{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261005-advance-20261006-r05-circle-0032",
  "pilot_number": "R0004-02982",
  "number": 2982,
  "family_id": "qf-circle-position",
  "title": {
    "zh-Hant": "第 2982 題：相對圓邊界的三種位置",
    "en": "Question 2982: Three Positions Relative to a Circle Boundary"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261005-advance-20261006-r05-circle-0032",
    "batch_id": "batch-20261005-advance-20261006-r05-circle-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-circle"
    ],
    "title_zh": "相對圓邊界的三種位置",
    "stem_zh": "在等單位正交座標系中，兩軸使用同一長度單位，圓心 O=(-6, -1)，非負半徑 r=5，圓 C={P:d(P,O)=r}。將 P=(-1, -1) 分為內、圓上或外；以到圓心距離與半徑比較。",
    "answer": {
      "kind": "short_text",
      "value": "圓上",
      "explanation_zh": "平方比較=(25, 25)。非負距離與半徑的順序等價於平方的順序；嚴格小於是內、相等是圓上、嚴格大於是外。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 1,
      "skills": [
        "相對圓邊界的三種位置"
      ]
    },
    "variant": {
      "family_id": "qf-circle-position",
      "seed": 202610065032,
      "parameters": {
        "O": [
          -6,
          -1
        ],
        "r": 5,
        "P": [
          -1,
          -1
        ],
        "frame": "euclidean_orthonormal_equal_units",
        "operation": "circle_position",
        "semantic_template": "qf-circle-position/{\"inputs\":[[\"0\",\"5\"],\"5\"],\"operation\":\"circle_position\"}"
      }
    },
    "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-r05-circle-0032",
    "batch_id": "batch-20261005-advance-20261006-r05-circle-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "505255659b4769acba1b4d56a423c6446fc8c61deb9b9e34c909c84f916498d2",
      "family_id": "qf-circle-position",
      "seed": 202610065032,
      "semantic_template": "qf-circle-position/{\"inputs\":[[\"0\",\"5\"],\"5\"],\"operation\":\"circle_position\"}"
    },
    "fields": {
      "title": "Three Positions Relative to a Circle Boundary",
      "stem": "Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-6, -1) and radius r=5≥0. Classify P=(-1, -1) as inside, on boundary, or outside.",
      "answer": {
        "kind": "short_text",
        "value": "on boundary",
        "explanation": "Compare squared distance 25 with r²=25: less means inside, equal means on boundary, and greater means outside. Here the result is on boundary."
      },
      "skills": [
        "Three Positions Relative to a Circle Boundary"
      ],
      "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，圓心 O=(-6, -1)，非負半徑 r=5，圓 C={P:d(P,O)=r}。將 P=(-1, -1) 分為內、圓上或外；以到圓心距離與半徑比較。",
        "en": "Use one equal-unit orthonormal Euclidean Cartesian frame. The circle boundary has center O=(-6, -1) and radius r=5≥0. Classify P=(-1, -1) as inside, on boundary, or outside."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "orthonormal equal-length units;r=5≥0",
      "after": "C={P:distance(P,O)=r} is a boundary,not a filled disk",
      "relation": "implies",
      "rule_id": "circle_boundary_convention",
      "conditions": [],
      "explanation": {
        "zh-Hant": "圓指到圓心距離恰為r的邊界；內部不是圓上的同義詞。半徑0採退化點集{O}，沒有圓內點，直徑長0。",
        "en": "The circle is the boundary at distance exactly r from the center,not the filled disk.For r=0 use the degenerate singleton{O},with no inside points and diameter length0."
      },
      "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": "radius squared := square(5)",
      "after": "radius squared = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非負半徑的平方是比較距離的門檻。",
        "en": "The square of the nonnegative radius is the distance-comparison threshold."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 radius squared，不以小數近似替代。",
          "en": "Recompute radius squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "5"
        ],
        "output": "25",
        "label": "radius squared"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "coordinate difference0 := sub(-1, -6)",
      "after": "coordinate difference0 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取點減圓心的座標差，不能直接平方絕對座標。",
        "en": "Subtract the center coordinate from the point coordinate;do not square absolute coordinates instead."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 coordinate difference0，不以小數近似替代。",
          "en": "Recompute coordinate difference0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "-6"
        ],
        "output": "5",
        "label": "coordinate difference0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "coordinate square0 := square(5)",
      "after": "coordinate square0 = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，負差仍給非負貢獻。",
        "en": "Square this coordinate difference;a negative difference still contributes a nonnegative square."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 coordinate square0，不以小數近似替代。",
          "en": "Recompute coordinate square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "5"
        ],
        "output": "25",
        "label": "coordinate square0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "coordinate difference1 := sub(-1, -1)",
      "after": "coordinate difference1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先取點減圓心的座標差，不能直接平方絕對座標。",
        "en": "Subtract the center coordinate from the point coordinate;do not square absolute coordinates instead."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 coordinate difference1，不以小數近似替代。",
          "en": "Recompute coordinate difference1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "0",
        "label": "coordinate difference1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "coordinate square1 := square(0)",
      "after": "coordinate square1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，負差仍給非負貢獻。",
        "en": "Square this coordinate difference;a negative difference still contributes a nonnegative square."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 coordinate square1，不以小數近似替代。",
          "en": "Recompute coordinate square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "0"
        ],
        "output": "0",
        "label": "coordinate square1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "squared distance := sum(25, 0)",
      "after": "squared distance = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正交且等單位時，由畢氏關係相加兩個平方差。",
        "en": "In an orthonormal equal-unit frame,the Pythagorean relation adds the squared differences."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 squared distance，不以小數近似替代。",
          "en": "Recompute squared distance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "25",
          "0"
        ],
        "output": "25",
        "label": "squared distance"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "d²=25;r²=25",
      "after": "requested classification=圓上",
      "relation": "implies",
      "rule_id": "nonnegative_squared_distance_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "距離與半徑皆非負，平方保序；相等才在邊界，嚴格小於為內、大於為外，不需小數開根。",
        "en": "Distance and radius are nonnegative,so squaring preserves their order.Equality means boundary,strictly smaller means inside,and larger means outside.No rounded square root is needed."
      },
      "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": "圓上",
    "squared_distance": "25",
    "radius_squared": "25"
  },
  "conclusion": {
    "canonical_value": "圓上",
    "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-r05-circle-foundations",
      "batch_path": "public/data/questions/batches/batch-20261005-advance-20261006-r05-circle-foundations/questions.jsonl",
      "batch_sha256": "f91314751461a37c852b683b0c62ae64ee964eaf2877eb78901259982b7dbe02",
      "original_line": 32,
      "original_line_sha256": "5b3d39fb86097bd508c7a44469e6791bd853dfcf8edda3d87f1dfc7a14b0b42d",
      "question_compact_sha256": "505255659b4769acba1b4d56a423c6446fc8c61deb9b9e34c909c84f916498d2",
      "english_row_compact_sha256": "d56dc7eb1036867770a548e53d1dfbf412332a2d91e59042d52ffa75c86777cf",
      "english_manifest_path": "public/data/i18n/editions/batch-20261005-advance-20261006-r05-circle-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261005-advance-20261006-r05-circle-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-circle": {
        "sha256": "f5666d01288a4c7140be59abad5dbf093293c9136d7d60dc752a79a497d99d2e",
        "english_sha256": "e3c801529b17bbcfc6b73fef34a12a234a6dedbbaf8f1e283d3371fd2c7290d0",
        "original_path": "public/data/mko/circle.json",
        "english_path": "public/data/i18n/editions/mko-circle.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b254245aa10315455f756d4620a6ac0243e0146bccea5d89e872f1359bb32a1d",
    "original_candidate_sha256": "bb4607b7a170fa4d1781bde0c6ed8637ee3b1de3a657e9576cb51a2a76c45fb3",
    "original_candidate_line_sha256": "bb4607b7a170fa4d1781bde0c6ed8637ee3b1de3a657e9576cb51a2a76c45fb3"
  },
  "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": 8,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "d646f2bd794eff208f1d3f58e42c0b4eaf0cac9c7564c7e7e57c5eed425a9205",
      "arithmetic_steps": 6
    },
    "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."
  }
}