{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r01-natural-number-0063",
  "pilot_number": "R0008-04813",
  "number": 4813,
  "family_id": "qf-density11-natural-interval-size",
  "title": {
    "zh-Hant": "第 4813 題：含零自然閉區間的大小",
    "en": "Question 4813: Natural Closed-Interval Cardinality"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r01-natural-number-0063",
    "batch_id": "batch-20261008-advance-20261011-r01-natural-number-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-natural-number"
    ],
    "title_zh": "含零自然閉區間的大小",
    "stem_zh": "N包含0；lo=0、hi=11。求 {n∈N | lo≤n≤hi} 的大小，端點均含。",
    "answer": {
      "kind": "expression",
      "value": "12",
      "explanation_zh": "N閉區間大小=12。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "含零自然閉區間的大小"
      ]
    },
    "variant": {
      "family_id": "qf-density11-natural-interval-size",
      "seed": 202610111063,
      "parameters": {
        "lo": 0,
        "hi": 11,
        "operation": "density11_natural_interval_size",
        "domain": "N_including_zero",
        "semantic_template": "qf-density11-natural-interval-size/[\"cardinality\",\"12\"]"
      }
    },
    "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-0063",
    "batch_id": "batch-20261008-advance-20261011-r01-natural-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "71a1fbdc59679151bc7fe8936ce171f4a09f8f3fad4de2dafb5e9a5befdd322d",
      "family_id": "qf-density11-natural-interval-size",
      "seed": 202610111063,
      "semantic_template": "qf-density11-natural-interval-size/[\"cardinality\",\"12\"]"
    },
    "fields": {
      "title": "Natural Closed-Interval Cardinality",
      "stem": "N includes 0. lo=0, hi=11. Find |{n∈N: lo≤n≤hi}|, including endpoints.",
      "answer": {
        "kind": "expression",
        "value": "12",
        "explanation": "Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 12."
      },
      "skills": [
        "Natural Closed-Interval Cardinality"
      ],
      "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；lo=0、hi=11。求 {n∈N | lo≤n≤hi} 的大小，端點均含。",
        "en": "N includes 0. lo=0, hi=11. Find |{n∈N: lo≤n≤hi}|, including endpoints."
      }
    }
  ],
  "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": "inclusive interval endpoint difference := sub(11, 0)",
      "after": "inclusive interval endpoint difference = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取上端減下端。",
        "en": "Subtract the lower endpoint from the upper endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inclusive interval endpoint difference，不以小數近似替代。",
          "en": "Recompute inclusive interval endpoint difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "0"
        ],
        "output": "11",
        "label": "inclusive interval endpoint difference"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "inclusive natural interval size := sum(11, 1)",
      "after": "inclusive natural interval size = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "閉區間含兩端，所以再加1。",
        "en": "The closed interval contains both endpoints,so add1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inclusive natural interval size，不以小數近似替代。",
          "en": "Recompute inclusive natural interval size with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "11",
          "1"
        ],
        "output": "12",
        "label": "inclusive natural interval size"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "lo=0;hi=11",
      "after": "natural interval size=12",
      "relation": "implies",
      "rule_id": "ordered_natural_interval_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "上端小於下端時集合為空且大小0，不能交換端點製造另一集合。",
        "en": "If the upper endpoint is below the lower,the set is empty with size0;do not swap endpoints to manufacture another set."
      },
      "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": 12
  },
  "conclusion": {
    "canonical_value": "12",
    "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": 63,
      "original_line_sha256": "dbc125d663898bca4deb4eecaebe96e7fb02b032b0957c85285aaac6a6c3e5e8",
      "question_compact_sha256": "71a1fbdc59679151bc7fe8936ce171f4a09f8f3fad4de2dafb5e9a5befdd322d",
      "english_row_compact_sha256": "f93e1ae85d9b1aafd74c4ba626500151f2aa621ce8960737617deed18080e5ff",
      "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": "bfffdbf53cb720d68d78e50de7674b9113578a76040ef51dd72bf9dbbe6bc8ba",
    "original_candidate_line_sha256": "bfffdbf53cb720d68d78e50de7674b9113578a76040ef51dd72bf9dbbe6bc8ba"
  },
  "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": "bd1ed85b09fe83633ab45ade15479ad36feb58bb856c0e76f4c1eaa01348b115",
      "arithmetic_steps": 2
    },
    "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."
  }
}