{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261011-advance-20261015-r02-ratio-proportion-0056",
  "pilot_number": "R0013-07081",
  "number": 7081,
  "family_id": "qf-density15-ratio-proportional-partition",
  "title": {
    "zh-Hant": "第 7081 題：同單位量的比例分配",
    "en": "Question 7081: Same-Unit Weighted Partition and Actual Ratio Guard"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261011-advance-20261015-r02-ratio-proportion-0056",
    "batch_id": "batch-20261011-advance-20261015-r02-ratio-proportion-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-ratio-proportion"
    ],
    "title_zh": "同單位量的比例分配",
    "stem_zh": "Q數值比協議。同單位有理量分配：正權重a=6、b=2、總量T=(5)/(2)。指定x=T*a/(a+b)，y=T*b/(a+b)。求 (x, y, x+y−T, bx−ay, 實際x:y合法資格, 比值或未定義)。",
    "answer": {
      "kind": "expression",
      "value": "(15/8, 5/8, 0/1, 0/1, 是, 3/1)",
      "explanation_zh": "同單位比例分配與比資格=(15/8, 5/8, 0/1, 0/1, 是, 3/1)。先分出x，再由總量減x取得y並核兩種殘差；實際比另核y非零。零總量不證實0:0合法比。零後項不因交乘相等而合法，有限比例非因果或幾何相似證明。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 1,
      "skills": [
        "同單位量的比例分配"
      ]
    },
    "variant": {
      "family_id": "qf-density15-ratio-proportional-partition",
      "seed": 202610152056,
      "parameters": {
        "a": 6,
        "b": 2,
        "total": [
          5,
          2
        ],
        "domain": "finite_Q_numeric_ratio_protocol",
        "operation": "density15_ratio_proportional_partition",
        "semantic_template": "qf-density15-ratio-proportional-partition/[\"15/8\",\"5/8\"]"
      }
    },
    "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": "2026-10-11",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261011-advance-20261015-r02-ratio-proportion-0056",
    "batch_id": "batch-20261011-advance-20261015-r02-ratio-proportion-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "7ab953754c52d737c8699b4bc3a3d699d7f6fac932f38167c7f0a7699b3dbce1",
      "family_id": "qf-density15-ratio-proportional-partition",
      "seed": 202610152056,
      "semantic_template": "qf-density15-ratio-proportional-partition/[\"15/8\",\"5/8\"]"
    },
    "fields": {
      "title": "Same-Unit Weighted Partition and Actual Ratio Guard",
      "stem": "Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=6, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined).",
      "answer": {
        "kind": "expression",
        "value": "(15/8, 5/8, 0/1, 0/1, Yes, 3/1)",
        "explanation": "Compute the partition and both defects, then guard the actual second component y; a zero total gives an illegal 0:0 ratio despite correct partition arithmetic. The exact result, retaining the requested order, is (15/8, 5/8, 0/1, 0/1, Yes, 3/1)."
      },
      "skills": [
        "Same-Unit Weighted Partition and Actual Ratio Guard"
      ],
      "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": "Q數值比協議。同單位有理量分配：正權重a=6、b=2、總量T=(5)/(2)。指定x=T*a/(a+b)，y=T*b/(a+b)。求 (x, y, x+y−T, bx−ay, 實際x:y合法資格, 比值或未定義)。",
        "en": "Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Positive weights a=6, b=2, same-unit nonnegative total T=5/2. Set x=T*a/(a+b), y=T*b/(a+b). Give (x, y, x+y−T, bx−ay, is the actual x:y legal?, x/y or undefined)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "exact rational numerical ratios;each denominator needs its own guard",
      "after": "cross-multiplied equality does not legalize a zero ratio denominator",
      "relation": "implies",
      "rule_id": "rational_ratio_definition_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "數值比不是因果或幾何相似的自動證明；交乘殘差0不會讓0後項合法，反向比要另核自己的後項。",
        "en": "A numerical ratio is not an automatic causal or geometric-similarity proof.Cross residual0 does not legalize a zero denominator,and the reverse ratio needs its own denominator guard."
      },
      "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": "positive weight sum := sum(6, 2)",
      "after": "positive weight sum = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個正權重和為正，分配分母合法。",
        "en": "The sum of two positive weights is positive and is a legal allocation denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 positive weight sum，不以小數近似替代。",
          "en": "Recompute positive weight sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6",
          "2"
        ],
        "output": "8",
        "label": "positive weight sum"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "total times first weight := mul(5/2, 6)",
      "after": "total times first weight = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "總量先乘第一權重。",
        "en": "Multiply the total by the first weight."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 total times first weight，不以小數近似替代。",
          "en": "Recompute total times first weight with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5/2",
          "6"
        ],
        "output": "15",
        "label": "total times first weight"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "first proportional part x := div(15, 8)",
      "after": "first proportional part x = 15/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再除正權重和得到第一份。",
        "en": "Divide by the positive weight sum to obtain the first part."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 first proportional part x，不以小數近似替代。",
          "en": "Recompute first proportional part x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15",
          "8"
        ],
        "output": "15/8",
        "label": "first proportional part x"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "second part from remaining total := sub(5/2, 15/8)",
      "after": "second part from remaining total = 5/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同單位總量扣第一份得到第二份。",
        "en": "Subtract the first part from the same-unit total to obtain the second."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 second part from remaining total，不以小數近似替代。",
          "en": "Recompute second part from remaining total with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5/2",
          "15/8"
        ],
        "output": "5/8",
        "label": "second part from remaining total"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "sum of allocated parts := sum(15/8, 5/8)",
      "after": "sum of allocated parts = 5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩份相加反核完整總量。",
        "en": "Add the two parts to audit the original total."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum of allocated parts，不以小數近似替代。",
          "en": "Recompute sum of allocated parts with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "15/8",
          "5/8"
        ],
        "output": "5/2",
        "label": "sum of allocated parts"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "allocation sum residual := sub(5/2, 5/2)",
      "after": "allocation sum residual = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "總和減原總量應為0。",
        "en": "The restored sum minus the original total must be0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 allocation sum residual，不以小數近似替代。",
          "en": "Recompute allocation sum residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5/2",
          "5/2"
        ],
        "output": "0",
        "label": "allocation sum residual"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "allocation b times x := mul(2, 15/8)",
      "after": "allocation b times x = 15/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "比例交乘第一項b·x。",
        "en": "Compute b·x as the first cross term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 allocation b times x，不以小數近似替代。",
          "en": "Recompute allocation b times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "15/8"
        ],
        "output": "15/4",
        "label": "allocation b times x"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "allocation a times y := mul(6, 5/8)",
      "after": "allocation a times y = 15/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "比例交乘第二項a·y。",
        "en": "Compute a·y as the second cross term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 allocation a times y，不以小數近似替代。",
          "en": "Recompute allocation a times y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "5/8"
        ],
        "output": "15/4",
        "label": "allocation a times y"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "allocation proportional residual := sub(15/4, 15/4)",
      "after": "allocation proportional residual = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "b·x−a·y應為0，仍不代替實際比值後項檢查。",
        "en": "The cross residual b·x−a·y must be0,but does not replace the actual ratio-denominator check."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 allocation proportional residual，不以小數近似替代。",
          "en": "Recompute allocation proportional residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "15/4",
          "15/4"
        ],
        "output": "0",
        "label": "allocation proportional residual"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "actual allocated ratio x/y := div(15/8, 5/8)",
      "after": "actual allocated ratio x/y = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "實際第二份非零才給x:y一個數值。",
        "en": "The actual second part must be nonzero to give x:y a numerical value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual allocated ratio x/y，不以小數近似替代。",
          "en": "Recompute actual allocated ratio x/y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15/8",
          "5/8"
        ],
        "output": "3",
        "label": "actual allocated ratio x/y"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "T=5/2;parts=(15/8,5/8);residuals=(0,0)",
      "after": "actual ratio legal=是;x/y=3/1",
      "relation": "implies",
      "rule_id": "zero_total_not_valid_zero_ratio",
      "conditions": [],
      "explanation": {
        "zh-Hant": "零總量可合法分成0與0，兩殘差也都0，但0:0仍沒有數值比；不由已知正權重略過實際後項。",
        "en": "A zero total validly allocates0 and0 with both residuals0,but0:0 has no numerical ratio.The positive declared weights do not waive the actual denominator check."
      },
      "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": [
      "15/8",
      "5/8",
      "0",
      "0",
      true,
      "3"
    ]
  },
  "conclusion": {
    "canonical_value": "(15/8, 5/8, 0/1, 0/1, 是, 3/1)",
    "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-20261011-advance-20261015-r02-ratio-proportion-foundations",
      "batch_path": "public/data/questions/batches/batch-20261011-advance-20261015-r02-ratio-proportion-foundations/questions.jsonl",
      "batch_sha256": "749548edc968c981dcb404243d5b1106d67599348b2a296460b0698a7f9441c7",
      "original_line": 56,
      "original_line_sha256": "7f58c5819978fd09ed633b46557f0ad17b44ee74c97924fd670116752493a82e",
      "question_compact_sha256": "7ab953754c52d737c8699b4bc3a3d699d7f6fac932f38167c7f0a7699b3dbce1",
      "english_row_compact_sha256": "f903898c17e8266bbd1197a834b74488ea5bbfe0beecf61604746126071c7f6a",
      "english_manifest_path": "public/data/i18n/editions/batch-20261011-advance-20261015-r02-ratio-proportion-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261011-advance-20261015-r02-ratio-proportion-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-ratio-proportion": {
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        "english_sha256": "f41a9985bd51a22f5108587c130e1ecaa9d44b5c929b8653cb23ff7ac3021771",
        "original_path": "public/data/mko/ratio-proportion.json",
        "english_path": "public/data/i18n/editions/mko-ratio-proportion.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "944e5a6e7ddc77eb28f9280c1b90da6c7083069b4790cbeb7488d009328efce8",
    "original_candidate_sha256": "f39c9e5bac81869677ec39c1c48d42bd89f7371dc9184aaba6388721eb40bfe1",
    "original_candidate_line_sha256": "f39c9e5bac81869677ec39c1c48d42bd89f7371dc9184aaba6388721eb40bfe1"
  },
  "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-p14",
    "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": "e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59",
      "question_first_witness_sha256": "0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 12,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "2cd76ddcf8ed4ae34c7bae12868751a589cfbebc86cf2e9fdc80f947cb4ae04c",
      "arithmetic_steps": 10
    },
    "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."
  }
}