{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261010-r03-distance-formula-0042",
  "pilot_number": "R0007-04017",
  "number": 4017,
  "family_id": "qf-density10-distance-candidate-defect",
  "title": {
    "zh-Hant": "第 4017 題：候選符號與有號平方缺陷",
    "en": "Question 4017: Candidate Sign and Squared-Distance Defect"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261010-r03-distance-formula-0042",
    "batch_id": "batch-20261008-advance-20261010-r03-distance-formula-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-distance-formula"
    ],
    "title_zh": "候選符號與有號平方缺陷",
    "stem_zh": "同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[16, 17]、整數候選c=19。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。",
    "answer": {
      "kind": "expression",
      "value": "(是, -184)",
      "explanation_zh": "候選非負與平方缺陷=(是, -184)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "候選符號與有號平方缺陷"
      ]
    },
    "variant": {
      "family_id": "qf-density10-distance-candidate-defect",
      "seed": 202610103042,
      "parameters": {
        "vector": [
          16,
          17
        ],
        "candidate": 19,
        "frame": "orthogonal_equal_unit",
        "operation": "density10_distance_candidate_defect",
        "domain": "integer_euclidean_plane",
        "semantic_template": "qf-density10-distance-candidate-defect/[[\"16\",\"17\"],19]"
      }
    },
    "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-20261010-r03-distance-formula-0042",
    "batch_id": "batch-20261008-advance-20261010-r03-distance-formula-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "321a90fa9d2d8d5050689f560cdbffc1e36083826d6a2a30549b254b1d17e1f6",
      "family_id": "qf-density10-distance-candidate-defect",
      "seed": 202610103042,
      "semantic_template": "qf-density10-distance-candidate-defect/[[\"16\",\"17\"],19]"
    },
    "fields": {
      "title": "Candidate Sign and Squared-Distance Defect",
      "stem": "In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[16, 17], integer candidate c=19. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.",
      "answer": {
        "kind": "expression",
        "value": "(Yes, -184)",
        "explanation": "Check sign and signed square defect separately; a negative candidate is not a distance even if its square defect is zero. The exact result, retaining the requested order, is (Yes, -184)."
      },
      "skills": [
        "Candidate Sign and Squared-Distance Defect"
      ],
      "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": "同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[16, 17]、整數候選c=19。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。",
        "en": "In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[16, 17], integer candidate c=19. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "integer vectors from a common origin;orthogonal equal-length units",
      "after": "L2(v)≥0 and L2(v)²=v₁²+v₂²",
      "relation": "implies",
      "rule_id": "euclidean_distance_frame",
      "conditions": [],
      "explanation": {
        "zh-Hant": "共同原點、正交及等單位是平方和公式的前提；向量座標可負，距離不可負。",
        "en": "A common origin,orthogonality and equal units support the sum-of-squares formula;coordinates may be negative but distances may not."
      },
      "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": "candidate vector coordinate1 squared := square(16)",
      "after": "candidate vector coordinate1 squared = 256",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先將此完整分量平方，保留負號在括號內。",
        "en": "Square the entire coordinate,including a negative sign inside the square."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate vector coordinate1 squared，不以小數近似替代。",
          "en": "Recompute candidate vector coordinate1 squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "16"
        ],
        "output": "256",
        "label": "candidate vector coordinate1 squared"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "candidate vector coordinate2 squared := square(17)",
      "after": "candidate vector coordinate2 squared = 289",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先將此完整分量平方，保留負號在括號內。",
        "en": "Square the entire coordinate,including a negative sign inside the square."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate vector coordinate2 squared，不以小數近似替代。",
          "en": "Recompute candidate vector coordinate2 squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "17"
        ],
        "output": "289",
        "label": "candidate vector coordinate2 squared"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "candidate vector L2 squared := sum(256, 289)",
      "after": "candidate vector L2 squared = 545",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個平方相加是距離平方，不是兩座標直接相加。",
        "en": "Add the coordinate squares to obtain squared distance,not a sum of the coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate vector L2 squared，不以小數近似替代。",
          "en": "Recompute candidate vector L2 squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "256",
          "289"
        ],
        "output": "545",
        "label": "candidate vector L2 squared"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "candidate squared := square(19)",
      "after": "candidate squared = 361",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選先平方，再與實際平方距離比較。",
        "en": "Square the candidate before comparing with squared distance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate squared，不以小數近似替代。",
          "en": "Recompute candidate squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "19"
        ],
        "output": "361",
        "label": "candidate squared"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "signed square defect := sub(361, 545)",
      "after": "signed square defect = -184",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "缺陷方向是c²−L2²，不取絕對值。",
        "en": "The signed defect is c²−L2²,not its absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 signed square defect，不以小數近似替代。",
          "en": "Recompute signed square defect with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "361",
          "545"
        ],
        "output": "-184",
        "label": "signed square defect"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "c=19;c≥0=是;c²−L2²=-184",
      "after": "report sign and defect separately",
      "relation": "implies",
      "rule_id": "distance_sign_and_defect",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一欄只問候選非負；完整距離資格另需缺陷0。負候選即使平方相等也不是距離。",
        "en": "The first field asks only whether the candidate is nonnegative;full distance validity also requires defect0.A negative candidate is not a distance even if its square matches."
      },
      "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": [
      true,
      "-184"
    ],
    "squared_length": "545"
  },
  "conclusion": {
    "canonical_value": "(是, -184)",
    "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-20261010-r03-distance-formula-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261010-r03-distance-formula-foundations/questions.jsonl",
      "batch_sha256": "817c2ffbb228a708e250aa13865b5f5cc70e482a1c54d82a63d493bd6bed1630",
      "original_line": 42,
      "original_line_sha256": "8c873e5b3fc11dd0ddb1cb04b24ff06350e85dad758698b42cbd5018de088658",
      "question_compact_sha256": "321a90fa9d2d8d5050689f560cdbffc1e36083826d6a2a30549b254b1d17e1f6",
      "english_row_compact_sha256": "f63dd4df244521f82f5fca3e3b2c0e8d3e427b90f2f79dd66d4db8d1a4385981",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261010-r03-distance-formula-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261010-r03-distance-formula-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-distance-formula": {
        "sha256": "153d0d0feabadaf33e08081677be0f81709df28ffa5bffaf0176a0f7875e5904",
        "english_sha256": "d297d8ae3afb0c4aa0a2de4cbd9e17c6355f2ae3f0fbf1510b9dc7cea3b27827",
        "original_path": "public/data/mko/distance-formula.json",
        "english_path": "public/data/i18n/editions/mko-distance-formula.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "de1fb611cf230f243ddd3334fbc53c9205cbe528b78346a10544972dede78b95",
    "original_candidate_sha256": "d577f5118d3de372df3f90156896a2bc65b1570018ac56b147764e1b23b22f9f",
    "original_candidate_line_sha256": "d577f5118d3de372df3f90156896a2bc65b1570018ac56b147764e1b23b22f9f"
  },
  "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-p8",
    "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": "faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a",
      "question_first_witness_sha256": "4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "9c6ad8f3258d31a2b4687cbbe001cead8ce2c2f101781fa5e808eb957b728d44",
      "arithmetic_steps": 5
    },
    "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."
  }
}