{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261011-advance-20261015-r02-ratio-proportion-0016",
  "pilot_number": "R0013-07041",
  "number": 7041,
  "family_id": "qf-density15-ratio-forward-reverse",
  "title": {
    "zh-Hant": "第 7041 題：數值比與反向資格",
    "en": "Question 7041: Forward Ratio and Independent Reverse Guard"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261011-advance-20261015-r02-ratio-proportion-0016",
    "batch_id": "batch-20261011-advance-20261015-r02-ratio-proportion-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-ratio-proportion"
    ],
    "title_zh": "數值比與反向資格",
    "stem_zh": "Q數值比協議。a=14、b=3且b≠0。求 (a/b, 反向資格, b/a或未定義, 前後乘積或未定義)。",
    "answer": {
      "kind": "expression",
      "value": "(14/3, 是, 3/14, 1/1)",
      "explanation_zh": "正向比與反向比資格=(14/3, 是, 3/14, 1/1)。反向比另核a非零，不由合法正向比略過反向後項。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": 2,
      "skills": [
        "數值比與反向資格"
      ]
    },
    "variant": {
      "family_id": "qf-density15-ratio-forward-reverse",
      "seed": 202610152016,
      "parameters": {
        "a": 14,
        "b": 3,
        "domain": "finite_Q_numeric_ratio_protocol",
        "operation": "density15_ratio_forward_reverse",
        "semantic_template": "qf-density15-ratio-forward-reverse/\"14/3\""
      }
    },
    "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": "2026-10-11",
      "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-0016",
    "batch_id": "batch-20261011-advance-20261015-r02-ratio-proportion-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "36863262824cc393f5614b9f5c9794af3c799235810af50de94e0f6ff518f69c",
      "family_id": "qf-density15-ratio-forward-reverse",
      "seed": 202610152016,
      "semantic_template": "qf-density15-ratio-forward-reverse/\"14/3\""
    },
    "fields": {
      "title": "Forward Ratio and Independent Reverse 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. a=14, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse or undefined).",
      "answer": {
        "kind": "expression",
        "value": "(14/3, Yes, 3/14, 1/1)",
        "explanation": "The reverse ratio separately requires a≠0; a legal forward ratio cannot waive the reverse denominator guard. The exact result, retaining the requested order, is (14/3, Yes, 3/14, 1/1)."
      },
      "skills": [
        "Forward Ratio and Independent Reverse 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。a=14、b=3且b≠0。求 (a/b, 反向資格, b/a或未定義, 前後乘積或未定義)。",
        "en": "Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. a=14, b=3≠0. Give (a/b, is the reverse ratio legal?, b/a or undefined, forward*reverse 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": "forward ratio a/b := div(14, 3)",
      "after": "forward ratio a/b = 14/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正向後項b已非零，先求正向比。",
        "en": "The forward denominator b is nonzero,so compute the forward ratio."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 forward ratio a/b，不以小數近似替代。",
          "en": "Recompute forward ratio a/b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "14",
          "3"
        ],
        "output": "14/3",
        "label": "forward ratio a/b"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "reverse ratio b/a := div(3, 14)",
      "after": "reverse ratio b/a = 3/14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "反向後項是a，也必須另行非零才可除。",
        "en": "The reverse denominator is a and must separately be nonzero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reverse ratio b/a，不以小數近似替代。",
          "en": "Recompute reverse ratio b/a with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "3",
          "14"
        ],
        "output": "3/14",
        "label": "reverse ratio b/a"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "forward times reverse := mul(14/3, 3/14)",
      "after": "forward times reverse = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩方向都已定義才相乘核倒數關係。",
        "en": "Multiply to audit reciprocity only when both directions are defined."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 forward times reverse，不以小數近似替代。",
          "en": "Recompute forward times reverse with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14/3",
          "3/14"
        ],
        "output": "1",
        "label": "forward times reverse"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "a=14;b=3;forward=14/3",
      "after": "reverse defined=是;reverse=3/14",
      "relation": "implies",
      "rule_id": "reverse_ratio_independent_denominator",
      "conditions": [],
      "explanation": {
        "zh-Hant": "a=0時合法正向比可為0，卻不能據此允許反向b/0；未定義乘積也不填成0。",
        "en": "When a=0,the valid forward ratio may be0,but this does not permit b/0 in reverse.Nor may the undefined product be assigned0."
      },
      "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": [
      "14/3",
      true,
      "3/14",
      "1"
    ]
  },
  "conclusion": {
    "canonical_value": "(14/3, 是, 3/14, 1/1)",
    "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": {
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      "batch_sha256": "749548edc968c981dcb404243d5b1106d67599348b2a296460b0698a7f9441c7",
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      "english_manifest_path": "public/data/i18n/editions/batch-20261011-advance-20261015-r02-ratio-proportion-foundations.en.json",
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        "original_path": "public/data/mko/ratio-proportion.json",
        "english_path": "public/data/i18n/editions/mko-ratio-proportion.en.json"
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "944e5a6e7ddc77eb28f9280c1b90da6c7083069b4790cbeb7488d009328efce8",
    "original_candidate_sha256": "b74353980efe2e02d347778b8cc30dfab65673f7c2b4c288e34e09e059424205",
    "original_candidate_line_sha256": "b74353980efe2e02d347778b8cc30dfab65673f7c2b4c288e34e09e059424205"
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  "historical_source_state": {
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      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
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    },
    "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": {
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    "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.",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}