{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261012-r01-mathematical-induction-0086",
  "pilot_number": "R0009-05461",
  "number": 5461,
  "family_id": "qf-density12-induction-range-obligations",
  "title": {
    "zh-Hant": "第 5461 題：有限已檢範圍的覆蓋與下一目標",
    "en": "Question 5461: Finite Coverage and Next Uncovered Index"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261012-r01-mathematical-induction-0086",
    "batch_id": "batch-20261008-advance-20261012-r01-mathematical-induction-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-mathematical-induction"
    ],
    "title_zh": "有限已檢範圍的覆蓋與下一目標",
    "stem_zh": "N從0起；有限歸納協議；已逐項檢查P(0)到P(9)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。",
    "answer": {
      "kind": "expression",
      "value": "(10, 9, 10)",
      "explanation_zh": "有限範圍覆蓋=(10, 9, 10)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "有限已檢範圍的覆蓋與下一目標"
      ]
    },
    "variant": {
      "family_id": "qf-density12-induction-range-obligations",
      "seed": 202610121086,
      "parameters": {
        "through": 9,
        "operation": "density12_induction_range_obligations",
        "domain": "finite_induction_protocol",
        "semantic_template": "qf-density12-induction-range-obligations/9"
      }
    },
    "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-0086",
    "batch_id": "batch-20261008-advance-20261012-r01-mathematical-induction-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "eb28f76186bd2ad9a4fd39ace24f91be2af2ff81a8ceaf91e75d6347424a4b90",
      "family_id": "qf-density12-induction-range-obligations",
      "seed": 202610121086,
      "semantic_template": "qf-density12-induction-range-obligations/9"
    },
    "fields": {
      "title": "Finite Coverage and Next Uncovered Index",
      "stem": "N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(9) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index).",
      "answer": {
        "kind": "expression",
        "value": "(10, 9, 10)",
        "explanation": "Inclusive zero-based coverage has one more index than its last index; adjacent steps equal that last index, and later indices remain unproved. The exact result, retaining the requested order, is (10, 9, 10)."
      },
      "skills": [
        "Finite Coverage and Next Uncovered Index"
      ],
      "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；已逐項檢查P(0)到P(9)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。",
        "en": "N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. P(0) through P(9) have been checked individually. Give (number of covered P indices, number of adjacent steps, next uncovered target index)."
      }
    }
  ],
  "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": "covered inclusive index count := sum(9, 1)",
      "after": "covered inclusive index count = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "P(0)到P(through)含兩端，共through+1個索引。",
        "en": "P(0) through P(through) includes both endpoints,so there are through+1 indices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 covered inclusive index count，不以小數近似替代。",
          "en": "Recompute covered inclusive index count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "1"
        ],
        "output": "10",
        "label": "covered inclusive index count"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "covered indices0,...,9",
      "after": "9 adjacent steps;next uncovered target=10",
      "relation": "implies",
      "rule_id": "finite_range_obligation_counts",
      "conditions": [],
      "explanation": {
        "zh-Hant": "步數等於末端索引，下一目標是through+1；只計算題目宣告的覆蓋，不冒稱已驗證各P。",
        "en": "The adjacent-step count equals the final index and the next target is through+1.This counts declared coverage,not independent verification of each P."
      },
      "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",
      9,
      "10"
    ]
  },
  "conclusion": {
    "canonical_value": "(10, 9, 10)",
    "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": "2c4d2a930f5e69af6997d1eda67562ef5208035b1ca63ff84393847fe647fd8a",
      "question_compact_sha256": "eb28f76186bd2ad9a4fd39ace24f91be2af2ff81a8ceaf91e75d6347424a4b90",
      "english_row_compact_sha256": "b979ed11ebc2937c2fe43157c3deabe4cec1a912e4763d04808ae5a0423d6b61",
      "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"
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    "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": "6cfda01e3075b431612d59c165486ddb7c3d1cbc8fcdc451529d9dc7fc6bd901",
    "original_candidate_line_sha256": "6cfda01e3075b431612d59c165486ddb7c3d1cbc8fcdc451529d9dc7fc6bd901"
  },
  "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-20261026-005-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": "62ded4cf774dd7ce0dcbc50f9941dee4e867180f964c57e244780ec4f7394ada",
      "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."
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}