{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261005-advance-20261006-r06-area-0087",
  "pilot_number": "R0004-02762",
  "number": 2762,
  "family_id": "qf-area-two-rectangle-union",
  "title": {
    "zh-Hant": "第 2762 題：兩個整數半開矩形聯集的面積",
    "en": "Question 2762: Area of a Union of Two Half-Open Integer Rectangles"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261005-advance-20261006-r06-area-0087",
    "batch_id": "batch-20261005-advance-20261006-r06-area-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-area"
    ],
    "title_zh": "兩個整數半開矩形聯集的面積",
    "stem_zh": "在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,3)×[0,4)，B=[0,3)×[0,4)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。",
    "answer": {
      "kind": "short_text",
      "value": "12",
      "explanation_zh": "面積=12 平方單位。本產生端把兩矩形分解為整數單位格，合併後去重計數；相離、觸邊、重疊、包含及相同矩形都按同一規則，不宣稱一般面積理論已形式化。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "兩個整數半開矩形聯集的面積"
      ]
    },
    "variant": {
      "family_id": "qf-area-two-rectangle-union",
      "seed": 202610066087,
      "parameters": {
        "frame": "euclidean_orthonormal_equal_units",
        "boundary": "half_open",
        "A": [
          0,
          3,
          0,
          4
        ],
        "B": [
          0,
          3,
          0,
          4
        ],
        "operation": "two_rectangle_union",
        "semantic_template": "qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"3\",\"0\",\"4\"],[\"0\",\"3\",\"0\",\"4\"]],\"operation\":\"two_rectangle_union\"}"
      }
    },
    "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": "2026-10-05",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261005-advance-20261006-r06-area-0087",
    "batch_id": "batch-20261005-advance-20261006-r06-area-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1c9a052221f71b0236de85f53789ddb0f21e803afc2ff8568502282dc486fae5",
      "family_id": "qf-area-two-rectangle-union",
      "seed": 202610066087,
      "semantic_template": "qf-area-two-rectangle-union/{\"inputs\":[[\"0\",\"3\",\"0\",\"4\"],[\"0\",\"3\",\"0\",\"4\"]],\"operation\":\"two_rectangle_union\"}"
    },
    "fields": {
      "title": "Area of a Union of Two Half-Open Integer Rectangles",
      "stem": "Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 3)×[0, 4) and B=[0, 3)×[0, 4) are half-open rectangles. Find area(A∪B) in squared length units.",
      "answer": {
        "kind": "short_text",
        "value": "12",
        "explanation": "For each axis, the overlap extent is max(0,min(upper endpoints)−max(lower endpoints)). Use area(A)+area(B)−area(A∩B) to obtain 12 squared units. Disjoint or merely touching rectangles have zero overlap area."
      },
      "skills": [
        "Area of a Union of Two Half-Open Integer Rectangles"
      ],
      "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": "在等單位正交座標系中，兩軸使用同一長度單位，半開矩形 A=[0,3)×[0,4)，B=[0,3)×[0,4)。求 A∪B 的面積平方單位數，端點有序且允許空矩形。",
        "en": "Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 3)×[0, 4) and B=[0, 3)×[0, 4) are half-open rectangles. Find area(A∪B) in squared length units."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "orthonormal equal-length coordinate units;half-open regions",
      "after": "area in squared length units",
      "relation": "implies",
      "rule_id": "half_open_area_convention",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩軸正交且使用相同長度單位；半開邊界避免相鄰單位格重複包含邊，面積單位是平方單位而非長度。",
        "en": "Axes are orthogonal with equal length units;half-open boundaries make adjacent unit cells disjoint.Area uses squared units,not length units."
      },
      "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": "A width := sub(3, 0)",
      "after": "A width = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有序端點的右減左是非負寬度，不把原點位移當面積。",
        "en": "Right minus left is the nonnegative width;translation of the origin contributes no area."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A width，不以小數近似替代。",
          "en": "Recompute A width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "0"
        ],
        "output": "3",
        "label": "A width"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "A height := sub(4, 0)",
      "after": "A height = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "上減下是非負高度，必使用同一矩形的端點。",
        "en": "Top minus bottom is the nonnegative height,using endpoints of the same rectangle."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A height，不以小數近似替代。",
          "en": "Recompute A height with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "0"
        ],
        "output": "4",
        "label": "A height"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "A area := mul(3, 4)",
      "after": "A area = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "矩形面積為寬乘高；任一零邊長表示空的半開矩形，面積0。",
        "en": "Rectangle area is width times height;a zero side makes the half-open rectangle empty,with area0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A area，不以小數近似替代。",
          "en": "Recompute A area with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "4"
        ],
        "output": "12",
        "label": "A area"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "B width := sub(3, 0)",
      "after": "B width = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有序端點的右減左是非負寬度，不把原點位移當面積。",
        "en": "Right minus left is the nonnegative width;translation of the origin contributes no area."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B width，不以小數近似替代。",
          "en": "Recompute B width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "0"
        ],
        "output": "3",
        "label": "B width"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "B height := sub(4, 0)",
      "after": "B height = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "上減下是非負高度，必使用同一矩形的端點。",
        "en": "Top minus bottom is the nonnegative height,using endpoints of the same rectangle."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B height，不以小數近似替代。",
          "en": "Recompute B height with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "0"
        ],
        "output": "4",
        "label": "B height"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "B area := mul(3, 4)",
      "after": "B area = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "矩形面積為寬乘高；任一零邊長表示空的半開矩形，面積0。",
        "en": "Rectangle area is width times height;a zero side makes the half-open rectangle empty,with area0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B area，不以小數近似替代。",
          "en": "Recompute B area with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "4"
        ],
        "output": "12",
        "label": "B area"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "overlap raw width := sub(3, 0)",
      "after": "overlap raw width = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交集左右界分別取較小右界與較大左界。",
        "en": "Use the smaller right endpoint and larger left endpoint for the intersection."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 overlap raw width，不以小數近似替代。",
          "en": "Recompute overlap raw width with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "0"
        ],
        "output": "3",
        "label": "overlap raw width"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "overlap raw height := sub(4, 0)",
      "after": "overlap raw height = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交集上下界同樣取內側界線。",
        "en": "Use the inner vertical endpoints in the same way."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 overlap raw height，不以小數近似替代。",
          "en": "Recompute overlap raw height with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "0"
        ],
        "output": "4",
        "label": "overlap raw height"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "raw overlap widths=(3,4)",
      "after": "nonnegative overlap widths=(3,4)",
      "relation": "implies",
      "rule_id": "empty_intersection_clamp",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相離或只觸邊時交集面積0；不能相乘兩個負寬度得到虛構正面積。",
        "en": "Disjoint or edge-touching rectangles have intersection area0;two negative raw widths must not create a fictitious positive area."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "intersection area := mul(3, 4)",
      "after": "intersection area = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先截取非負交長，再相乘求交集面積。",
        "en": "Multiply the clamped nonnegative intersection lengths."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 intersection area，不以小數近似替代。",
          "en": "Recompute intersection area with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "4"
        ],
        "output": "12",
        "label": "intersection area"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "sum of rectangle areas := sum(12, 12)",
      "after": "sum of rectangle areas = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把兩個矩形面積相加，此時交集被算了兩次。",
        "en": "First add the rectangle areas;the intersection is then counted twice."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum of rectangle areas，不以小數近似替代。",
          "en": "Recompute sum of rectangle areas with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "12"
        ],
        "output": "24",
        "label": "sum of rectangle areas"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "union area := sub(24, 12)",
      "after": "union area = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用有限容斥減去一次交集，使每個區域只計一次。",
        "en": "Finite inclusion-exclusion subtracts the intersection once,so each region is counted once."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 union area，不以小數近似替代。",
          "en": "Recompute union area with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "24",
          "12"
        ],
        "output": "12",
        "label": "union area"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "A∪B=(A\\B) disjoint-union B",
      "after": "area(A∪B)=12",
      "relation": "implies",
      "rule_id": "rectangle_union_inclusion_exclusion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "包含、相同、空矩形、相離或交疊皆適用，不能在有交集時直接相加。",
        "en": "The rule covers containment,equal or empty rectangles,disjointness and overlap;plain addition is invalid when positive-area overlap exists."
      },
      "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",
    "rectangle_areas": [
      "12",
      "12"
    ],
    "overlap_widths": [
      "3",
      "4"
    ],
    "intersection_area": "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-20261005-advance-20261006-r06-area-foundations",
      "batch_path": "public/data/questions/batches/batch-20261005-advance-20261006-r06-area-foundations/questions.jsonl",
      "batch_sha256": "ae5e68190a9fbefa299612f8a40e9c091f12108ad9798ec677bf34c4df61c45e",
      "original_line": 87,
      "original_line_sha256": "32d02066c65c810544c975b9591894205ecc4587d6a3819a0564fd02bb1fb572",
      "question_compact_sha256": "1c9a052221f71b0236de85f53789ddb0f21e803afc2ff8568502282dc486fae5",
      "english_row_compact_sha256": "518a12dbde3b6574fb7da0684d8e814e16746cb958452a0b7710bcf5b179a7f5",
      "english_manifest_path": "public/data/i18n/editions/batch-20261005-advance-20261006-r06-area-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261005-advance-20261006-r06-area-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-area": {
        "sha256": "dd727d645461b67b46876e9fedc6a33242ba8cb9ef182cf6bd74290b45ac1b58",
        "english_sha256": "3c701b4dcaf9e9568dd937b5470cb18f890bd452abc9d2716d75a66401416371",
        "original_path": "public/data/mko/area.json",
        "english_path": "public/data/i18n/editions/mko-area.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b254245aa10315455f756d4620a6ac0243e0146bccea5d89e872f1359bb32a1d",
    "original_candidate_sha256": "463c7bc000077eddbe74355fb42655073c97a050193d35754572d29ea80667b5",
    "original_candidate_line_sha256": "463c7bc000077eddbe74355fb42655073c97a050193d35754572d29ea80667b5"
  },
  "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-p5",
    "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": "312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56",
      "question_first_witness_sha256": "831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 14,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "6971f5d975c62204e643f40bf16e215a838f2316dcdc9c6851fc4b6d54bab917",
      "arithmetic_steps": 11
    },
    "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."
  }
}