{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261005-advance-20261006-r06-area-0059",
  "pilot_number": "R0004-02784",
  "number": 2784,
  "family_id": "qf-area-unit-cell-union",
  "title": {
    "zh-Hant": "第 2784 題：半開單位格聯集的面積",
    "en": "Question 2784: Area of a Half-Open Unit-Cell Union"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261005-advance-20261006-r06-area-0059",
    "batch_id": "batch-20261005-advance-20261006-r06-area-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-area"
    ],
    "title_zh": "半開單位格聯集的面積",
    "stem_zh": "在等單位正交座標系中，兩軸使用同一長度單位，半開單位格由左下角整數座標列 [[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1],[-1,-1],[0,-1],[-2,-2]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。",
    "answer": {
      "kind": "short_text",
      "value": "8",
      "explanation_zh": "面積=8 平方單位。相同座標格只計一次，互異半開單位格不重疊；空格子聯集面積零，排列或布局不創造額外面積。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "半開單位格聯集的面積"
      ]
    },
    "variant": {
      "family_id": "qf-area-unit-cell-union",
      "seed": 202610066059,
      "parameters": {
        "frame": "euclidean_orthonormal_equal_units",
        "boundary": "half_open",
        "cells": [
          [
            -2,
            -2
          ],
          [
            -1,
            -2
          ],
          [
            0,
            -2
          ],
          [
            1,
            -2
          ],
          [
            2,
            -2
          ],
          [
            -2,
            -1
          ],
          [
            -1,
            -1
          ],
          [
            0,
            -1
          ],
          [
            -2,
            -2
          ]
        ],
        "operation": "unit_cell_union",
        "semantic_template": "qf-area-unit-cell-union/{\"inputs\":[8],\"operation\":\"unit_cell_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-0059",
    "batch_id": "batch-20261005-advance-20261006-r06-area-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "68015c04c7cbd571de4aeb0a4ae7a5ea789997b2886285bb177f41728f7f8a62",
      "family_id": "qf-area-unit-cell-union",
      "seed": 202610066059,
      "semantic_template": "qf-area-unit-cell-union/{\"inputs\":[8],\"operation\":\"unit_cell_union\"}"
    },
    "fields": {
      "title": "Area of a Half-Open Unit-Cell Union",
      "stem": "Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.",
      "answer": {
        "kind": "short_text",
        "value": "8",
        "explanation": "Deduplicate coordinate pairs and count the distinct unit cells once each, giving 8 squared units. Repeated listings add no area, and an empty list has area zero."
      },
      "skills": [
        "Area of a Half-Open Unit-Cell Union"
      ],
      "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，半開單位格由左下角整數座標列 [[-2,-2],[-1,-2],[0,-2],[1,-2],[2,-2],[-2,-1],[-1,-1],[0,-1],[-2,-2]] 指定，每格為 [i,i+1)×[j,j+1)。求這些格子聯集的面積平方單位數；空列與重複列格均可出現。",
        "en": "Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared 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": "[[-2, -2], [-1, -2], [0, -2], [1, -2], [2, -2], [-2, -1], [-1, -1], [0, -1], [-2, -2]]",
      "after": "[[-2, -2], [-2, -1], [-1, -2], [-1, -1], [0, -2], [0, -1], [1, -2], [2, -2]]",
      "relation": "implies",
      "rule_id": "unit_cell_deduplication",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個整數左下角唯一指定一個半開單位格，同一格重複出現只算一次。",
        "en": "Each integer lower-left corner identifies one half-open unit cell;repeated listings of that cell count only once."
      },
      "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": "[[-2, -2], [-2, -1], [-1, -2], [-1, -1], [0, -2], [0, -1], [1, -2], [2, -2]]",
      "after": "8 distinct, pairwise disjoint unit cells",
      "relation": "implies",
      "rule_id": "disjoint_unit_cell_area",
      "conditions": [],
      "explanation": {
        "zh-Hant": "不同整數座標的半開單位格不交，每格面積1；有限可加性給聯集面積，空聯集則為0。",
        "en": "Distinct integer half-open unit cells are disjoint and each has area1;finite additivity gives union area,and an empty union has area0."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "unit-cell union area := sum(1, 1, 1, 1, 1, 1, 1, 1)",
      "after": "unit-cell union area = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把每個互異格的1平方單位相加；空列的和為0。",
        "en": "Add one square unit per distinct cell;the empty sum is0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 unit-cell union area，不以小數近似替代。",
          "en": "Recompute unit-cell union area with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1"
        ],
        "output": "8",
        "label": "unit-cell union area"
      }
    }
  ],
  "math_model": {
    "result": "8",
    "distinct_cells": [
      [
        -2,
        -2
      ],
      [
        -2,
        -1
      ],
      [
        -1,
        -2
      ],
      [
        -1,
        -1
      ],
      [
        0,
        -2
      ],
      [
        0,
        -1
      ],
      [
        1,
        -2
      ],
      [
        2,
        -2
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "8",
    "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": 59,
      "original_line_sha256": "761ff1b18d06065713053d179ff3df1cdfb74284f8a4f5d1aa82e6581c6e7423",
      "question_compact_sha256": "68015c04c7cbd571de4aeb0a4ae7a5ea789997b2886285bb177f41728f7f8a62",
      "english_row_compact_sha256": "4d608b0301964cf7eac0715348c118722674ea020707211f9b3661a69023465a",
      "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": "ef724f7908617d886d22ac0f6d39ebdb2e6e4319e373e4187577eeb828afcedc",
    "original_candidate_line_sha256": "ef724f7908617d886d22ac0f6d39ebdb2e6e4319e373e4187577eeb828afcedc"
  },
  "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": 4,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "92823cfcbfbc87b1eae82f8178a1cb0a7cfaf7a1001356fdc127bc86b0498c48",
      "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."
  }
}