{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261008-r01-sequence-limit-0039",
  "pilot_number": "R0056-28739",
  "number": 28739,
  "family_id": "qf-sequence-limit-least-cutoff",
  "title": {
    "zh-Hant": "第 28739 題：含截止項的最小自然尾界",
    "en": "Question 28739: Least Natural Cutoff Including the Cutoff Term"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261008-r01-sequence-limit-0039",
    "batch_id": "batch-20261007-advance-20261008-r01-sequence-limit-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-sequence-limit"
    ],
    "title_zh": "含截止項的最小自然尾界",
    "stem_zh": "從0起算實數列 a_n=L+(-14)/(n+2)，n∈N0，d≥1。ε=4/2>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。",
    "answer": {
      "kind": "short_text",
      "value": "6",
      "explanation_zh": "最小尾界N=6。產生端逐一尋找首個滿足嚴格界的自然索引；倒數誤差單調，不把等號接受或略過截止項。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "含截止項的最小自然尾界"
      ]
    },
    "variant": {
      "family_id": "qf-sequence-limit-least-cutoff",
      "seed": 202610081039,
      "parameters": {
        "c": -14,
        "d": 2,
        "eps_n": 4,
        "eps_d": 2,
        "operation": "least_cutoff",
        "domain": "naturals_from_zero",
        "semantic_template": "qf-sequence-limit-least-cutoff/{\"inputs\":[\"14\",\"2\",\"2/1\"],\"operation\":\"least_cutoff\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-07",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261007-advance-20261008-r01-sequence-limit-0039",
    "batch_id": "batch-20261007-advance-20261008-r01-sequence-limit-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "3e02a0e12ba7dd462995c034ce8f17760387e2ae2d7da87e479add0566d3842b",
      "family_id": "qf-sequence-limit-least-cutoff",
      "seed": 202610081039,
      "semantic_template": "qf-sequence-limit-least-cutoff/{\"inputs\":[\"14\",\"2\",\"2/1\"],\"operation\":\"least_cutoff\"}"
    },
    "fields": {
      "title": "Least Natural Cutoff Including the Cutoff Term",
      "stem": "For an arbitrary fixed real L, u(n)=L+(-14)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=4/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε.",
      "answer": {
        "kind": "short_text",
        "value": "6",
        "explanation": "The nonincreasing tail error reduces this model to |c|q<p(N+d). Its least cutoff is max(0,floor(|c|q/p)−d+1)=6. Equality fails the strict bound; a zero tail permits N=0."
      },
      "skills": [
        "Least Natural Cutoff Including the Cutoff Term"
      ],
      "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": "從0起算實數列 a_n=L+(-14)/(n+2)，n∈N0，d≥1。ε=4/2>0，求最小 N∈N0，使所有 n≥N 都有 |a_n−L|<ε。",
        "en": "For an arbitrary fixed real L, u(n)=L+(-14)/(n+2) on natural indices starting at zero, with d=2≥1. For ε=4/2>0, find the least natural N≥0 for which every n≥N satisfies |u(n)−L|<ε."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-sequence-limit-least-cutoff, least_cutoff]",
      "after": "derived = [least_nonnegative_strict_tail_cutoff]",
      "relation": "implies",
      "rule_id": "R56_source_answer_role_and_EXACT_family_interface",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本題輸出依原scalar／set／是非／fraction角色；有理數整數保留/1。不加未宣告domain，不改中英來源、公式或題目。",
        "en": "Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。",
          "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [-14, 2, naturals_from_zero]",
      "after": "derived = [14, u(n)=L+c/(n+d), |u(n)-L|=|c|/(n+d)]",
      "relation": "implies",
      "rule_id": "Sequence_WHOLE_law_and_positive_increasing_denominator",
      "conditions": [],
      "explanation": {
        "zh-Hant": "題設給所有自然數n>=0的完整law，d>0使n+d正且隨n增大。取絕對值先消L再得誤差，不從有限prefix猜limit；error／cutoff題的L只是任意固定實數，非新增參數。",
        "en": "The premise is the WHOLE law for every n>=0,not a finite prefix.With d>0,the denominator is positive and increasing.Subtract L before taking the absolute value.In error/cutoff tasks L is an arbitrary fixed real,not an invented input parameter."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。",
          "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Sequence ORIGINAL absolute c times epsilon denominator := mul(14, 2)",
      "after": "Sequence ORIGINAL absolute c times epsilon denominator = 28",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Sequence ORIGINAL absolute c times epsilon denominator，不以小數近似替代。",
          "en": "Recompute Sequence ORIGINAL absolute c times epsilon denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "2"
        ],
        "output": "28",
        "label": "Sequence ORIGINAL absolute c times epsilon denominator"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [28, 4]",
      "after": "derived = [7, 0, left=quot*eps_n+rem, 0<=rem<eps_n]",
      "relation": "implies",
      "rule_id": "Sequence_EXACT_floor_quotient_remainder_NO_float_or_ceil",
      "conditions": [],
      "explanation": {
        "zh-Hant": "因en>0，可精確整除得floor與remainder；不能浮點round或改成ceil。衍生乘積／quotient可以超SAFE，只把原輸入維持SAFE。",
        "en": "Because eps_n>0,integer division yields the exact floor and remainder.No floating rounding or ceil replacement is allowed.Derived products/quotients may exceed SAFE;the raw inputs alone are SAFE-bounded."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。",
          "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Sequence least ORIGINAL N plus d := sum(6, 2)",
      "after": "Sequence least ORIGINAL N plus d = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Sequence least ORIGINAL N plus d，不以小數近似替代。",
          "en": "Recompute Sequence least ORIGINAL N plus d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "2"
        ],
        "output": "8",
        "label": "Sequence least ORIGINAL N plus d"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Sequence least ORIGINAL epsilon numerator times N plus d := mul(4, 8)",
      "after": "Sequence least ORIGINAL epsilon numerator times N plus d = 32",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Sequence least ORIGINAL epsilon numerator times N plus d，不以小數近似替代。",
          "en": "Recompute Sequence least ORIGINAL epsilon numerator times N plus d with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "8"
        ],
        "output": "32",
        "label": "Sequence least ORIGINAL epsilon numerator times N plus d"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source roles = [-14, 2, 4, 2, 28, 7]",
      "after": "derived = [6, 32, 28, 是]",
      "relation": "implies",
      "rule_id": "Sequence_LEAST_N_strict_tail_INCLUDING_N_AND_previous_failure",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全尾從N本身開始；正分母單調使全部n>=N合格iff |c|*ed<en*(N+d)。最小整數N=max(0,floor(|c|*ed/en)-d+1)。N>0才驗N-1失敗，N=0不造負index；derived N不限SAFE。",
        "en": "The tail INCLUDES N.Monotonic positive denominators make every n>=N qualify IFF |c|*ed<en*(N+d).The least integer is max(0,floor(|c|*ed/en)-d+1).Only when N>0 test failure at N-1;do not invent a negative index for N=0.Derived N is not SAFE-capped."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。",
          "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "6"
  },
  "conclusion": {
    "canonical_value": "6",
    "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-20261007-advance-20261008-r01-sequence-limit-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261008-r01-sequence-limit-foundations/questions.jsonl",
      "batch_sha256": "124ae93c1492492eb4f3fc0395814be15a2792916927f6f1ab8aaa2eadb2c2b1",
      "original_line": 39,
      "original_line_sha256": "5d7fc11bc152ee059a5fea0042b74ae0f4f00564a4a53d87b8f97e03def6da45",
      "question_compact_sha256": "3e02a0e12ba7dd462995c034ce8f17760387e2ae2d7da87e479add0566d3842b",
      "english_row_compact_sha256": "c64657776fd8a124045f3c9f46b5603cebc0f723e8c2e84a9e2f512d69764e26",
      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261008-r01-sequence-limit-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261008-r01-sequence-limit-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-sequence-limit": {
        "sha256": "3018ee44ad1a41db719f4c5275fd215d436e1dc4e73fdc44993604a211486fff",
        "english_sha256": "ee964b7eba696e940cf9793cbba93912244a2bd9ed11ec3f4e647fafa8fd38ae",
        "original_path": "public/data/mko/sequence-limit.json",
        "english_path": "public/data/i18n/editions/mko-sequence-limit.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "e340023047850507ee3fe2ae1466d606cfcbd26849aa3bb168d53248f38097e5",
    "original_candidate_sha256": "7767bc5dce0b354b9cba7790e6817704f12e4ec4dc8794b1ac7c7e040f1e072b",
    "original_candidate_line_sha256": "7767bc5dce0b354b9cba7790e6817704f12e4ec4dc8794b1ac7c7e040f1e072b"
  },
  "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-20261026-010-p57",
    "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": "8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8",
      "question_first_witness_sha256": "3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "3fbca07714c802ae592f17256e84b6aef76e029cac66d198604ee50d642fa1b6",
      "arithmetic_steps": 3
    },
    "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."
  }
}