{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261027-r03-complete-metric-space-0068",
  "pilot_number": "R0024-12818",
  "number": 12818,
  "family_id": "qf-mod27-complete-alternating-noncauchy",
  "title": {
    "zh-Hant": "第 12818 題：完備度量空間",
    "en": "Question 12818: A Non-Cauchy Sequence Does Not Contradict Space Completeness"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261027-r03-complete-metric-space-0068",
    "batch_id": "batch-20261014-advance-20261027-r03-complete-metric-space-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-complete-metric-space"
    ],
    "title_zh": "完備度量空間",
    "stem_zh": "完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"left\":1,\"right\":2,\"N\":35}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(1, 2, 35, 36, 14, 14, 是, 否)",
      "explanation_zh": "Wave4根節點核對=(1, 2, 35, 36, 14, 14, 是, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；有限metric用最小正距離作全稱證書；抽樣序列或有限尾段不能證一般完備。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "完備度量空間"
      ]
    },
    "variant": {
      "family_id": "qf-mod27-complete-alternating-noncauchy",
      "seed": 202610273068,
      "parameters": {
        "n": 4,
        "metric": [
          [
            0,
            8,
            12,
            14
          ],
          [
            8,
            0,
            14,
            16
          ],
          [
            12,
            14,
            0,
            20
          ],
          [
            14,
            16,
            20,
            0
          ]
        ],
        "left": 1,
        "right": 2,
        "N": 35,
        "domain": "Certified_complete_finite_metric_space",
        "operation": "complete-alternating-noncauchy",
        "semantic_template": "qf-mod27-complete-alternating-noncauchy/{\"N\":35,\"left\":1,\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"n\":4,\"right\":2}"
      }
    },
    "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、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式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-20261027-r03-complete-metric-space-0068",
    "batch_id": "batch-20261014-advance-20261027-r03-complete-metric-space-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "7ee0e714b7486965faac10b41bdc9bc015cda7e73e9340590363b42702ac0a0f",
      "family_id": "qf-mod27-complete-alternating-noncauchy",
      "seed": 202610273068,
      "semantic_template": "qf-mod27-complete-alternating-noncauchy/{\"N\":35,\"left\":1,\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"n\":4,\"right\":2}"
    },
    "fields": {
      "title": "A Non-Cauchy Sequence Does Not Contradict Space Completeness",
      "stem": "Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 14, 16], [12, 14, 0, 20], [14, 16, 20, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=1 and right=2, with arbitrary proposed cutoff N=35. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.",
      "answer": {
        "kind": "expression",
        "value": "(1, 2, 35, 36, 14, 14, Yes, No)",
        "explanation": "The adjacent terms at N and N+1 are the two distinct points in one order, so symmetry gives distance=epsilon, not distance<epsilon. The same fixed positive epsilon defeats every cutoff: counterexample=Yes and cauchy=No. Being non-Cauchy, this sequence does not refute completeness of the space, which concerns only Cauchy sequences. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 2, 35, 36, 14, 14, Yes, No)."
      },
      "skills": [
        "A Non-Cauchy Sequence Does Not Contradict Space Completeness"
      ],
      "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": "完備性量化空間內每個Cauchy序列。操作=complete-alternating-noncauchy；資料={\"n\":4,\"metric\":[[0,8,12,14],[8,0,14,16],[12,14,0,20],[14,16,20,0]],\"left\":1,\"right\":2,\"N\":35}。請依 [left,right,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。",
        "en": "Use the complete qualified finite metric table=[[0, 8, 12, 14], [8, 0, 14, 16], [12, 14, 0, 20], [14, 16, 20, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=1 and right=2, with arbitrary proposed cutoff N=35. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "carrier Fin(4);complete nonnegative integer metric candidate=[[0, 8, 12, 14], [8, 0, 14, 16], [12, 14, 0, 20], [14, 16, 20, 0]]",
      "after": "zero iff equal=是;symmetric=是",
      "relation": "implies",
      "rule_id": "complete_metric_separation_and_symmetry",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核整表的非負、對角與非對角分離、對稱，不只核本題用到的兩格；至少兩點是本最小正距離協定前提，不是單點空間不完備。",
        "en": "First qualify nonnegativity,diagonal/off-diagonal separation and symmetry on the entire table,not only queried cells.At least two points is a premise of this minimum-positive-distance protocol,not a claim that a singleton is incomplete."
      },
      "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": "endpoints=[0, 0];direct distance=0;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[0, 16, 24, 28];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "endpoints=[0, 1];direct distance=8;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[8, 8, 26, 30];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "endpoints=[0, 2];direct distance=12;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[12, 22, 12, 34];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "endpoints=[0, 3];direct distance=14;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[14, 24, 32, 14];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "endpoints=[1, 0];direct distance=8;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[8, 8, 26, 30];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "endpoints=[1, 1];direct distance=0;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[16, 0, 28, 32];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "endpoints=[1, 2];direct distance=14;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[20, 14, 14, 36];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "endpoints=[1, 3];direct distance=16;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[22, 16, 34, 16];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "endpoints=[2, 0];direct distance=12;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[12, 22, 12, 34];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "endpoints=[2, 1];direct distance=14;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[20, 14, 14, 36];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "endpoints=[2, 2];direct distance=0;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[24, 28, 0, 40];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "endpoints=[2, 3];direct distance=20;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[26, 30, 20, 20];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "endpoints=[3, 0];direct distance=14;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[14, 24, 32, 14];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "endpoints=[3, 1];direct distance=16;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[22, 16, 34, 16];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "endpoints=[3, 2];direct distance=20;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[26, 30, 20, 20];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "endpoints=[3, 3];direct distance=0;all intermediate labels=[0, 1, 2, 3]",
      "after": "all two-leg distances=[28, 32, 40, 0];direct<=every two-leg=是",
      "relation": "implies",
      "rule_id": "complete_metric_all_triangle_intermediates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對每個有序端點對遍歷每個中間點，三角條件要全部成立；完整有限核查不是幾個三角形樣本。",
        "en": "For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles."
      },
      "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": "all off-diagonal positive distances=[8, 12, 14, 8, 14, 16, 12, 14, 20, 14, 16, 20]",
      "after": "minimum positive distance delta=8>0",
      "relation": "implies",
      "rule_id": "finite_metric_positive_minimum_separation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限非空正距離清單有正最小值，這是後續全稱Cauchy證書的空間尺度，不是從某條序列估出的門檻。",
        "en": "A finite nonempty list of positive distances has a positive minimum.This is the space-wide scale for the universal Cauchy certificate,not a threshold guessed from a sequence."
      },
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "adjacent alternating tail index := sum(35, 1)",
      "after": "adjacent alternating tail index = 36",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 adjacent alternating tail index，不以小數近似替代。",
          "en": "Recompute adjacent alternating tail index with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "35",
          "1"
        ],
        "output": "36",
        "label": "adjacent alternating tail index"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "explicit infinite sequence alternates labels=[1, 2];arbitrary cutoff=35",
      "after": "indices=[35, 36];epsilon=d(left,right)=14>0;actual adjacent distance=14=epsilon",
      "relation": "implies",
      "rule_id": "metric_alternation_fixed_epsilon_strict_failure",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相鄰兩項就是這兩點的某一順序，對稱性給相同距離。固定epsilon=d>0，距離等於epsilon就違反嚴格小於；這個構造對每個N有效。",
        "en": "Adjacent terms are these two points in one order,and symmetry gives the same distance.Fix positive epsilon=d.Equality with epsilon violates strict inequality,and this construction works for every cutoff N."
      },
      "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": "every cutoff defeated at fixed epsilon=14",
      "after": "counterexample=是;Cauchy=否;non-Cauchy sequence does not refute space completeness",
      "relation": "implies",
      "rule_id": "nonCauchy_not_counterexample_to_completeness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完備性只要求Cauchy序列有極限，非Cauchy序列不在前提內。最後欄沿用來源已更正的Cauchy欄名，不是not_cauchy。",
        "en": "Completeness concerns only Cauchy sequences.A non-Cauchy sequence does not meet that premise.The final field uses the corrected source Cauchy label,not not_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,
      2,
      35,
      36,
      14,
      14,
      true,
      false
    ]
  },
  "conclusion": {
    "canonical_value": "(1, 2, 35, 36, 14, 14, 是, 否)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "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-20261027-r03-complete-metric-space-foundations",
      "batch_path": "public/data/questions/errata/ocme-source-label-complete-metric-cauchy-20261015/questions.jsonl",
      "batch_sha256": "3e51402839b678d86535201a10926109f587c20aca57a54eb054fba9dcc90d61",
      "original_line": 68,
      "original_line_sha256": "9a0a92e3d3bc25501e2b428b2c0ff6d149b50cc40da734770aac4b6d81669e81",
      "question_compact_sha256": "7ee0e714b7486965faac10b41bdc9bc015cda7e73e9340590363b42702ac0a0f",
      "english_row_compact_sha256": "462f1600e9dcce7dd3e4089cd07ef45df9ce8db4c1e07b7ab74210698ccc1e20",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261027-r03-complete-metric-space-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261027-r03-complete-metric-space-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-complete-metric-space": {
        "sha256": "783e45783b1dc07ccecb89f0c76b128f6c4a9a3da036e1678e908f2250530c73",
        "english_sha256": "505d9fc575b5de4a835e628a71c116b9f8b9518c9e79dc126f4b2e6f7d011b6d",
        "original_path": "public/data/mko/complete-metric-space.json",
        "english_path": "public/data/i18n/editions/mko-complete-metric-space.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "17a80c8482f64498e5a88bd4368d0a11b92d45c259784a0a536a29fbb98662d5",
    "original_candidate_sha256": "2fe9fa4549a530cf632d7653000fa1dc1cd5139f14d71a1c00ff899268b26930",
    "original_candidate_line_sha256": "2fe9fa4549a530cf632d7653000fa1dc1cd5139f14d71a1c00ff899268b26930"
  },
  "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-p25",
    "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": "1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35",
      "question_first_witness_sha256": "05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 21,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "d2c3d2ddb1743a32c0ed4e4597e76c60fdc99402e10e0d06d78332f6878e45a9",
      "arithmetic_steps": 1
    },
    "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."
  }
}