{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261009-advance-20261013-r03-rational-number-0005",
  "pilot_number": "R0011-06455",
  "number": 6455,
  "family_id": "qf-density13-rational-sum-difference-pair",
  "title": {
    "zh-Hant": "第 6455 題：有理和與有序差",
    "en": "Question 6455: Exact Ordered Rational Sum and Difference"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261009-advance-20261013-r03-rational-number-0005",
    "batch_id": "batch-20261009-advance-20261013-r03-rational-number-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-rational-number"
    ],
    "title_zh": "有理和與有序差",
    "stem_zh": "精確有理數協議；q≠0，a=-8/2、b=5/5。依序求 (a+b, a−b)，均寫最簡n/d。",
    "answer": {
      "kind": "expression",
      "value": "(-3/1, -5/1)",
      "explanation_zh": "有理和與差=(-3/1, -5/1)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 1,
      "skills": [
        "有理和與有序差"
      ]
    },
    "variant": {
      "family_id": "qf-density13-rational-sum-difference-pair",
      "seed": 202610133005,
      "parameters": {
        "a": [
          -8,
          2
        ],
        "b": [
          5,
          5
        ],
        "operation": "density13_rational_sum_difference_pair",
        "domain": "finite_rational_representation_protocol",
        "semantic_template": "qf-density13-rational-sum-difference-pair/[\"-4/1\",\"1/1\"]"
      }
    },
    "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": "2026-10-09",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261009-advance-20261013-r03-rational-number-0005",
    "batch_id": "batch-20261009-advance-20261013-r03-rational-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "54fbc98e3c9ce4befebb77ea24254b2f23d5af8b6a67ccbbaf843eb8aa2d726d",
      "family_id": "qf-density13-rational-sum-difference-pair",
      "seed": 202610133005,
      "semantic_template": "qf-density13-rational-sum-difference-pair/[\"-4/1\",\"1/1\"]"
    },
    "fields": {
      "title": "Exact Ordered Rational Sum and Difference",
      "stem": "Exact rational representations have nonzero denominators; a=-8/2, b=5/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(-3/1, -5/1)",
        "explanation": "Use common denominators for addition and subtraction, then reduce; exact finite instances are not a proof of the entire rational structure. The exact result, retaining the requested order, is (-3/1, -5/1)."
      },
      "skills": [
        "Exact Ordered Rational Sum and Difference"
      ],
      "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": "精確有理數協議；q≠0，a=-8/2、b=5/5。依序求 (a+b, a−b)，均寫最簡n/d。",
        "en": "Exact rational representations have nonzero denominators; a=-8/2, b=5/5. Negative input denominators are allowed. Give (a+b, a−b). Use exact reduced fractions with positive denominators, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw a=[-8, 2];raw b=[5, 5]",
      "after": "a=-4/1;b=1/1",
      "relation": "implies",
      "rule_id": "nonzero_signed_rational_representation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分母只要求非零，負分母有效；約為正分母既約分數以作精確運算，但保留原始表示與來源。",
        "en": "Original denominators need only be nonzero;negative denominators are valid.Normalize to reduced positive-denominator fractions for exact arithmetic while retaining the original representations and sources."
      },
      "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": "rational sum := sum(-4, 1)",
      "after": "rational sum = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "精確相加後用正分母最簡值。",
        "en": "Add exactly and report the reduced positive-denominator value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rational sum，不以小數近似替代。",
          "en": "Recompute rational sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-4",
          "1"
        ],
        "output": "-3",
        "label": "rational sum"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "rational ordered difference := sub(-4, 1)",
      "after": "rational ordered difference = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原序求a−b，不取絕對值。",
        "en": "Subtract in the original order a−b,not absolute difference."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rational ordered difference，不以小數近似替代。",
          "en": "Recompute rational ordered difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-4",
          "1"
        ],
        "output": "-5",
        "label": "rational ordered difference"
      }
    }
  ],
  "math_model": {
    "result": [
      "-3",
      "-5"
    ]
  },
  "conclusion": {
    "canonical_value": "(-3/1, -5/1)",
    "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-20261009-advance-20261013-r03-rational-number-foundations",
      "batch_path": "public/data/questions/batches/batch-20261009-advance-20261013-r03-rational-number-foundations/questions.jsonl",
      "batch_sha256": "87bdbac39c0ce10a6a71861c70e3b36ec58f0ff44a6888e386a7015f76fc685e",
      "original_line": 5,
      "original_line_sha256": "06588eb78cba683012a0bd116c01359a3451d3d4be53d6c9139ce4720cf05407",
      "question_compact_sha256": "54fbc98e3c9ce4befebb77ea24254b2f23d5af8b6a67ccbbaf843eb8aa2d726d",
      "english_row_compact_sha256": "34ff0e4aeddf4b434c4ecd28725a162af38d05a56679512d8c2e9eac8cd015a0",
      "english_manifest_path": "public/data/i18n/editions/batch-20261009-advance-20261013-r03-rational-number-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261009-advance-20261013-r03-rational-number-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-rational-number": {
        "sha256": "f1b52aad1867316fd94d6e07d6e94237a202d6769454898a0831d8d1bb89f23f",
        "english_sha256": "7f9330a95c6c6cb3df1c1696d1ffa62bb4442ff453291c4176febb6d0d71ff5a",
        "original_path": "public/data/mko/rational-number.json",
        "english_path": "public/data/i18n/editions/mko-rational-number.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "5b2ce631281eb24b279908f47c6de66fa60d326520852879d324c4f3ac653970",
    "original_candidate_sha256": "468f38924539496466ab210256a16fa4d3c0ad4917022a0ffcef93ebc2df76ab",
    "original_candidate_line_sha256": "468f38924539496466ab210256a16fa4d3c0ad4917022a0ffcef93ebc2df76ab"
  },
  "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-20261026-005-p12",
    "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": "b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80",
      "question_first_witness_sha256": "93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 3,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "e7900663a5a063cdc6c0b3225f8a89625ed0a8634e77b6a0a3ff42caa56c9afb",
      "arithmetic_steps": 2
    },
    "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."
  }
}