{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261026-r03-cauchy-sequence-0057",
  "pilot_number": "R0023-12157",
  "number": 12157,
  "family_id": "qf-mod26-cauchy-reciprocal-epsilon-witness",
  "title": {
    "zh-Hant": "第 12157 題：Cauchy序列",
    "en": "Question 12157: A Strict Reciprocal Bound for the Selected Epsilon"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261026-r03-cauchy-sequence-0057",
    "batch_id": "batch-20261013-advance-20261026-r03-cauchy-sequence-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-cauchy-sequence"
    ],
    "title_zh": "Cauchy序列",
    "stem_zh": "保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":7,\"epsilon\":[1,9]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(7, 1/9, 3, 1/10, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)",
      "explanation_zh": "結構深度核對=(7, 1/9, 3, 1/10, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "Cauchy序列"
      ]
    },
    "variant": {
      "family_id": "qf-mod26-cauchy-reciprocal-epsilon-witness",
      "seed": 202610263057,
      "parameters": {
        "shift": 7,
        "epsilon": [
          1,
          9
        ],
        "domain": "Certified_real_Cauchy_sequence_slices",
        "operation": "cauchy-reciprocal-epsilon-witness",
        "semantic_template": "qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,9],\"shift\":7}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、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-20261026-r03-cauchy-sequence-0057",
    "batch_id": "batch-20261013-advance-20261026-r03-cauchy-sequence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "861096207467dc84ebcab4f79452b659a90c834beb9f4603fbd1c0ea04f72ef9",
      "family_id": "qf-mod26-cauchy-reciprocal-epsilon-witness",
      "seed": 202610263057,
      "semantic_template": "qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,9],\"shift\":7}"
    },
    "fields": {
      "title": "A Strict Reciprocal Bound for the Selected Epsilon",
      "stem": "Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)<epsilon. No limit or completeness premise is assumed. A finite recorded tail alone does not prove this infinite statement. The infinite model is x_n=1/(n+shift), with shift=7, epsilon=1/9. Give (shift,epsilon,least nonnegative N with 1/(N+shift)<epsilon,bound,m,n>=N implies distance<=1/(N+shift)<epsilon,cauchy_for_epsilon?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(7, 1/9, 3, 1/10, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)",
        "explanation": "Retain the strict inequality when choosing N. For every two later indices the reciprocal terms lie between zero and the stated upper bound, so their distance is at most that bound. The output certifies this chosen epsilon and the explicit model, not an arbitrary sequence. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (7, 1/9, 3, 1/10, m,n>=N implies distance<=1/(N+shift)<epsilon, Yes)."
      },
      "skills": [
        "A Strict Reciprocal Bound for the Selected Epsilon"
      ],
      "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": "保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":7,\"epsilon\":[1,9]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。",
        "en": "Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)<epsilon. No limit or completeness premise is assumed. A finite recorded tail alone does not prove this infinite statement. The infinite model is x_n=1/(n+shift), with shift=7, epsilon=1/9. Give (shift,epsilon,least nonnegative N with 1/(N+shift)<epsilon,bound,m,n>=N implies distance<=1/(N+shift)<epsilon,cauchy_for_epsilon?). Use exact reduced rational outputs with positive denominators; omit /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "epsilon reciprocal threshold := div(9, 1)",
      "after": "epsilon reciprocal threshold = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 epsilon reciprocal threshold，不以小數近似替代。",
          "en": "Recompute epsilon reciprocal threshold with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "9",
          "1"
        ],
        "output": "9",
        "label": "epsilon reciprocal threshold"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "positive epsilon=1/9;reciprocal threshold=9",
      "after": "floor threshold=9;9*1<=9<(9+1)*1",
      "relation": "implies",
      "rule_id": "exact_integer_threshold_floor",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1/(N+shift)<a/b等價於N+shift>b/a，因a及分母皆正；先取精確下整數以處理嚴格邊界。",
        "en": "Since a and denominators are positive,1/(N+shift)<a/b is equivalent to N+shift>b/a.Use the exact floor to handle the strict boundary."
      },
      "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": "unclamped reciprocal cutoff := sum(9, -7, 1)",
      "after": "unclamped reciprocal cutoff = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 unclamped reciprocal cutoff，不以小數近似替代。",
          "en": "Recompute unclamped reciprocal cutoff with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "-7",
          "1"
        ],
        "output": "3",
        "label": "unclamped reciprocal cutoff"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "reciprocal tail bound denominator := sum(3, 7)",
      "after": "reciprocal tail bound denominator = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reciprocal tail bound denominator，不以小數近似替代。",
          "en": "Recompute reciprocal tail bound denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "7"
        ],
        "output": "10",
        "label": "reciprocal tail bound denominator"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "reciprocal tail bound := div(1, 10)",
      "after": "reciprocal tail bound = 1/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reciprocal tail bound，不以小數近似替代。",
          "en": "Recompute reciprocal tail bound with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "10"
        ],
        "output": "1/10",
        "label": "reciprocal tail bound"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "previous cutoff denominator := sum(2, 7)",
      "after": "previous cutoff denominator = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 previous cutoff denominator，不以小數近似替代。",
          "en": "Recompute previous cutoff denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "7"
        ],
        "output": "9",
        "label": "previous cutoff denominator"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "previous cutoff bound := div(1, 9)",
      "after": "previous cutoff bound = 1/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依指定原始角色及已算中間量精確運算，不替換模型、重排配對或以近似數判斷。",
        "en": "Compute exactly from the specified source roles and established intermediates,without substituting models,reordering pairings or comparing rounded values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 previous cutoff bound，不以小數近似替代。",
          "en": "Recompute previous cutoff bound with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "9"
        ],
        "output": "1/9",
        "label": "previous cutoff bound"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "N=3;bound=1/10;previous bound=1/9;epsilon=1/9",
      "after": "current bound is strict;previous cutoff fails;N is least nonnegative",
      "relation": "implies",
      "rule_id": "reciprocal_least_cutoff_strict_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正倒數隨索引遞減，前一門檻仍≥epsilon，故所有更小非負索引都失敗；保留等號失敗情況。",
        "en": "Positive reciprocals decrease with the index.The previous bound is still≥epsilon,so every smaller nonnegative cutoff fails.Equality remains a failure."
      },
      "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": "explicit x_n=1/(n+7);N=3;bound=1/10<epsilon=1/9",
      "after": "all m,n>=N:both terms in[0,bound],so distance<=bound<epsilon",
      "relation": "implies",
      "rule_id": "reciprocal_all_tail_pairs_not_samples",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意兩個尾項都介於0與界之間，故距離至多該界。門檻公式對任意正epsilon可構造；輸出欄只證本次選定epsilon的明示模型。",
        "en": "Any two tail terms lie between0 and the bound,so their distance is at most that bound.The cutoff formula works for arbitrary positive epsilon;the output flag certifies this selected epsilon in the explicit model."
      },
      "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": [
      7,
      "1/9",
      3,
      "1/10",
      "m,n>=N implies distance<=1/(N+shift)<epsilon",
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(7, 1/9, 3, 1/10, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)",
    "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."
  },
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      "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."
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    "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.",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}