{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261009-advance-20261013-r03-rational-number-0080",
  "pilot_number": "R0011-06480",
  "number": 6480,
  "family_id": "qf-density13-rational-target-missing-addend",
  "title": {
    "zh-Hant": "第 6480 題：缺失加數與精確回代",
    "en": "Question 6480: Missing Rational Addend and Substitution Defect"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261009-advance-20261013-r03-rational-number-0080",
    "batch_id": "batch-20261009-advance-20261013-r03-rational-number-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-rational-number"
    ],
    "title_zh": "缺失加數與精確回代",
    "stem_zh": "精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-15/6。依序求 (x, x+a+b−target)。",
    "answer": {
      "kind": "expression",
      "value": "(-8/3, 0/1)",
      "explanation_zh": "缺失加數與回代差=(-8/3, 0/1)。從目標扣兩個已知加數再回代；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": [
        "缺失加數與精確回代"
      ]
    },
    "variant": {
      "family_id": "qf-density13-rational-target-missing-addend",
      "seed": 202610133080,
      "parameters": {
        "a": [
          1,
          2
        ],
        "b": [
          -1,
          3
        ],
        "target": [
          -15,
          6
        ],
        "operation": "density13_rational_target_missing_addend",
        "domain": "finite_rational_representation_protocol",
        "semantic_template": "qf-density13-rational-target-missing-addend/\"-8/3\""
      }
    },
    "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-0080",
    "batch_id": "batch-20261009-advance-20261013-r03-rational-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "88a75ee11cf2be7f94503d26017a41ed05af2aafa1b61c2541603d050e107d14",
      "family_id": "qf-density13-rational-target-missing-addend",
      "seed": 202610133080,
      "semantic_template": "qf-density13-rational-target-missing-addend/\"-8/3\""
    },
    "fields": {
      "title": "Missing Rational Addend and Substitution Defect",
      "stem": "Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -15/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(-8/3, 0/1)",
        "explanation": "Subtract both known addends from the target, then independently substitute to check the exact defect. The exact result, retaining the requested order, is (-8/3, 0/1)."
      },
      "skills": [
        "Missing Rational Addend and Substitution 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": "精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-15/6。依序求 (x, x+a+b−target)。",
        "en": "Exact rational representations have nonzero denominators; a=1/2, b=-1/3. Negative input denominators are allowed. The target is -15/6 and x+a+b=target. Give (x, x+a+b−target). Use exact reduced fractions with positive denominators, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw a=[1, 2];raw b=[-1, 3]",
      "after": "a=1/2;b=-1/3",
      "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": "target minus a := sub(-5/2, 1/2)",
      "after": "target minus a = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由x+a+b=target先扣a。",
        "en": "From x+a+b=target,subtract a first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 target minus a，不以小數近似替代。",
          "en": "Recompute target minus a with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-5/2",
          "1/2"
        ],
        "output": "-3",
        "label": "target minus a"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "missing rational addend := sub(-3, -1/3)",
      "after": "missing rational addend = -8/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再扣b得到唯一有理加數x。",
        "en": "Then subtract b to obtain the unique rational addend x."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 missing rational addend，不以小數近似替代。",
          "en": "Recompute missing rational addend with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-3",
          "-1/3"
        ],
        "output": "-8/3",
        "label": "missing rational addend"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "reverse x+a := sum(-8/3, 1/2)",
      "after": "reverse x+a = -13/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "將恢復的x代回加a。",
        "en": "Substitute the recovered x back and add a."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reverse x+a，不以小數近似替代。",
          "en": "Recompute reverse x+a with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-8/3",
          "1/2"
        ],
        "output": "-13/6",
        "label": "reverse x+a"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "reverse x+a+b := sum(-13/6, -1/3)",
      "after": "reverse x+a+b = -5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加原b，恢復原式左側。",
        "en": "Add the original b to reconstruct the original left side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reverse x+a+b，不以小數近似替代。",
          "en": "Recompute reverse x+a+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-13/6",
          "-1/3"
        ],
        "output": "-5/2",
        "label": "reverse x+a+b"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "missing addend substitution residual := sub(-5/2, -5/2)",
      "after": "missing addend substitution residual = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "回代值減原目標，應精確為0。",
        "en": "Subtract the original target from the substituted value;the result must be exactly0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 missing addend substitution residual，不以小數近似替代。",
          "en": "Recompute missing addend substitution residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-5/2",
          "-5/2"
        ],
        "output": "0",
        "label": "missing addend substitution residual"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "x=-8/3;target=-5/2",
      "after": "substitution residual=0/1",
      "relation": "implies",
      "rule_id": "rational_missing_addend_reverse_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "移項後以完整回代核對，不僅複製目標或把0殘差當未定義值。",
        "en": "Audit the rearrangement by full substitution,not by copying the target or confusing zero residual with undefinedness."
      },
      "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": [
      "-8/3",
      "0"
    ]
  },
  "conclusion": {
    "canonical_value": "(-8/3, 0/1)",
    "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",
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      "question_compact_sha256": "88a75ee11cf2be7f94503d26017a41ed05af2aafa1b61c2541603d050e107d14",
      "english_row_compact_sha256": "44a4925798d0ba67e95f795b5395e110809bb39e4e362227f8e53f13cddf7427",
      "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": {
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        "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": "784dd49b12d5cc4ffae26bfe7496b65b308cf8741912996740412bfae488ec1f",
    "original_candidate_line_sha256": "784dd49b12d5cc4ffae26bfe7496b65b308cf8741912996740412bfae488ec1f"
  },
  "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": {
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      "question_first_witness_sha256": "93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55"
    },
    "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": "cdb7d77a8999a1b6e716cc16a37dcd27cedde47b85488ec5652370d613c43ebb",
      "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."
  }
}