{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261026-r03-cauchy-sequence-0009",
  "pilot_number": "R0023-12134",
  "number": 12134,
  "family_id": "qf-mod26-cauchy-finite-tail-audit",
  "title": {
    "zh-Hant": "第 12134 題：Cauchy序列",
    "en": "Question 12134: A Finite Tail Audit Cannot Establish General Cauchy Behavior"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261026-r03-cauchy-sequence-0009",
    "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-finite-tail-audit；資料={\"sequence\":[[1,10],[1,11],[1,12],[1,13],[1,14]],\"epsilon\":[1,6],\"N\":2}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([1/10, 1/11, 1/12, 1/13, 1/14], 1/6, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)",
      "explanation_zh": "結構深度核對=([1/10, 1/11, 1/12, 1/13, 1/14], 1/6, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)；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": 2,
      "skills": [
        "Cauchy序列"
      ]
    },
    "variant": {
      "family_id": "qf-mod26-cauchy-finite-tail-audit",
      "seed": 202610263009,
      "parameters": {
        "sequence": [
          [
            1,
            10
          ],
          [
            1,
            11
          ],
          [
            1,
            12
          ],
          [
            1,
            13
          ],
          [
            1,
            14
          ]
        ],
        "epsilon": [
          1,
          6
        ],
        "N": 2,
        "domain": "Certified_real_Cauchy_sequence_slices",
        "operation": "cauchy-finite-tail-audit",
        "semantic_template": "qf-mod26-cauchy-finite-tail-audit/{\"N\":2,\"epsilon\":[1,6],\"sequence\":[[1,10],[1,11],[1,12],[1,13],[1,14]]}"
      }
    },
    "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-0009",
    "batch_id": "batch-20261013-advance-20261026-r03-cauchy-sequence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2ee1366aff5c0ff43d0d529ff1678bf30c78e18197704e8fbbd86076547a06ab",
      "family_id": "qf-mod26-cauchy-finite-tail-audit",
      "seed": 202610263009,
      "semantic_template": "qf-mod26-cauchy-finite-tail-audit/{\"N\":2,\"epsilon\":[1,6],\"sequence\":[[1,10],[1,11],[1,12],[1,13],[1,14]]}"
    },
    "fields": {
      "title": "A Finite Tail Audit Cannot Establish General Cauchy Behavior",
      "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. Recorded rational sequence=[1/10, 1/11, 1/12, 1/13, 1/14], epsilon=1/6, cutoff N=2; only listed zero-based indices are available. Give (normalized sequence,epsilon,N,all bad [m,n,distance] records in ordered-index order,finite_tail_pass?,general Cauchy established by this audit?,finite-tail-insufficient-for-general-Cauchy). A bad distance is at least epsilon. Use exact reduced rational outputs with positive denominators; omit /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "([1/10, 1/11, 1/12, 1/13, 1/14], 1/6, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)",
        "explanation": "Enumerate every ordered pair of listed indices at or beyond N. An empty tail passes vacuously. The general_cauchy No means not established by this finite audit, not a proof that every infinite continuation is non-Cauchy. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1/10, 1/11, 1/12, 1/13, 1/14], 1/6, 2, [], Yes, No, finite-tail-insufficient-for-general-Cauchy)."
      },
      "skills": [
        "A Finite Tail Audit Cannot Establish General Cauchy Behavior"
      ],
      "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-finite-tail-audit；資料={\"sequence\":[[1,10],[1,11],[1,12],[1,13],[1,14]],\"epsilon\":[1,6],\"N\":2}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 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. Recorded rational sequence=[1/10, 1/11, 1/12, 1/13, 1/14], epsilon=1/6, cutoff N=2; only listed zero-based indices are available. Give (normalized sequence,epsilon,N,all bad [m,n,distance] records in ordered-index order,finite_tail_pass?,general Cauchy established by this audit?,finite-tail-insufficient-for-general-Cauchy). A bad distance is at least epsilon. Use exact reduced rational outputs with positive denominators; omit /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw sequence=[[1, 10], [1, 11], [1, 12], [1, 13], [1, 14]];raw epsilon=[1, 6]",
      "after": "normalized sequence=[1/10, 1/11, 1/12, 1/13, 1/14];positive epsilon=1/6;cutoff=2",
      "relation": "implies",
      "rule_id": "finite_Cauchy_exact_data_and_cutoff",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分母非零，正規化後epsilon必須正；只核紀錄中的索引，N可等於長度而得到空尾段。",
        "en": "Original denominators are nonzero and normalized epsilon must be positive.Only recorded indices are audited;N may equal the length,leaving an empty tail."
      },
      "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": "recorded pair2,2 signed difference := sub(1/12, 1/12)",
      "after": "recorded pair2,2 signed difference = 0",
      "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 recorded pair2,2 signed difference，不以小數近似替代。",
          "en": "Recompute recorded pair2,2 signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/12",
          "1/12"
        ],
        "output": "0",
        "label": "recorded pair2,2 signed difference"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "ordered recorded indices=[2, 2];signed difference=0;epsilon=1/6",
      "after": "absolute distance=0;strictly below epsilon=是",
      "relation": "implies",
      "rule_id": "finite_Cauchy_each_ordered_tail_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。",
        "en": "Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms."
      },
      "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": "recorded pair2,3 signed difference := sub(1/12, 1/13)",
      "after": "recorded pair2,3 signed difference = 1/156",
      "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 recorded pair2,3 signed difference，不以小數近似替代。",
          "en": "Recompute recorded pair2,3 signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/12",
          "1/13"
        ],
        "output": "1/156",
        "label": "recorded pair2,3 signed difference"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "ordered recorded indices=[2, 3];signed difference=1/156;epsilon=1/6",
      "after": "absolute distance=1/156;strictly below epsilon=是",
      "relation": "implies",
      "rule_id": "finite_Cauchy_each_ordered_tail_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。",
        "en": "Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms."
      },
      "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": "recorded pair2,4 signed difference := sub(1/12, 1/14)",
      "after": "recorded pair2,4 signed difference = 1/84",
      "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 recorded pair2,4 signed difference，不以小數近似替代。",
          "en": "Recompute recorded pair2,4 signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/12",
          "1/14"
        ],
        "output": "1/84",
        "label": "recorded pair2,4 signed difference"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "ordered recorded indices=[2, 4];signed difference=1/84;epsilon=1/6",
      "after": "absolute distance=1/84;strictly below epsilon=是",
      "relation": "implies",
      "rule_id": "finite_Cauchy_each_ordered_tail_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。",
        "en": "Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms."
      },
      "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": "recorded pair3,2 signed difference := sub(1/13, 1/12)",
      "after": "recorded pair3,2 signed difference = -1/156",
      "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 recorded pair3,2 signed difference，不以小數近似替代。",
          "en": "Recompute recorded pair3,2 signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/13",
          "1/12"
        ],
        "output": "-1/156",
        "label": "recorded pair3,2 signed difference"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "ordered recorded indices=[3, 2];signed difference=-1/156;epsilon=1/6",
      "after": "absolute distance=1/156;strictly below epsilon=是",
      "relation": "implies",
      "rule_id": "finite_Cauchy_each_ordered_tail_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。",
        "en": "Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms."
      },
      "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": "recorded pair3,3 signed difference := sub(1/13, 1/13)",
      "after": "recorded pair3,3 signed difference = 0",
      "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 recorded pair3,3 signed difference，不以小數近似替代。",
          "en": "Recompute recorded pair3,3 signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/13",
          "1/13"
        ],
        "output": "0",
        "label": "recorded pair3,3 signed difference"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "ordered recorded indices=[3, 3];signed difference=0;epsilon=1/6",
      "after": "absolute distance=0;strictly below epsilon=是",
      "relation": "implies",
      "rule_id": "finite_Cauchy_each_ordered_tail_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。",
        "en": "Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms."
      },
      "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": "recorded pair3,4 signed difference := sub(1/13, 1/14)",
      "after": "recorded pair3,4 signed difference = 1/182",
      "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 recorded pair3,4 signed difference，不以小數近似替代。",
          "en": "Recompute recorded pair3,4 signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/13",
          "1/14"
        ],
        "output": "1/182",
        "label": "recorded pair3,4 signed difference"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "ordered recorded indices=[3, 4];signed difference=1/182;epsilon=1/6",
      "after": "absolute distance=1/182;strictly below epsilon=是",
      "relation": "implies",
      "rule_id": "finite_Cauchy_each_ordered_tail_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。",
        "en": "Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "recorded pair4,2 signed difference := sub(1/14, 1/12)",
      "after": "recorded pair4,2 signed difference = -1/84",
      "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 recorded pair4,2 signed difference，不以小數近似替代。",
          "en": "Recompute recorded pair4,2 signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/14",
          "1/12"
        ],
        "output": "-1/84",
        "label": "recorded pair4,2 signed difference"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "ordered recorded indices=[4, 2];signed difference=-1/84;epsilon=1/6",
      "after": "absolute distance=1/84;strictly below epsilon=是",
      "relation": "implies",
      "rule_id": "finite_Cauchy_each_ordered_tail_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。",
        "en": "Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms."
      },
      "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": "recorded pair4,3 signed difference := sub(1/14, 1/13)",
      "after": "recorded pair4,3 signed difference = -1/182",
      "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 recorded pair4,3 signed difference，不以小數近似替代。",
          "en": "Recompute recorded pair4,3 signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/14",
          "1/13"
        ],
        "output": "-1/182",
        "label": "recorded pair4,3 signed difference"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "ordered recorded indices=[4, 3];signed difference=-1/182;epsilon=1/6",
      "after": "absolute distance=1/182;strictly below epsilon=是",
      "relation": "implies",
      "rule_id": "finite_Cauchy_each_ordered_tail_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。",
        "en": "Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms."
      },
      "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": "recorded pair4,4 signed difference := sub(1/14, 1/14)",
      "after": "recorded pair4,4 signed difference = 0",
      "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 recorded pair4,4 signed difference，不以小數近似替代。",
          "en": "Recompute recorded pair4,4 signed difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/14",
          "1/14"
        ],
        "output": "0",
        "label": "recorded pair4,4 signed difference"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "ordered recorded indices=[4, 4];signed difference=0;epsilon=1/6",
      "after": "absolute distance=0;strictly below epsilon=是",
      "relation": "implies",
      "rule_id": "finite_Cauchy_each_ordered_tail_pair",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用絕對距離與嚴格小於，等於epsilon也失敗；按完整有序索引對輸出，不只比相鄰項。",
        "en": "Use absolute distance and strict inequality;distance equal to epsilon fails.Check all ordered tail-index pairs,not only adjacent terms."
      },
      "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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "all bad recorded pairs=[]",
      "after": "finite recorded tail passes=是;general Cauchy established=否",
      "relation": "implies",
      "rule_id": "finite_tail_not_quantified_infinite_Cauchy",
      "conditions": [],
      "explanation": {
        "zh-Hant": "空尾段可空泛通過，但有限紀錄沒有給∀epsilon存在N且所有無限尾對的義務。此No是未證，不是說每種無限接續都非Cauchy。",
        "en": "An empty tail can pass vacuously.Finite records do not supply the infinite for-every-epsilon/eventual-all-pairs obligation.No here means not established,not that every infinite continuation is non-Cauchy."
      },
      "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": [
      [
        "1/10",
        "1/11",
        "1/12",
        "1/13",
        "1/14"
      ],
      "1/6",
      2,
      [],
      true,
      false,
      "有限尾段不足以證一般Cauchy"
    ]
  },
  "conclusion": {
    "canonical_value": "([1/10, 1/11, 1/12, 1/13, 1/14], 1/6, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)",
    "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-20261026-r03-cauchy-sequence-foundations",
      "batch_path": "public/data/questions/errata/ocme-source-label-cauchy-20261015/questions.jsonl",
      "batch_sha256": "3e823e6e797b4fbcd90701bb27e1ad8774db27113ea171f652a5cef11aa196b6",
      "original_line": 9,
      "original_line_sha256": "bec48533b2849b7213e1974bdb69e3fb2405533d0a7343c9a4c8a4f6f350688f",
      "question_compact_sha256": "2ee1366aff5c0ff43d0d529ff1678bf30c78e18197704e8fbbd86076547a06ab",
      "english_row_compact_sha256": "d0bac3a8f895a8264c8894e782fe22cc32e219f9ecddd311e1857ddd25dade1f",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261026-r03-cauchy-sequence-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261026-r03-cauchy-sequence-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-cauchy-sequence": {
        "sha256": "f833bc771b9e0a259993b3560086625a99bfce9069edf8b08acd37a09b08c004",
        "english_sha256": "99682dd196d4595b954c7f7161c3deb478e8b66887dc9eba68ebb95204765b5f",
        "original_path": "public/data/mko/cauchy-sequence.json",
        "english_path": "public/data/i18n/editions/mko-cauchy-sequence.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c6ae588be9bc8b3dd8461be32c073abfa25a0f827b4602cf28bf291ce532d507",
    "original_candidate_sha256": "e5c6849c0af4ddce7b7f8da46e4cc6916403eac10e8d84493a04772488108a24",
    "original_candidate_line_sha256": "e5c6849c0af4ddce7b7f8da46e4cc6916403eac10e8d84493a04772488108a24"
  },
  "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-p24",
    "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": "f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff",
      "question_first_witness_sha256": "49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 20,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "bee08726937276b54f5236a29ee87f4bad5ba092ae00ed85ac4a4cf7ae925904",
      "arithmetic_steps": 9
    },
    "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."
  }
}