{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261011-advance-20261015-r02-ratio-proportion-0039",
  "pilot_number": "R0013-07014",
  "number": 7014,
  "family_id": "qf-density15-ratio-complete-point-law",
  "title": {
    "zh-Hant": "第 7014 題：三點同參數比例核對",
    "en": "Question 7014: Complete Three-Point Proportion Law and Ratio Values"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261011-advance-20261015-r02-ratio-proportion-0039",
    "batch_id": "batch-20261011-advance-20261015-r02-ratio-proportion-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-ratio-proportion"
    ],
    "title_zh": "三點同參數比例核對",
    "stem_zh": "Q數值比協議。給定k=(1)/(2)、完整三點(x,y)=[[0, 0], [1, 13], [2, -12]]。依原序求 (y−kx三殘差, 同k資格三值, y/x三值或未定義)，x=0時仍核y是否為0。",
    "answer": {
      "kind": "expression",
      "value": "([0/1, 25/2, -13/1], [是, 否, 否], [未定義, 13/1, -6/1])",
      "explanation_zh": "完整三點比例律與殘差=([0/1, 25/2, -13/1], [是, 否, 否], [未定義, 13/1, -6/1])。逐點算殘差；比值再分開核x非零。有限點不能推論所有輸入。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": [
        "三點同參數比例核對"
      ]
    },
    "variant": {
      "family_id": "qf-density15-ratio-complete-point-law",
      "seed": 202610152039,
      "parameters": {
        "k": [
          1,
          2
        ],
        "pairs": [
          [
            0,
            0
          ],
          [
            1,
            13
          ],
          [
            2,
            -12
          ]
        ],
        "domain": "finite_Q_numeric_ratio_protocol",
        "operation": "density15_ratio_complete_point_law",
        "semantic_template": "qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"25/2\",\"13/1\"],[\"-13/1\",\"-6/1\"]]"
      }
    },
    "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": "2026-10-11",
      "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-0039",
    "batch_id": "batch-20261011-advance-20261015-r02-ratio-proportion-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "701a5f3789f561eb2a674db053b8015450402b5af41bf78a7d7d967166be50b1",
      "family_id": "qf-density15-ratio-complete-point-law",
      "seed": 202610152039,
      "semantic_template": "qf-density15-ratio-complete-point-law/[[\"0/1\",\"undefined\"],[\"25/2\",\"13/1\"],[\"-13/1\",\"-6/1\"]]"
    },
    "fields": {
      "title": "Complete Three-Point Proportion Law and Ratio Values",
      "stem": "Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=1/2, complete ordered pairs (x,y)=[[0, 0], [1, 13], [2, -12]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0.",
      "answer": {
        "kind": "expression",
        "value": "([0/1, 25/2, -13/1], [Yes, No, No], [undefined, 13/1, -6/1])",
        "explanation": "Check every point’s exact defect, then separately require nonzero x for its ratio; agreement on these finite points does not establish the law for every input. The exact result, retaining the requested order, is ([0/1, 25/2, -13/1], [Yes, No, No], [undefined, 13/1, -6/1])."
      },
      "skills": [
        "Complete Three-Point Proportion Law and Ratio Values"
      ],
      "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。給定k=(1)/(2)、完整三點(x,y)=[[0, 0], [1, 13], [2, -12]]。依原序求 (y−kx三殘差, 同k資格三值, y/x三值或未定義)，x=0時仍核y是否為0。",
        "en": "Use exact numeric ratios in Q; the second component must be nonzero. A zero cross-product defect cannot legalize a zero second component. k=1/2, complete ordered pairs (x,y)=[[0, 0], [1, 13], [2, -12]]. Give (three y−kx defects, three same-k eligibility values, three y/x values or undefined). At x=0 still test whether y=0."
      }
    }
  ],
  "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": "point0 kx := mul(1/2, 0)",
      "after": "point0 kx = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先代入此點x求k·x，不先把y/x當成已定義。",
        "en": "First evaluate k·x at this point,without assuming y/x is defined."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 kx，不以小數近似替代。",
          "en": "Recompute point0 kx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "0"
        ],
        "output": "0",
        "label": "point0 kx"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "point0 law residual := sub(0, 0)",
      "after": "point0 law residual = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "y−kx為此點比例律殘差。",
        "en": "The point's proportional-law residual is y−kx."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 law residual，不以小數近似替代。",
          "en": "Recompute point0 law residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "point0 law residual"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "point=(0,0);k=1/2;residual=0",
      "after": "law matches=是;ratio=未定義",
      "relation": "implies",
      "rule_id": "finite_point_law_and_ratio_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x=0時仍可核y是否0，但比值y/x未定義；三個點符合不能推論未列出的所有輸入。",
        "en": "At x=0 one can still test whether y=0,but y/x is undefined.Agreement at these three points does not establish the law on unlisted inputs."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "point1 kx := mul(1/2, 1)",
      "after": "point1 kx = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先代入此點x求k·x，不先把y/x當成已定義。",
        "en": "First evaluate k·x at this point,without assuming y/x is defined."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 kx，不以小數近似替代。",
          "en": "Recompute point1 kx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1"
        ],
        "output": "1/2",
        "label": "point1 kx"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "point1 law residual := sub(13, 1/2)",
      "after": "point1 law residual = 25/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "y−kx為此點比例律殘差。",
        "en": "The point's proportional-law residual is y−kx."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 law residual，不以小數近似替代。",
          "en": "Recompute point1 law residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "13",
          "1/2"
        ],
        "output": "25/2",
        "label": "point1 law residual"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "point1 y/x := div(13, 1)",
      "after": "point1 y/x = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只有此點x非零才求y/x。",
        "en": "Evaluate y/x only when this point's x is nonzero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 y/x，不以小數近似替代。",
          "en": "Recompute point1 y/x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "13",
          "1"
        ],
        "output": "13",
        "label": "point1 y/x"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "point=(1,13);k=1/2;residual=25/2",
      "after": "law matches=否;ratio=13/1",
      "relation": "implies",
      "rule_id": "finite_point_law_and_ratio_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x=0時仍可核y是否0，但比值y/x未定義；三個點符合不能推論未列出的所有輸入。",
        "en": "At x=0 one can still test whether y=0,but y/x is undefined.Agreement at these three points does not establish the law on unlisted inputs."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "point2 kx := mul(1/2, 2)",
      "after": "point2 kx = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先代入此點x求k·x，不先把y/x當成已定義。",
        "en": "First evaluate k·x at this point,without assuming y/x is defined."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 kx，不以小數近似替代。",
          "en": "Recompute point2 kx with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "2"
        ],
        "output": "1",
        "label": "point2 kx"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "point2 law residual := sub(-12, 1)",
      "after": "point2 law residual = -13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "y−kx為此點比例律殘差。",
        "en": "The point's proportional-law residual is y−kx."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 law residual，不以小數近似替代。",
          "en": "Recompute point2 law residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-12",
          "1"
        ],
        "output": "-13",
        "label": "point2 law residual"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "point2 y/x := div(-12, 2)",
      "after": "point2 y/x = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只有此點x非零才求y/x。",
        "en": "Evaluate y/x only when this point's x is nonzero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 y/x，不以小數近似替代。",
          "en": "Recompute point2 y/x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-12",
          "2"
        ],
        "output": "-6",
        "label": "point2 y/x"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "point=(2,-12);k=1/2;residual=-13",
      "after": "law matches=否;ratio=-6/1",
      "relation": "implies",
      "rule_id": "finite_point_law_and_ratio_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x=0時仍可核y是否0，但比值y/x未定義；三個點符合不能推論未列出的所有輸入。",
        "en": "At x=0 one can still test whether y=0,but y/x is undefined.Agreement at these three points does not establish the law on unlisted inputs."
      },
      "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": [
      [
        "0",
        "25/2",
        "-13"
      ],
      [
        true,
        false,
        false
      ],
      [
        "未定義",
        "13",
        "-6"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([0/1, 25/2, -13/1], [是, 否, 否], [未定義, 13/1, -6/1])",
    "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": 39,
      "original_line_sha256": "e7ab6e9925c653b2c0863fee7d664769240ef196025157d8179da7577cee021f",
      "question_compact_sha256": "701a5f3789f561eb2a674db053b8015450402b5af41bf78a7d7d967166be50b1",
      "english_row_compact_sha256": "8ca7bc7b13306311a0a534ac3990d344414814e786620ae5bb083738e0b74049",
      "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": "cac7fa1ad31af0abcf68127b798101d6f832f647ca238fc68a84d39adad3a749",
    "original_candidate_line_sha256": "cac7fa1ad31af0abcf68127b798101d6f832f647ca238fc68a84d39adad3a749"
  },
  "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",
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}