-weierstrass-theorem-real"
    ],
    "title_zh": "實數列Bolzano－Weierstrass定理",
    "stem_zh": "whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[15,16],\"coefficient\":15,\"shift\":16,\"parity\":0,\"epsilon\":[1,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(0, k(j)=2j+0, 15/16, 1/24, 173, 2835/2896, 是, forall_j_ge_J_symbolic_subsequence_tail)",
      "explanation_zh": "Wave6第三層核對=(0, k(j)=2j+0, 15/16, 1/24, 173, 2835/2896, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "實數列Bolzano－Weierstrass定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod40-bw-oscillatory-tail",
      "seed": 202611092065,
      "parameters": {
        "limit": [
          15,
          16
        ],
        "coefficient": 15,
        "shift": 16,
        "parity": 0,
        "epsilon": [
          1,
          24
        ],
        "domain": "Symbolic_bounded_real_sequence_subsequence",
        "operation": "bw-oscillatory-tail",
        "semantic_template": "qf-mod40-bw-oscillatory-tail/{\"coefficient\":15,\"epsilon\":[1,24],\"limit\":[15,16],\"parity\":0,\"shift\":16}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0065",
    "batch_id": "batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f9441d5dc84682a7ddd05d996b83870131d23f2239b66aec72d131c77f76dc5e",
      "family_id": "qf-mod40-bw-oscillatory-tail",
      "seed": 202611092065,
      "semantic_template": "qf-mod40-bw-oscillatory-tail/{\"coefficient\":15,\"epsilon\":[1,24],\"limit\":[15,16],\"parity\":0,\"shift\":16}"
    },
    "fields": {
      "title": "Parity Subsequences Need a Strict All-Later Epsilon Witness",
      "stem": "Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=15/16, c=15>0, shift=16≥1, parity=0 and epsilon=1/24>0. The infinite index rule is k(j)=2j+parity for every j≥0. Give (parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(0, k(j)=2j+0, 15/16, 1/24, 173, 2835/2896, Yes, forall_j_ge_J_symbolic_subsequence_tail)",
        "explanation": "Find the least nonnegative J such that c/(2J+parity+shift)<epsilon. Equality still fails the strict inequality. Report a at k(J), not a_J; increasing denominators establish the same bound for every j≥J. Both parity subsequences converge to L. The exact result, retaining the requested order, is (0, k(j)=2j+0, 15/16, 1/24, 173, 2835/2896, Yes, forall_j_ge_J_symbolic_subsequence_tail)."
      },
      "skills": [
        "Parity Subsequences Need a Strict All-Later Epsilon Witness"
      ],
      "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": "whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[15,16],\"coefficient\":15,\"shift\":16,\"parity\":0,\"epsilon\":[1,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use a_n=L+(−1)^n*c/(n+shift) for every n≥0, L=15/16, c=15>0, shift=16≥1, parity=0 and epsilon=1/24>0. The infinite index rule is k(j)=2j+parity for every j≥0. Give (parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=bw-oscillatory-tail;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[parity, index_rule, limit, epsilon, J, value_at_k_J, strict_tail_certificate, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記保留；此處直接列有序輸出欄位，不依賴JavaScript。欄位或tag不是額外數學證據，固定No保留原聲稱範圍。",
        "en": "Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope."
      },
      "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": "ORIGINAL a_n=15/16+(-1)^n*(15)/(n+16) for EVERY n>=0;parity=0",
      "after": "k(j)=2j+0 for ALL j>=0;k(j+1)-k(j)=2>0;selected sign=1;whole sequence bounded by |L|+c/shift",
      "relation": "implies",
      "rule_id": "bw_parity_whole_law_and_infinite_strict_index",
      "conditions": [],
      "explanation": {
        "zh-Hant": "奇偶索引是定義在全部自然數的嚴格遞增映射；取到的符號固定，誤差是c/(2j+parity+shift)。整數列有界但是否單調不是此子數列證書的前提。",
        "en": "The parity rule is an infinite strictly increasing index map.Selected signs are fixed and errors equal c/(2j+parity+shift).The whole sequence is bounded;its monotonicity is not a premise for this subsequence certificate."
      },
      "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": "bw c over epsilon := div(15, 1/24)",
      "after": "bw c over epsilon = 360",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bw c over epsilon，不以小數近似替代。",
          "en": "Recompute bw c over epsilon with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15",
          "1/24"
        ],
        "output": "360",
        "label": "bw c over epsilon"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "bw index multiple := mul(2, 173)",
      "after": "bw index multiple = 346",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bw index multiple，不以小數近似替代。",
          "en": "Recompute bw index multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "173"
        ],
        "output": "346",
        "label": "bw index multiple"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "bw actual source index := sum(0, 346)",
      "after": "bw actual source index = 346",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bw actual source index，不以小數近似替代。",
          "en": "Recompute bw actual source index with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "346"
        ],
        "output": "346",
        "label": "bw actual source index"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "bw witness denominator := sum(346, 16)",
      "after": "bw witness denominator = 362",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bw witness denominator，不以小數近似替代。",
          "en": "Recompute bw witness denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "346",
          "16"
        ],
        "output": "362",
        "label": "bw witness denominator"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "bw witness distance := div(15, 362)",
      "after": "bw witness distance = 15/362",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bw witness distance，不以小數近似替代。",
          "en": "Recompute bw witness distance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15",
          "362"
        ],
        "output": "15/362",
        "label": "bw witness distance"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "bw predecessor distance := div(15, 360)",
      "after": "bw predecessor distance = 1/24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bw predecessor distance，不以小數近似替代。",
          "en": "Recompute bw predecessor distance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15",
          "360"
        ],
        "output": "1/24",
        "label": "bw predecessor distance"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "bw:c=15,shift=16,epsilon=1/24;index(q)=2q+0;c/epsilon=360;floor=360",
      "after": "least nonnegative witness=173;actual source index=346;distance=15/362<epsilon;predecessor distance=1/24;no predecessor evaluated when witness0",
      "relation": "implies",
      "rule_id": "tail_strict_least_index_not_equality_or_negative_predecessor",
      "conditions": [],
      "explanation": {
        "zh-Hant": "嚴格小於等價分母大於c/epsilon，整數分母需至少floor(c/epsilon)+1，再按stride向上取整且截到0。N>0才驗前一項仍≥epsilon；相等尚未通過。",
        "en": "Strict distance<epsilon means the denominator exceeds c/epsilon,so its integer threshold is floor(c/epsilon)+1.Round the affine index upward and clamp only the witness to0.Check the predecessor only when positive;equality still fails."
      },
      "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": "bw:positive c=15,epsilon=1/24,shift=16,stride=2,offset=0;least witness=173",
      "after": "for EVERY integer q>=173,denominator=2q+0+16>=witness denominator>0,so 0<distance<=(15/362)<epsilon",
      "relation": "implies",
      "rule_id": "tail_all_later_quantified_from_positive_denominator_growth",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這一步是所有後續索引的代數不等式，不是只看一個數值猜收斂。分母正且遞增，固定正分子使誤差遞減；前一項檢查只證最小性。",
        "en": "This is an algebraic inequality for every later index,not convergence guessed from one evaluated term.Positive increasing denominators and a fixed positive numerator control the whole tail;the predecessor check proves minimality only."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "bw at selected k(J) denominator := sum(346, 16)",
      "after": "bw at selected k(J) denominator = 362",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bw at selected k(J) denominator，不以小數近似替代。",
          "en": "Recompute bw at selected k(J) denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "346",
          "16"
        ],
        "output": "362",
        "label": "bw at selected k(J) denominator"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "bw at selected k(J) unsigned tail := div(15, 362)",
      "after": "bw at selected k(J) unsigned tail = 15/362",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bw at selected k(J) unsigned tail，不以小數近似替代。",
          "en": "Recompute bw at selected k(J) unsigned tail with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15",
          "362"
        ],
        "output": "15/362",
        "label": "bw at selected k(J) unsigned tail"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "bw at selected k(J) signed value := sum(15/16, 15/362)",
      "after": "bw at selected k(J) signed value = 2835/2896",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bw at selected k(J) signed value，不以小數近似替代。",
          "en": "Recompute bw at selected k(J) signed value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "15/16",
          "15/362"
        ],
        "output": "2835/2896",
        "label": "bw at selected k(J) signed value"
      }
    }
  ],
  "math_model": {
    "result": [
      0,
      "k(j)=2j+0",
      "15/16",
      "1/24",
      173,
      "2835/2896",
      true,
      "forall_j_ge_J_symbolic_subsequence_tail"
    ]
  },
  "conclusion": {
    "canonical_value": "(0, k(j)=2j+0, 15/16, 1/24, 173, 2835/2896, 是, forall_j_ge_J_symbolic_subsequence_tail)",
    "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-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations/questions.jsonl",
      "batch_sha256": "cd386460b627b8305543a8fb6674bbe6aba93e951a5d4ce260bdc15d107b10c1",
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