{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261026-r03-cauchy-sequence-0041",
  "pilot_number": "R0023-12191",
  "number": 12191,
  "family_id": "qf-mod26-cauchy-ultimately-constant-certificate",
  "title": {
    "zh-Hant": "第 12191 題：Cauchy序列",
    "en": "Question 12191: An Explicit Infinite Constant Tail Supplies a Cauchy Witness"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261026-r03-cauchy-sequence-0041",
    "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-ultimately-constant-certificate；資料={\"prefix\":[[3,1],[16,2],[-2,1]],\"tail\":[-1,2]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([3, 8, -2], -1/2, 3, 任意epsilon>0取N=prefix長度, 是)",
      "explanation_zh": "結構深度核對=([3, 8, -2], -1/2, 3, 任意epsilon>0取N=prefix長度, 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": 2,
      "skills": [
        "Cauchy序列"
      ]
    },
    "variant": {
      "family_id": "qf-mod26-cauchy-ultimately-constant-certificate",
      "seed": 202610263041,
      "parameters": {
        "prefix": [
          [
            3,
            1
          ],
          [
            16,
            2
          ],
          [
            -2,
            1
          ]
        ],
        "tail": [
          -1,
          2
        ],
        "domain": "Certified_real_Cauchy_sequence_slices",
        "operation": "cauchy-ultimately-constant-certificate",
        "semantic_template": "qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[3,1],[16,2],[-2,1]],\"tail\":[-1,2]}"
      }
    },
    "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、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"
    }
  },
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    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261026-r03-cauchy-sequence-0041",
    "batch_id": "batch-20261013-advance-20261026-r03-cauchy-sequence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "318b99d80f3b645775fa37dc2ca18e76481e1b1f3dea840640c48b01345f1be9",
      "family_id": "qf-mod26-cauchy-ultimately-constant-certificate",
      "seed": 202610263041,
      "semantic_template": "qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[3,1],[16,2],[-2,1]],\"tail\":[-1,2]}"
    },
    "fields": {
      "title": "An Explicit Infinite Constant Tail Supplies a Cauchy Witness",
      "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 real sequence has finite prefix=[3/1, 16/2, -2/1] followed forever by tail=-1/2. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "([3, 8, -2], -1/2, 3, for any epsilon>0 choose N=prefix length, Yes)",
        "explanation": "For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 8, -2], -1/2, 3, for any epsilon>0 choose N=prefix length, Yes)."
      },
      "skills": [
        "An Explicit Infinite Constant Tail Supplies a Cauchy 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": "保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[3,1],[16,2],[-2,1]],\"tail\":[-1,2]}。請依 [prefix,tail,N,witness,cauchy] 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 real sequence has finite prefix=[3/1, 16/2, -2/1] followed forever by tail=-1/2. Give (normalized prefix,tail,N,witness,cauchy?). Use exact reduced rational outputs with positive denominators; omit /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "prefix=[3, 8, -2];all infinite later terms explicitly=-1/2",
      "after": "for every epsilon>0 choose N=3;all m,n>=N have distance0<epsilon",
      "relation": "implies",
      "rule_id": "explicit_infinite_constant_tail_Cauchy_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是題目明示永遠常值的尾段，不是由有限樣本猜規律。相同尾值差0，N與epsilon無關仍對每個正epsilon有效。",
        "en": "The problem explicitly declares an infinite constant tail;it is not inferred from a sample.Equal tail values give distance0,and the epsilon-independent cutoff works for every positive epsilon."
      },
      "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": [
      [
        "3",
        "8",
        "-2"
      ],
      "-1/2",
      3,
      "任意epsilon>0取N=prefix長度",
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([3, 8, -2], -1/2, 3, 任意epsilon>0取N=prefix長度, 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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    "original_candidate_sha256": "dc454c255980315d7083dbdd2c4d44a255ef45acb76dbe3b87a90c7768e63ac2",
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      "formalization": "not_formalized",
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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."
  },
  "revision": {
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    "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.",
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}