{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261026-r03-cauchy-sequence-0099",
  "pilot_number": "R0023-12124",
  "number": 12124,
  "family_id": "qf-mod26-cauchy-alternating-counterexample",
  "title": {
    "zh-Hant": "第 12124 題：Cauchy序列",
    "en": "Question 12124: Adjacent Opposite Terms Refute Cauchy Behavior"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261026-r03-cauchy-sequence-0099",
    "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-alternating-counterexample；資料={\"amplitude\":24,\"N\":47}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(24, 47, 48, 24, 48, 是, 否)",
      "explanation_zh": "結構深度核對=(24, 47, 48, 24, 48, 是, 否)；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": 3,
      "skills": [
        "Cauchy序列"
      ]
    },
    "variant": {
      "family_id": "qf-mod26-cauchy-alternating-counterexample",
      "seed": 202610263099,
      "parameters": {
        "amplitude": 24,
        "N": 47,
        "domain": "Certified_real_Cauchy_sequence_slices",
        "operation": "cauchy-alternating-counterexample",
        "semantic_template": "qf-mod26-cauchy-alternating-counterexample/{\"N\":47,\"amplitude\":24}"
      }
    },
    "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-0099",
    "batch_id": "batch-20261013-advance-20261026-r03-cauchy-sequence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "77820513e5beec9080bf675763ba55f0333d16e0386333b09314e2abc4024ff4",
      "family_id": "qf-mod26-cauchy-alternating-counterexample",
      "seed": 202610263099,
      "semantic_template": "qf-mod26-cauchy-alternating-counterexample/{\"N\":47,\"amplitude\":24}"
    },
    "fields": {
      "title": "Adjacent Opposite Terms Refute 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. The infinite sequence alternates between amplitude and its negative, with positive amplitude=24; arbitrary proposed cutoff N=47. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property.",
      "answer": {
        "kind": "expression",
        "value": "(24, 47, 48, 24, 48, Yes, No)",
        "explanation": "The adjacent terms at N and N+1 are both beyond the cutoff and have distance twice the amplitude, hence fail the strict epsilon bound. The same construction works for every N at one fixed positive epsilon: counterexample=Yes and cauchy=No. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (24, 47, 48, 24, 48, Yes, No)."
      },
      "skills": [
        "Adjacent Opposite Terms Refute 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-alternating-counterexample；資料={\"amplitude\":24,\"N\":47}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,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 sequence alternates between amplitude and its negative, with positive amplitude=24; arbitrary proposed cutoff N=47. Give [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Use next_index=N+1 and epsilon=amplitude. The final No is the Cauchy flag, not the negated property."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "opposite adjacent tail distance := mul(2, 24)",
      "after": "opposite adjacent tail distance = 48",
      "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 opposite adjacent tail distance，不以小數近似替代。",
          "en": "Recompute opposite adjacent tail distance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "24"
        ],
        "output": "48",
        "label": "opposite adjacent tail distance"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "explicit x_n=(24)*(-1)^n;arbitrary proposed cutoff N=47;epsilon=24",
      "after": "indices=[47, 48];opposite signs give distance=48>=epsilon;counterexample=是;Cauchy=否",
      "relation": "implies",
      "rule_id": "alternating_fixed_epsilon_every_cutoff_counterexample",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定正epsilon=A，對任意N取N與N+1，它們皆在尾段且符號相反，距離2A≥A；故每個N都失敗。最後旗標是Cauchy，不是not_cauchy，沿用已更正來源欄名。",
        "en": "Fix positive epsilon=A.For any cutoff use N and N+1,both in the tail with opposite signs and distance2A≥A.Thus every cutoff fails.The last flag is Cauchy,not not_cauchy,following the corrected source protocol."
      },
      "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": [
      24,
      47,
      48,
      "24",
      48,
      true,
      false
    ]
  },
  "conclusion": {
    "canonical_value": "(24, 47, 48, 24, 48, 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."
  },
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      "original_line": 99,
      "original_line_sha256": "b274eaf4f6e0952f045ede896da9337f666cecc739b90b5ecfab7f95af6ab074",
      "question_compact_sha256": "77820513e5beec9080bf675763ba55f0333d16e0386333b09314e2abc4024ff4",
      "english_row_compact_sha256": "89a220121616e00649c4b1363568c38265cb19cfb9e90ed1d1ec3ad93275e3b6",
      "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"
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    "mko_bindings": {
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        "original_path": "public/data/mko/cauchy-sequence.json",
        "english_path": "public/data/i18n/editions/mko-cauchy-sequence.en.json"
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    "selected_part_sha256": "c6ae588be9bc8b3dd8461be32c073abfa25a0f827b4602cf28bf291ce532d507",
    "original_candidate_sha256": "e4cb93f7a1b105c5d3a9847862363a69349e88cb4a446714034e71161279c886",
    "original_candidate_line_sha256": "e4cb93f7a1b105c5d3a9847862363a69349e88cb4a446714034e71161279c886"
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  "historical_source_state": {
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      "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": {
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      "question_first_witness_sha256": "49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18"
    },
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    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 2,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
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    },
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    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}