{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-r06-function-graph-0070",
  "pilot_number": "R0019-10220",
  "number": 10220,
  "family_id": "qf-function-graph-finite-set",
  "title": {
    "zh-Hant": "第 10220 題：列出有限函數圖形",
    "en": "Question 10220: Listing a Finite Function Graph"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-r06-function-graph-0070",
    "batch_id": "batch-20261003-r06-function-graph-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-function-graph"
    ],
    "title_zh": "列出有限函數圖形",
    "stem_zh": "固定直角座標系，定義域 D={7, 8, 9}，f:D→R，f(x)=(2)x+(2)。按第一座標遞增列出完整函數圖形，重複點只列一次。",
    "answer": {
      "kind": "expression",
      "value": "{(7, 16), (8, 18), (9, 20)}",
      "explanation_zh": "圖形是 {(7, 16), (8, 18), (9, 20)}。對定義域的每個輸入各列 (x,f(x))；空定義域的圖形為 ∅。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "列出有限函數圖形"
      ]
    },
    "variant": {
      "family_id": "qf-function-graph-finite-set",
      "seed": 202610036070,
      "parameters": {
        "A": [
          7,
          8,
          9
        ],
        "m": 2,
        "b": 2,
        "domain": "reals",
        "operation": "graph_set",
        "semantic_template": "qf-function-graph-finite-set/{\"A\":[7,8,9],\"b\":2,\"domain\":\"reals\",\"m\":2,\"operation\":\"graph_set\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication"
      ],
      "warning_zh": "僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-03",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261003-r06-function-graph-0070",
    "batch_id": "batch-20261003-r06-function-graph-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "79bc2782ec2e1599bc9e3e47b813dc9521919129e28bf12a63bfe9ce1acbb821",
      "family_id": "qf-function-graph-finite-set",
      "seed": 202610036070,
      "semantic_template": "qf-function-graph-finite-set/{\"A\":[7,8,9],\"b\":2,\"domain\":\"reals\",\"m\":2,\"operation\":\"graph_set\"}"
    },
    "fields": {
      "title": "Listing a Finite Function Graph",
      "stem": "In a fixed Cartesian coordinate system, D={7, 8, 9} and f:D→R is given by f(x)=(2)x+(2). List the complete graph in increasing order of first coordinate, listing identical points once.",
      "answer": {
        "kind": "expression",
        "value": "{(7, 16), (8, 18), (9, 20)}",
        "explanation": "For every input in D, list (x,f(x)). The graph is {(7, 16), (8, 18), (9, 20)}; an empty domain gives the empty graph ∅."
      },
      "skills": [
        "Listing a Finite Function Graph"
      ],
      "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": "固定直角座標系，定義域 D={7, 8, 9}，f:D→R，f(x)=(2)x+(2)。按第一座標遞增列出完整函數圖形，重複點只列一次。",
        "en": "In a fixed Cartesian coordinate system, D={7, 8, 9} and f:D→R is given by f(x)=(2)x+(2). List the complete graph in increasing order of first coordinate, listing identical points once."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw domain=[7, 8, 9]",
      "after": "sorted distinct declared domain=[7, 8, 9]",
      "relation": "implies",
      "rule_id": "function_graph_declared_domain_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本題定義域是明示有限整數子集，不自動變成全R；輸入清單須本來相異。座標第一個是輸入，第二個是輸出。",
        "en": "The domain is the declared finite integer subset,not automatically all R.The input list must already be distinct.The first coordinate is input,the second 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": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "domain input7 m*x := mul(2, 7)",
      "after": "domain input7 m*x = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 domain input7 m*x，不以小數近似替代。",
          "en": "Recompute domain input7 m*x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "7"
        ],
        "output": "14",
        "label": "domain input7 m*x"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "domain input7 m*x+b := sum(14, 2)",
      "after": "domain input7 m*x+b = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 domain input7 m*x+b，不以小數近似替代。",
          "en": "Recompute domain input7 m*x+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "14",
          "2"
        ],
        "output": "16",
        "label": "domain input7 m*x+b"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "legal input=7;formula value=16",
      "after": "graph point=(7, 16)",
      "relation": "implies",
      "rule_id": "each_legal_affine_graph_point",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依每個合法輸入代值各列一個有序對，不在原域外延長圖形。",
        "en": "Evaluate once per legal input and list its ordered point without extending the graph outside its declared 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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "domain input8 m*x := mul(2, 8)",
      "after": "domain input8 m*x = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 domain input8 m*x，不以小數近似替代。",
          "en": "Recompute domain input8 m*x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "8"
        ],
        "output": "16",
        "label": "domain input8 m*x"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "domain input8 m*x+b := sum(16, 2)",
      "after": "domain input8 m*x+b = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 domain input8 m*x+b，不以小數近似替代。",
          "en": "Recompute domain input8 m*x+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "2"
        ],
        "output": "18",
        "label": "domain input8 m*x+b"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "legal input=8;formula value=18",
      "after": "graph point=(8, 18)",
      "relation": "implies",
      "rule_id": "each_legal_affine_graph_point",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依每個合法輸入代值各列一個有序對，不在原域外延長圖形。",
        "en": "Evaluate once per legal input and list its ordered point without extending the graph outside its declared 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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "domain input9 m*x := mul(2, 9)",
      "after": "domain input9 m*x = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 domain input9 m*x，不以小數近似替代。",
          "en": "Recompute domain input9 m*x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "9"
        ],
        "output": "18",
        "label": "domain input9 m*x"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "domain input9 m*x+b := sum(18, 2)",
      "after": "domain input9 m*x+b = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始操作數及指定次序精確計算，保留有號值與分母條件。",
        "en": "Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 domain input9 m*x+b，不以小數近似替代。",
          "en": "Recompute domain input9 m*x+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "18",
          "2"
        ],
        "output": "20",
        "label": "domain input9 m*x+b"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "legal input=9;formula value=20",
      "after": "graph point=(9, 20)",
      "relation": "implies",
      "rule_id": "each_legal_affine_graph_point",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依每個合法輸入代值各列一個有序對，不在原域外延長圖形。",
        "en": "Evaluate once per legal input and list its ordered point without extending the graph outside its declared 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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "complete legal-input points=[[7, 16], [8, 18], [9, 20]]",
      "after": "finite graph={(7, 16), (8, 18), (9, 20)}",
      "relation": "implies",
      "rule_id": "complete_finite_graph_set",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依第一座標遞增列完整集合；域空時圖形為∅，不是杜撰(0,0)。",
        "en": "List the complete set in increasing first-coordinate order.An empty domain gives∅,not an invented(0,0)."
      },
      "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": [
      [
        7,
        "16"
      ],
      [
        8,
        "18"
      ],
      [
        9,
        "20"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "{(7, 16), (8, 18), (9, 20)}",
    "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-20261003-r06-function-graph-foundations",
      "batch_path": "public/data/questions/batches/batch-20261003-r06-function-graph-foundations/questions.jsonl",
      "batch_sha256": "672c24a566f33e7b609e784d83ad5359ac74260f1462edb0d78ab9f52ada37d0",
      "original_line": 70,
      "original_line_sha256": "0d0f9114c302b9a64f476342c0dec9116bc30f8cceec5686ecd831e23472d2ae",
      "question_compact_sha256": "79bc2782ec2e1599bc9e3e47b813dc9521919129e28bf12a63bfe9ce1acbb821",
      "english_row_compact_sha256": "5962a756e5ae3beb68f81c517b346692202a239ba65a5d3d616fff6872c6b91e",
      "english_manifest_path": "public/data/i18n/editions/batch-20261003-r06-function-graph-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261003-r06-function-graph-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-function-graph": {
        "sha256": "6a0892c9773f3366d44da22a063aa3c99163828135681c4e15719eda4e3638be",
        "english_sha256": "4c1bb95586ab87e347b417ee2f0798ef6a6326dd0c2100e111a3f89dd223f2f2",
        "original_path": "public/data/mko/function-graph.json",
        "english_path": "public/data/i18n/editions/mko-function-graph.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "7359cb2346bdb6bf38b7ca4a5fd5b5a99c3007a711f34346754d57b77b2dfcc8",
    "original_candidate_sha256": "54e662e52227a56a6f50c1f998149c4f9fea0df3ab3570ab01049e74d1368ae7",
    "original_candidate_line_sha256": "54e662e52227a56a6f50c1f998149c4f9fea0df3ab3570ab01049e74d1368ae7",
    "mko_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c",
      "previous_english_sha256": {
        "mko-function-graph": "0ff654e81592a8027c0bbed1f3702328cf47f1020be1c3c105f81fa67658fa21"
      }
    }
  },
  "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-20261026-006-p20",
    "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": "eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5",
      "question_first_witness_sha256": "3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419"
    },
    "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": "af8ed6dd9e239961959f05c779b7f3d5a0ebd4c09d958d3f60f546442ea79458",
      "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."
  }
}