{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261021-r01-mathematical-representation-0016",
  "pilot_number": "R0004-02616",
  "number": 2616,
  "family_id": "qf-app21-representation-affine-table",
  "title": {
    "zh-Hant": "第 2616 題：數學表徵語義橋",
    "en": "Question 2616: A Rule-to-Table Bridge on an Explicit Finite Domain"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261021-r01-mathematical-representation-0016",
    "batch_id": "batch-20261013-advance-20261021-r01-mathematical-representation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-mathematical-representation"
    ],
    "title_zh": "數學表徵語義橋",
    "stem_zh": "明示文字規則、符號或有限表格。操作=representation-affine-table；資料={\"a\":-2,\"b\":3,\"xs\":[-2,-1,1]}。請依 [a,b,xs,table] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(-2, 3, [-2, -1, 1], [[-2, 7], [-1, 5], [1, 1]])",
      "explanation_zh": "應用擴展核對=(-2, 3, [-2, -1, 1], [[-2, 7], [-1, 5], [1, 1]])；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不由圖形外觀補值；有限表徵一致不證未列域上的等價。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "數學表徵語義橋"
      ]
    },
    "variant": {
      "family_id": "qf-app21-representation-affine-table",
      "seed": 202610211016,
      "parameters": {
        "a": -2,
        "b": 3,
        "xs": [
          -2,
          -1,
          1
        ],
        "domain": "Finite_representation_bridge",
        "operation": "representation-affine-table",
        "semantic_template": "qf-app21-representation-affine-table/{\"a\":-2,\"b\":3,\"xs\":[-2,-1,1]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261021-r01-mathematical-representation-0016",
    "batch_id": "batch-20261013-advance-20261021-r01-mathematical-representation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "11687c9e4d41ab680678c66beb05a4572d39ff011401e2d044d33b740b777ad0",
      "family_id": "qf-app21-representation-affine-table",
      "seed": 202610211016,
      "semantic_template": "qf-app21-representation-affine-table/{\"a\":-2,\"b\":3,\"xs\":[-2,-1,1]}"
    },
    "fields": {
      "title": "A Rule-to-Table Bridge on an Explicit Finite Domain",
      "stem": "Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-2, b=3, inputs=[-2, -1, 1]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row).",
      "answer": {
        "kind": "expression",
        "value": "(-2, 3, [-2, -1, 1], [[-2, 7], [-1, 5], [1, 1]])",
        "explanation": "Remove repeated inputs, sort numerically and evaluate the rule at every declared input. Do not fill unlisted points from a picture. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, 3, [-2, -1, 1], [[-2, 7], [-1, 5], [1, 1]])."
      },
      "skills": [
        "A Rule-to-Table Bridge on an Explicit Finite Domain"
      ],
      "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": "明示文字規則、符號或有限表格。操作=representation-affine-table；資料={\"a\":-2,\"b\":3,\"xs\":[-2,-1,1]}。請依 [a,b,xs,table] 欄序回答。",
        "en": "Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-2, b=3, inputs=[-2, -1, 1]; f(x)=a*x+b. Give (a,b,sorted distinct inputs,every [x,f(x)] row)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared symbolic rule or complete finite table",
      "after": "keep the stated domain and representation distinct",
      "relation": "implies",
      "rule_id": "representation_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只由明示規則或表格計算；有限表格的一致性不自動證明未列出論域上的函數等同。",
        "en": "Use only the declared rule or table;agreement on a finite table alone does not establish function equality on an unlisted domain."
      },
      "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": "[-2, -1, 1]",
      "after": "[-2, -1, 1]",
      "relation": "implies",
      "rule_id": "distinct_numeric_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "定義域按數值去重排序；重複輸入不產生額外函數值。",
        "en": "Deduplicate and numerically order the domain;repeating an input creates no additional function value."
      },
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "x=-2 ax := mul(-2, -2)",
      "after": "x=-2 ax = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先算乘法a·x，避免把a(x+b)誤作原規則。",
        "en": "Evaluate a·x first;do not replace the rule by a(x+b)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x=-2 ax，不以小數近似替代。",
          "en": "Recompute x=-2 ax with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-2"
        ],
        "output": "4",
        "label": "x=-2 ax"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "x=-2 ax+b := sum(4, 3)",
      "after": "x=-2 ax+b = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加截距b，完整代入此輸入。",
        "en": "Then add the intercept b,fully substituting this input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x=-2 ax+b，不以小數近似替代。",
          "en": "Recompute x=-2 ax+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "4",
          "3"
        ],
        "output": "7",
        "label": "x=-2 ax+b"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "x=-2; ax+b=7",
      "after": "[-2, 7]",
      "relation": "implies",
      "rule_id": "representation_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個輸入恰對應一個算出的輸出，座標對的順序是輸入再輸出。",
        "en": "Each input is paired with its calculated output;the ordered pair lists input before output."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "x=-1 ax := mul(-2, -1)",
      "after": "x=-1 ax = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先算乘法a·x，避免把a(x+b)誤作原規則。",
        "en": "Evaluate a·x first;do not replace the rule by a(x+b)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x=-1 ax，不以小數近似替代。",
          "en": "Recompute x=-1 ax with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-1"
        ],
        "output": "2",
        "label": "x=-1 ax"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "x=-1 ax+b := sum(2, 3)",
      "after": "x=-1 ax+b = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加截距b，完整代入此輸入。",
        "en": "Then add the intercept b,fully substituting this input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x=-1 ax+b，不以小數近似替代。",
          "en": "Recompute x=-1 ax+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "3"
        ],
        "output": "5",
        "label": "x=-1 ax+b"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "x=-1; ax+b=5",
      "after": "[-1, 5]",
      "relation": "implies",
      "rule_id": "representation_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個輸入恰對應一個算出的輸出，座標對的順序是輸入再輸出。",
        "en": "Each input is paired with its calculated output;the ordered pair lists input before output."
      },
      "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": "x=1 ax := mul(-2, 1)",
      "after": "x=1 ax = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先算乘法a·x，避免把a(x+b)誤作原規則。",
        "en": "Evaluate a·x first;do not replace the rule by a(x+b)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x=1 ax，不以小數近似替代。",
          "en": "Recompute x=1 ax with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "1"
        ],
        "output": "-2",
        "label": "x=1 ax"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "x=1 ax+b := sum(-2, 3)",
      "after": "x=1 ax+b = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加截距b，完整代入此輸入。",
        "en": "Then add the intercept b,fully substituting this input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 x=1 ax+b，不以小數近似替代。",
          "en": "Recompute x=1 ax+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "3"
        ],
        "output": "1",
        "label": "x=1 ax+b"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "x=1; ax+b=1",
      "after": "[1, 1]",
      "relation": "implies",
      "rule_id": "representation_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個輸入恰對應一個算出的輸出，座標對的順序是輸入再輸出。",
        "en": "Each input is paired with its calculated output;the ordered pair lists input before output."
      },
      "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": [
      -2,
      3,
      [
        -2,
        -1,
        1
      ],
      [
        [
          -2,
          "7"
        ],
        [
          -1,
          "5"
        ],
        [
          1,
          "1"
        ]
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(-2, 3, [-2, -1, 1], [[-2, 7], [-1, 5], [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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    "question_binding": {
      "batch_id": "batch-20261013-advance-20261021-r01-mathematical-representation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261021-r01-mathematical-representation-foundations/questions.jsonl",
      "batch_sha256": "c3d5564b7f52318668739030fffceffb4febaa2d3d1229aa1ac4a96d58a778b8",
      "original_line": 16,
      "original_line_sha256": "694c907eb421160daedf7a3622893e3cd2f6b1ef7a52f6602e310c411702e5ce",
      "question_compact_sha256": "11687c9e4d41ab680678c66beb05a4572d39ff011401e2d044d33b740b777ad0",
      "english_row_compact_sha256": "6c8cee265c57f43ae1b44121141b287bcca754745b473692b7b7e252ed0c5435",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261021-r01-mathematical-representation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261021-r01-mathematical-representation-foundations.en.jsonl"
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    "mko_bindings": {
      "mko-mathematical-representation": {
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        "original_path": "public/data/mko/mathematical-representation.json",
        "english_path": "public/data/i18n/editions/mko-mathematical-representation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b254245aa10315455f756d4620a6ac0243e0146bccea5d89e872f1359bb32a1d",
    "original_candidate_sha256": "fdca4e4f3bf0da60e4d4c7374a2974c6148d28031a3997824bffdf2ffcd75bc4",
    "original_candidate_line_sha256": "fdca4e4f3bf0da60e4d4c7374a2974c6148d28031a3997824bffdf2ffcd75bc4"
  },
  "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-20261026-004-p5",
    "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": "312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56",
      "question_first_witness_sha256": "831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 11,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "973f3b8824cf83e309147df9fe583d8de909509288177f3e3815648f5636e2c9",
      "arithmetic_steps": 6
    },
    "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."
  }
}