{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-r06-function-graph-0099",
  "pilot_number": "R0019-10299",
  "number": 10299,
  "family_id": "qf-function-graph-table-validity",
  "title": {
    "zh-Hant": "第 10299 題：有限總函數圖形合法性",
    "en": "Question 10299: Validity of a Finite Total-Function Graph"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-r06-function-graph-0099",
    "batch_id": "batch-20261003-r06-function-graph-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-function-graph"
    ],
    "title_zh": "有限總函數圖形合法性",
    "stem_zh": "明示定義域 D={39, 40}、陪域 B={13, 14, 15}。有序對集合 {(39, 13), (40, 53)} 是否正好是某個總函數 D→B 的圖形？",
    "answer": {
      "kind": "short_text",
      "value": "否",
      "explanation_zh": "總函數=否。須讓定義域每個輸入恰有一個陪域內輸出，且不得多出域外點；垂直線至多一點不保證總定義性。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "有限總函數圖形合法性"
      ]
    },
    "variant": {
      "family_id": "qf-function-graph-table-validity",
      "seed": 202610036099,
      "parameters": {
        "A": [
          39,
          40
        ],
        "B": [
          13,
          14,
          15
        ],
        "pairs": [
          [
            39,
            13
          ],
          [
            40,
            53
          ]
        ],
        "domain": "reals",
        "operation": "graph_validity",
        "semantic_template": "qf-function-graph-table-validity/{\"A\":[39,40],\"B\":[13,14,15],\"domain\":\"reals\",\"operation\":\"graph_validity\",\"pairs\":[[39,13],[40,53]]}"
      }
    },
    "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-0099",
    "batch_id": "batch-20261003-r06-function-graph-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "cfde61bdadf517cc963547229980b556bcad23f7204df476af9ab5606b2d5799",
      "family_id": "qf-function-graph-table-validity",
      "seed": 202610036099,
      "semantic_template": "qf-function-graph-table-validity/{\"A\":[39,40],\"B\":[13,14,15],\"domain\":\"reals\",\"operation\":\"graph_validity\",\"pairs\":[[39,13],[40,53]]}"
    },
    "fields": {
      "title": "Validity of a Finite Total-Function Graph",
      "stem": "The declared domain is D={39, 40} and codomain is B={13, 14, 15}. Is the ordered-pair set {(39, 13), (40, 53)} exactly the graph of a total function D→B?",
      "answer": {
        "kind": "short_text",
        "value": "No",
        "explanation": "Check exactly one output in B for every input in D and exclude points outside D; identical repeated pairs represent the same set element. The answer is No. At most one point per vertical line alone does not establish totality on D."
      },
      "skills": [
        "Validity of a Finite Total-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": "明示定義域 D={39, 40}、陪域 B={13, 14, 15}。有序對集合 {(39, 13), (40, 53)} 是否正好是某個總函數 D→B 的圖形？",
        "en": "The declared domain is D={39, 40} and codomain is B={13, 14, 15}. Is the ordered-pair set {(39, 13), (40, 53)} exactly the graph of a total function D→B?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw domain=[39, 40]",
      "after": "sorted distinct declared domain=[39, 40]",
      "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": "codomain=[13, 14, 15];raw pair rows=[[39, 13], [40, 53]]",
      "after": "ordered-pair set=[[39, 13], [40, 53]]",
      "relation": "implies",
      "rule_id": "graph_set_duplicate_pairs_count_once",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此協定把圖形當集合：完全相同的有序對去重，但同一x的不同y不能去掉。這不同於要求每輸入恰一原始列的表格格式。",
        "en": "This protocol treats the graph as a set:deduplicate identical pairs,but retain different y values at one x.This differs from a raw-row protocol requiring one original row per input."
      },
      "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": "pair=[39, 13];domain=[39, 40];codomain=[13, 14, 15]",
      "after": "both coordinates typed=是",
      "relation": "implies",
      "rule_id": "graph_every_pair_type_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整張集合每對都須在D×B，不因某些輸入已找到合法點而忽略域外或陪域外點。",
        "en": "Every pair must lie in D×B.Do not ignore outside-domain or outside-codomain pairs after finding legal points for some 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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "pair=[40, 53];domain=[39, 40];codomain=[13, 14, 15]",
      "after": "both coordinates typed=否",
      "relation": "implies",
      "rule_id": "graph_every_pair_type_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整張集合每對都須在D×B，不因某些輸入已找到合法點而忽略域外或陪域外點。",
        "en": "Every pair must lie in D×B.Do not ignore outside-domain or outside-codomain pairs after finding legal points for some 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": "declared input=39;normalized pair set=[[39, 13], [40, 53]]",
      "after": "distinct output fiber=[13];count=1",
      "relation": "implies",
      "rule_id": "graph_each_declared_input_exactly_one",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個宣告輸入須恰一不同輸出；最多一點只給單值，不保證沒有缺輸入。",
        "en": "Each declared input needs exactly one distinct output.At most one point gives single-valuedness,not totality."
      },
      "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": "declared input=40;normalized pair set=[[39, 13], [40, 53]]",
      "after": "distinct output fiber=[53];count=1",
      "relation": "implies",
      "rule_id": "graph_each_declared_input_exactly_one",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個宣告輸入須恰一不同輸出；最多一點只給單值，不保證沒有缺輸入。",
        "en": "Each declared input needs exactly one distinct output.At most one point gives single-valuedness,not totality."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "all pair type flags=[是, 否];source fiber sizes=[1, 1]",
      "after": "total function graph=否",
      "relation": "implies",
      "rule_id": "total_graph_type_and_totality_joint_gate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合型與全定義性共同決定。空域配空圖合法，但額外域外點不合法；合法空函數不要求陪域非空。",
        "en": "Typing and totality are joint conditions.An empty domain with an empty graph is valid,but extra outside pairs are not.The valid empty function does not require a nonempty codomain."
      },
      "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": false
  },
  "conclusion": {
    "canonical_value": "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": {
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      "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"
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    "original_candidate_line_sha256": "fe04d43acab975ebba9c6cf6bc75950ee92ff2d830af2d9524e7169a55a10558",
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  "historical_source_state": {
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      "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": {
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    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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    "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."
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}