{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261012-r01-mathematical-induction-0011",
  "pilot_number": "R0009-05486",
  "number": 5486,
  "family_id": "qf-density12-induction-successor-index-pair",
  "title": {
    "zh-Hant": "第 5486 題：歸納假設與後繼目標索引",
    "en": "Question 5486: Induction-Hypothesis and Successor Indices"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261012-r01-mathematical-induction-0011",
    "batch_id": "batch-20261008-advance-20261012-r01-mathematical-induction-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-mathematical-induction"
    ],
    "title_zh": "歸納假設與後繼目標索引",
    "stem_zh": "N從0起；有限歸納協議；目前n=10。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。",
    "answer": {
      "kind": "expression",
      "value": "(10, 11)",
      "explanation_zh": "假設與後繼目標索引=(10, 11)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "歸納假設與後繼目標索引"
      ]
    },
    "variant": {
      "family_id": "qf-density12-induction-successor-index-pair",
      "seed": 202610121011,
      "parameters": {
        "n": 10,
        "operation": "density12_induction_successor_index_pair",
        "domain": "finite_induction_protocol",
        "semantic_template": "qf-density12-induction-successor-index-pair/10"
      }
    },
    "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": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261012-r01-mathematical-induction-0011",
    "batch_id": "batch-20261008-advance-20261012-r01-mathematical-induction-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "a9ce00daf1e4218484c2adc30c88aadf61137169b83590d76166a26a880f4b55",
      "family_id": "qf-density12-induction-successor-index-pair",
      "seed": 202610121011,
      "semantic_template": "qf-density12-induction-successor-index-pair/10"
    },
    "fields": {
      "title": "Induction-Hypothesis and Successor Indices",
      "stem": "N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=10. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds.",
      "answer": {
        "kind": "expression",
        "value": "(10, 11)",
        "explanation": "Compute the successor of the nonnegative index; identifying scope does not prove the proposition. The exact result, retaining the requested order, is (10, 11)."
      },
      "skills": [
        "Induction-Hypothesis and Successor Indices"
      ],
      "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": "N從0起；有限歸納協議；目前n=10。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。",
        "en": "N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. n=10. Give (index n of hypothesis P, index n+1 of the successor target), without claiming P holds."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "natural indices start at0;finite induction protocol",
      "after": "index bookkeeping and finite values are not proofs of all P(n)",
      "relation": "implies",
      "rule_id": "finite_induction_claim_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "自然數索引不要求序列值非負；有限範圍或後繼索引的核對不自動提供命題真值或所有自然數的歸納證明。",
        "en": "Natural indices do not require nonnegative sequence values.Checking a finite range or a successor index does not supply proposition truth or an induction proof over all naturals."
      },
      "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": "successor target index := sum(10, 1)",
      "after": "successor target index = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從當前非負索引加1得到後繼目標。",
        "en": "Add1 to the current natural index to obtain the successor target."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 successor target index，不以小數近似替代。",
          "en": "Recompute successor target index with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "1"
        ],
        "output": "11",
        "label": "successor target index"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "hypothesis index=10",
      "after": "target index=11",
      "relation": "implies",
      "rule_id": "hypothesis_vs_successor_target",
      "conditions": [],
      "explanation": {
        "zh-Hant": "P(n)是歸納假設的位置，P(n+1)是待證目標；列出兩個索引不表示任一命題已成立。",
        "en": "P(n) marks the induction hypothesis and P(n+1) the target;listing their indices does not assert either proposition."
      },
      "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": [
      10,
      11
    ]
  },
  "conclusion": {
    "canonical_value": "(10, 11)",
    "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": {
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      "batch_id": "batch-20261008-advance-20261012-r01-mathematical-induction-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261012-r01-mathematical-induction-foundations/questions.jsonl",
      "batch_sha256": "4ff961382c9b0e9265156817fa51ccd4f7623e1942ae117f50b21701e1edb9f1",
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      "original_line_sha256": "eae2f31565577d728a9daa7a1990da8663a6b9e5646996fcc33f978c1f2473d9",
      "question_compact_sha256": "a9ce00daf1e4218484c2adc30c88aadf61137169b83590d76166a26a880f4b55",
      "english_row_compact_sha256": "3fc63e02a70a5704d4f961e368e14f54b1c333939f6928674a54aaba799672a5",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261012-r01-mathematical-induction-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261012-r01-mathematical-induction-foundations.en.jsonl"
    },
    "mko_bindings": {
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        "original_path": "public/data/mko/mathematical-induction.json",
        "english_path": "public/data/i18n/editions/mko-mathematical-induction.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "3132b1200be7a5de38191badc52fb1727d4dd9d4c2f26818de2e8a76426d2d3f",
    "original_candidate_sha256": "46758c06bce16329f269982d41370a2c8ea4275849c8280432b155c33f5dfc30",
    "original_candidate_line_sha256": "46758c06bce16329f269982d41370a2c8ea4275849c8280432b155c33f5dfc30"
  },
  "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-20261026-005-p10",
    "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": "71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56",
      "question_first_witness_sha256": "5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 3,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "87670ede8a59683a92011b685d4c17dd02316ff542df700740b945e8b1ee9e18",
      "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."
  }
}