{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r01-natural-number-0020",
  "pilot_number": "R0008-04895",
  "number": 4895,
  "family_id": "qf-density11-natural-successor-window",
  "title": {
    "zh-Hant": "第 4895 題：含零自然數的後繼窗",
    "en": "Question 4895: Inclusive Natural Successor Window"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r01-natural-number-0020",
    "batch_id": "batch-20261008-advance-20261011-r01-natural-number-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-natural-number"
    ],
    "title_zh": "含零自然數的後繼窗",
    "stem_zh": "N包含0；n=18、k=1。列含起點n並做k次後繼的完整窗，按次序寫整數列表。",
    "answer": {
      "kind": "expression",
      "value": "[18, 19]",
      "explanation_zh": "含起點後繼窗=[18, 19]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "含零自然數的後繼窗"
      ]
    },
    "variant": {
      "family_id": "qf-density11-natural-successor-window",
      "seed": 202610111020,
      "parameters": {
        "n": 18,
        "k": 1,
        "operation": "density11_natural_successor_window",
        "domain": "N_including_zero",
        "semantic_template": "qf-density11-natural-successor-window/[18,1]"
      }
    },
    "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-20261011-r01-natural-number-0020",
    "batch_id": "batch-20261008-advance-20261011-r01-natural-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "5aaae4cae8964b73feafba0263930ca072b342d2c0a4b3ca11584bb5288c9919",
      "family_id": "qf-density11-natural-successor-window",
      "seed": 202610111020,
      "semantic_template": "qf-density11-natural-successor-window/[18,1]"
    },
    "fields": {
      "title": "Inclusive Natural Successor Window",
      "stem": "N includes 0. n=18, k=1. List the complete ordered window beginning at n and applying the successor k times.",
      "answer": {
        "kind": "expression",
        "value": "[18, 19]",
        "explanation": "Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [18, 19]."
      },
      "skills": [
        "Inclusive Natural Successor Window"
      ],
      "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包含0；n=18、k=1。列含起點n並做k次後繼的完整窗，按次序寫整數列表。",
        "en": "N includes 0. n=18, k=1. List the complete ordered window beginning at n and applying the successor k times."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "N={0,1,2,...}",
      "after": "0 belongs;negative and nonintegral rational values do not",
      "relation": "implies",
      "rule_id": "natural_zero_convention",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本題自然數包含0；與有理數值、部分前驅及閉區間的規則分開判定。",
        "en": "This convention includes0 in N;distinguish membership,rational values,partial predecessors and closed intervals."
      },
      "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": "initial n=18;successor count k=1",
      "after": "[18]",
      "relation": "implies",
      "rule_id": "successor_window_initial",
      "conditions": [],
      "explanation": {
        "zh-Hant": "含起點，k=0仍有一項；不是先做一次後繼再開始列。",
        "en": "Include the initial point;even k=0 has one item.Do not apply a successor before starting the list."
      },
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "successor step1 := sum(18, 1)",
      "after": "successor step1 = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只把上一個自然數加1。",
        "en": "Add1 to the preceding natural at each step."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 successor step1，不以小數近似替代。",
          "en": "Recompute successor step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "18",
          "1"
        ],
        "output": "19",
        "label": "successor step1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "[18, 19]",
      "after": "1 successors and 2 total entries",
      "relation": "implies",
      "rule_id": "successor_window_length",
      "conditions": [],
      "explanation": {
        "zh-Hant": "做k次後繼得到k+1項，保留完整順序及起點。",
        "en": "Applying k successors produces k+1 entries,retaining the full order and initial point."
      },
      "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": [
      18,
      19
    ]
  },
  "conclusion": {
    "canonical_value": "[18, 19]",
    "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-20261008-advance-20261011-r01-natural-number-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261011-r01-natural-number-foundations/questions.jsonl",
      "batch_sha256": "ccaa1444bd0d6cb142b205d76ade261dd201940cf4971f1370270518e95618a0",
      "original_line": 20,
      "original_line_sha256": "7f648ec22f47cce72d467553b753fb6a1cfdd9b082e9e38222f89ba395fc9a8f",
      "question_compact_sha256": "5aaae4cae8964b73feafba0263930ca072b342d2c0a4b3ca11584bb5288c9919",
      "english_row_compact_sha256": "83838ec0265b117236f6807834846aa13d58382e4c1da5ef16fa6245c1a989e2",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261011-r01-natural-number-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261011-r01-natural-number-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-natural-number": {
        "sha256": "222b72502fb661f3372012f602fd2f1eeef754b9e2069498c6b24234778730c4",
        "english_sha256": "f7444a7bfebcfed594478ed5e7c0f36cb1985114de58b5b473beddd1961c0bbc",
        "original_path": "public/data/mko/natural-number.json",
        "english_path": "public/data/i18n/editions/mko-natural-number.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6dc34e24b0f43c50e49365ca1a1153175f8661d5290c9236df5527bb047157eb",
    "original_candidate_sha256": "9b7abe7037489861d74060959d9ae2b825787ca16edcfa5cefd2fa8843d18915",
    "original_candidate_line_sha256": "9b7abe7037489861d74060959d9ae2b825787ca16edcfa5cefd2fa8843d18915"
  },
  "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-20261026-004-p9",
    "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": "b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289",
      "question_first_witness_sha256": "caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 4,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "575acd35e5016a102418a79b88a5f2a7227d1e6df217bbd483824a7cc83a25a7",
      "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."
  }
}