{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260923-real-rational-order-0009",
  "pilot_number": "R0068-34884",
  "number": 34884,
  "family_id": "qf-real-rational-order",
  "title": {
    "zh-Hant": "第 34884 題：有理點的精確排序",
    "en": "Question 34884: Exact Ordering of Rational Points"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260923-real-rational-order-0009",
    "batch_id": "batch-20260923-real-number-rational-points",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-real-number"
    ],
    "title_zh": "有理點的精確排序",
    "stem_zh": "有理點 a=-2/1、b=1/2。用交叉乘積判斷 a 與 b 的關係（<、= 或 >）。",
    "answer": {
      "kind": "short_text",
      "value": "<",
      "explanation_zh": "-2×2 < 1×1，故 -2/1 < 1/2。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 1,
      "skills": [
        "有理數",
        "交叉乘積"
      ]
    },
    "variant": {
      "family_id": "qf-real-rational-order",
      "seed": 20261032,
      "parameters": {
        "left": [
          -2,
          1
        ],
        "right": [
          1,
          2
        ],
        "operation": "cross_multiplication_order",
        "semantic_template": "qf-real-rational-order/{\"left\":[-2,1],\"operation\":\"cross_multiplication_order\",\"right\":[1,2]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "schema_pending",
        "independent_recomputation_pending"
      ],
      "warning_zh": "本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "GPT-5.6 Terra",
      "generated_at": "2026-09-23",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20260923-real-rational-order-0009",
    "batch_id": "batch-20260923-real-number-rational-points",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "e9183cce176a55042818d9126b0d1ef488f201d31c1d4326794a965545dde881",
      "family_id": "qf-real-rational-order",
      "seed": 20261032,
      "semantic_template": "qf-real-rational-order/{\"left\":[-2,1],\"operation\":\"cross_multiplication_order\",\"right\":[1,2]}"
    },
    "fields": {
      "title": "Exact Ordering of Rational Points",
      "stem": "The rational points are a=-2/1 and b=1/2. Use cross multiplication to determine whether a<b, a=b, or a>b.",
      "answer": {
        "kind": "short_text",
        "value": "<",
        "explanation": "-2×2 < 1×1, so -2/1 < 1/2."
      },
      "skills": [
        "rational numbers",
        "cross multiplication"
      ],
      "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": "有理點 a=-2/1、b=1/2。用交叉乘積判斷 a 與 b 的關係（<、= 或 >）。",
        "en": "The rational points are a=-2/1 and b=1/2. Use cross multiplication to determine whether a<b, a=b, or a>b."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-real-rational-order, [left, operation, right]]",
      "after": "derived = [cross_multiplication_order, no-invented-domain]",
      "relation": "implies",
      "rule_id": "R68_BANK_specific_keys_operations_and_numeric_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按來源分bank：legacy raw SAFE/衍生BigInt無界，BASE Number.isInteger不偷加SAFE/百萬界。underscore operation、domain及寬/嚴keys各依本bank；原JSON完整保存。",
        "en": "Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本題的原始已知量、論域與本步條件；保留原題/英文/MKO/AST/狀態，不把單向推論倒成等價，也不把有限核對當獨立審查或一般形式證明。",
          "en": "Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator := div(-2, 1)",
      "after": "Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作數/次序精確運算；乘/減/除明示兩操作數，legacy BigInt衍生量不重套raw SAFE界，BASE Number來源的型別/表示條件另驗，不以浮點近似湊答案。",
        "en": "Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator，不以小數近似替代。",
          "en": "Recompute Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-2",
        "label": "Real ORIGINAL left point ORIGINAL exact numerator over nonzero denominator"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [Real ORIGINAL left point, [-2, 1]]",
      "after": "derived = [[-2, 1], -2/1]",
      "relation": "implies",
      "rule_id": "Real_BASE_fraction_sign_gcd_normalization_NO_invented_SAFEcap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "BASE normalizeFraction要整數num/den且den非0，不偷加SAFE/百萬/一億界。先正den再gcd約分，零為0/1；此衍生表示不改原raw pair。",
        "en": "BASE normalizeFraction requires INTEGER components and nonzero denominator,without invented SAFE/million/100million caps.Normalize denominator sign and gcd;zero is0/1.This derived representation does not alter the ORIGINAL raw pair."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本題的原始已知量、論域與本步條件；保留原題/英文/MKO/AST/狀態，不把單向推論倒成等價，也不把有限核對當獨立審查或一般形式證明。",
          "en": "Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator := div(1, 2)",
      "after": "Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作數/次序精確運算；乘/減/除明示兩操作數，legacy BigInt衍生量不重套raw SAFE界，BASE Number來源的型別/表示條件另驗，不以浮點近似湊答案。",
        "en": "Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator，不以小數近似替代。",
          "en": "Recompute Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "2"
        ],
        "output": "1/2",
        "label": "Real ORIGINAL right point ORIGINAL exact numerator over nonzero denominator"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [Real ORIGINAL right point, [1, 2]]",
      "after": "derived = [[1, 2], 1/2]",
      "relation": "implies",
      "rule_id": "Real_BASE_fraction_sign_gcd_normalization_NO_invented_SAFEcap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "BASE normalizeFraction要整數num/den且den非0，不偷加SAFE/百萬/一億界。先正den再gcd約分，零為0/1；此衍生表示不改原raw pair。",
        "en": "BASE normalizeFraction requires INTEGER components and nonzero denominator,without invented SAFE/million/100million caps.Normalize denominator sign and gcd;zero is0/1.This derived representation does not alter the ORIGINAL raw pair."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本題的原始已知量、論域與本步條件；保留原題/英文/MKO/AST/狀態，不把單向推論倒成等價，也不把有限核對當獨立審查或一般形式證明。",
          "en": "Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Real ORIGINAL normalized left numerator times right positive denominator := mul(-2, 2)",
      "after": "Real ORIGINAL normalized left numerator times right positive denominator = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作數/次序精確運算；乘/減/除明示兩操作數，legacy BigInt衍生量不重套raw SAFE界，BASE Number來源的型別/表示條件另驗，不以浮點近似湊答案。",
        "en": "Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Real ORIGINAL normalized left numerator times right positive denominator，不以小數近似替代。",
          "en": "Recompute Real ORIGINAL normalized left numerator times right positive denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "2"
        ],
        "output": "-4",
        "label": "Real ORIGINAL normalized left numerator times right positive denominator"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Real ORIGINAL normalized right numerator times left positive denominator := mul(1, 1)",
      "after": "Real ORIGINAL normalized right numerator times left positive denominator = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原操作數/次序精確運算；乘/減/除明示兩操作數，legacy BigInt衍生量不重套raw SAFE界，BASE Number來源的型別/表示條件另驗，不以浮點近似湊答案。",
        "en": "Calculate exactly from ORIGINAL ordered operands;mul/sub/div have EXPLICITLY TWO operands.Legacy derived BigInts are not recapped at RAW SAFE;check BASE Number type/representation conditions separately.Do not use floating approximations to force an answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Real ORIGINAL normalized right numerator times left positive denominator，不以小數近似替代。",
          "en": "Recompute Real ORIGINAL normalized right numerator times left positive denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Real ORIGINAL normalized right numerator times left positive denominator"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "source roles = [-2/1, 1/2]",
      "after": "derived = [-4, 1, <]",
      "relation": "implies",
      "rule_id": "Real_ORDER_normalize_sign_BEFORE_crossproduct_inequality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先把兩den化正才不會錯用不等式方向，然後比較全部交叉乘積。比的是原有理點，不以浮點小數近似判別。",
        "en": "Normalize BOTH denominator signs BEFORE cross-product order comparison to preserve inequality direction.Compare the ORIGINAL rational points exactly,not floating approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向核對本題的原始已知量、論域與本步條件；保留原題/英文/MKO/AST/狀態，不把單向推論倒成等價，也不把有限核對當獨立審查或一般形式證明。",
          "en": "Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "<"
  },
  "conclusion": {
    "canonical_value": "<",
    "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-20260923-real-number-rational-points",
      "batch_path": "public/data/questions/batches/batch-20260923-real-number-rational-points/questions.jsonl",
      "batch_sha256": "d19b574fb612be66646603f4bbeec96578a1d948a33ab097ee9b84b2e171e5e9",
      "original_line": 34,
      "original_line_sha256": "31475e8f5c7c4da4af30850058b98f54b5154662c2c8e1502340cebf16d03611",
      "question_compact_sha256": "e9183cce176a55042818d9126b0d1ef488f201d31c1d4326794a965545dde881",
      "english_row_compact_sha256": "790464ad98034b16b784b654adc00a1ee960a2f91ee458da75425ddf41718313",
      "english_manifest_path": "public/data/i18n/editions/batch-20260923-real-number-rational-points.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20260923-real-number-rational-points.en.jsonl"
    },
    "mko_bindings": {
      "mko-real-number": {
        "sha256": "2bc8c0a1defae2276ce50808a0761bf4d62f26892e443fcab79fc8090113d880",
        "english_sha256": "5dde700ff98e1ae0b35a5924253f8ad22515aa33c9a17deab8b7fa9a35e1ef5c",
        "original_path": "public/data/mko/real-number.json",
        "english_path": "public/data/i18n/editions/mko-real-number.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "345142be27643acca9d0fec2422ac7311b85bf142709391cd6b5b3a33c03cf8e",
    "original_candidate_sha256": "8715f6688368290c078a4ef8eb053984688d6e8c975bfec27c00a295aca288ab",
    "original_candidate_line_sha256": "8715f6688368290c078a4ef8eb053984688d6e8c975bfec27c00a295aca288ab"
  },
  "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-20261026-012-p69",
    "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": "e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed",
      "question_first_witness_sha256": "8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229"
    },
    "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": "e4ccd04cc36b379397633fe2a8699ea9d86e50c9db35c38b44369160199280b0",
      "arithmetic_steps": 4
    },
    "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."
  }
}