{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261011-advance-20261015-r02-ratio-proportion-0100",
  "pilot_number": "R0013-07075",
  "number": 7075,
  "family_id": "qf-density15-ratio-missing-component-guard",
  "title": {
    "zh-Hant": "第 7075 題：候選比例的後項與交乘",
    "en": "Question 7075: Candidate Proportion Guards and Cross-Product Defect"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261011-advance-20261015-r02-ratio-proportion-0100",
    "batch_id": "batch-20261011-advance-20261015-r02-ratio-proportion-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-ratio-proportion"
    ],
    "title_zh": "候選比例的後項與交乘",
    "stem_zh": "Q數值比協議。待核候選a:b=x:d，a=12、b=3、d=2、候選x=(25)/(2)。求 (兩後項合法資格, 合法時解x*=ad/b或未定義, ad−b*x, 原比例成立資格)，候選後項未先假定合法。",
    "answer": {
      "kind": "expression",
      "value": "(是, 8/1, -27/2, 否)",
      "explanation_zh": "候選比例後項資格與交乘=(是, 8/1, -27/2, 否)。先核b與d，交乘殘差和原比例資格分開；非法後項不被零殘差恢復。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": 3,
      "skills": [
        "候選比例的後項與交乘"
      ]
    },
    "variant": {
      "family_id": "qf-density15-ratio-missing-component-guard",
      "seed": 202610152100,
      "parameters": {
        "a": 12,
        "b": 3,
        "d": 2,
        "x": [
          25,
          2
        ],
        "domain": "finite_Q_numeric_ratio_protocol",
        "operation": "density15_ratio_missing_component_guard",
        "semantic_template": "qf-density15-ratio-missing-component-guard/[true,\"8/1\",\"-27/2\"]"
      }
    },
    "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-0100",
    "batch_id": "batch-20261011-advance-20261015-r02-ratio-proportion-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4bb846364bb2316ad0130dd6d63777e7d44aad3ae4a63040f75c155a0c9254ea",
      "family_id": "qf-density15-ratio-missing-component-guard",
      "seed": 202610152100,
      "semantic_template": "qf-density15-ratio-missing-component-guard/[true,\"8/1\",\"-27/2\"]"
    },
    "fields": {
      "title": "Candidate Proportion Guards and Cross-Product Defect",
      "stem": "Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=12, b=3, d=2, x=25/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?).",
      "answer": {
        "kind": "expression",
        "value": "(Yes, 8/1, -27/2, No)",
        "explanation": "Check b and d before solving or asserting the proportion; an algebraically zero cross-product defect does not restore illegal ratios. The exact result, retaining the requested order, is (Yes, 8/1, -27/2, No)."
      },
      "skills": [
        "Candidate Proportion Guards and Cross-Product 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": "Q數值比協議。待核候選a:b=x:d，a=12、b=3、d=2、候選x=(25)/(2)。求 (兩後項合法資格, 合法時解x*=ad/b或未定義, ad−b*x, 原比例成立資格)，候選後項未先假定合法。",
        "en": "Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. Candidate a:b=x:d has a=12, b=3, d=2, x=25/2. Give (are both second components legal?, legal solution x*=ad/b or undefined, ad−b*x, does the original proportion hold?)."
      }
    }
  ],
  "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": "cross product ad := mul(12, 2)",
      "after": "cross product ad = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先算交乘第一項ad，即使原比例不合法，此多項式項仍可計算。",
        "en": "Compute the cross term ad;this polynomial term exists even if the original ratio is illegal."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 cross product ad，不以小數近似替代。",
          "en": "Recompute cross product ad with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "2"
        ],
        "output": "24",
        "label": "cross product ad"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "cross product b times candidate := mul(3, 25/2)",
      "after": "cross product b times candidate = 75/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二項用題目原候選x，不以另求出的x*覆寫。",
        "en": "Use the original candidate x in the second term,not a substituted solved x*."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 cross product b times candidate，不以小數近似替代。",
          "en": "Recompute cross product b times candidate with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "25/2"
        ],
        "output": "75/2",
        "label": "cross product b times candidate"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ad minus b times candidate := sub(24, 75/2)",
      "after": "ad minus b times candidate = -27/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留交乘有號殘差，與原兩後項資格分開。",
        "en": "Keep the signed cross residual separate from eligibility of both original denominators."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ad minus b times candidate，不以小數近似替代。",
          "en": "Recompute ad minus b times candidate with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "24",
          "75/2"
        ],
        "output": "-27/2",
        "label": "ad minus b times candidate"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "legal missing component x star := div(24, 3)",
      "after": "legal missing component x star = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只有原比例兩後項都合法時，才由ad/b反解。",
        "en": "Only when both original denominators are legal,solve for x* by ad/b."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 legal missing component x star，不以小數近似替代。",
          "en": "Recompute legal missing component x star with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "24",
          "3"
        ],
        "output": "8",
        "label": "legal missing component x star"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "b=3;d=2;candidate=25/2;cross residual=-27/2",
      "after": "original ratios legal=是;x*=8/1;candidate valid=否",
      "relation": "implies",
      "rule_id": "missing_ratio_component_original_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "b或d為0時原比例不合法，即使交乘殘差0也不恢復；解x*與核原候選是兩件事。",
        "en": "If b or d is0,the original proportion is illegal even when the cross residual is0.Solving for x* and auditing the supplied candidate are distinct tasks."
      },
      "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": [
      true,
      "8",
      "-27/2",
      false
    ]
  },
  "conclusion": {
    "canonical_value": "(是, 8/1, -27/2, 否)",
    "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": 100,
      "original_line_sha256": "a04d0069604b1aada0042ebc918c03e9f0141a8d2ab65af8de999c1e28f1fff4",
      "question_compact_sha256": "4bb846364bb2316ad0130dd6d63777e7d44aad3ae4a63040f75c155a0c9254ea",
      "english_row_compact_sha256": "51960d0bc4e41121f3db5a90828f48149435a99d1cd35872243aa3fd6e304c73",
      "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": {
        "sha256": "b0200b9dd1e1ceb4c2b34f37f0279a1abaf31c3021957de6cd9ea72919da47ec",
        "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": "f301043be79b8db226851962d336463142f0aa5261618be143e61482d5a6f04f",
    "original_candidate_line_sha256": "f301043be79b8db226851962d336463142f0aa5261618be143e61482d5a6f04f"
  },
  "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": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "deef154246f04617586fa485b6a3548cd109d5a880361759565c10e54ead04ad",
      "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."
  }
}