{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261012-r07-function-mapping-0029",
  "pilot_number": "R0010-05579",
  "number": 5579,
  "family_id": "qf-density12-mapping-range-codomain-defect",
  "title": {
    "zh-Hant": "第 5579 題：值域與未用陪域",
    "en": "Question 5579: Actual Range and Unused Codomain"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261012-r07-function-mapping-0029",
    "batch_id": "batch-20261008-advance-20261012-r07-function-mapping-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-function-mapping"
    ],
    "title_zh": "值域與未用陪域",
    "stem_zh": "完整有限總映射 f:Fin(1)→Fin(8)，索引從0；images=[0]。依序列 (遞增實際值域, 遞增未使用的Fin(m)陪域元素, 值域是否等於整個陪域)。",
    "answer": {
      "kind": "expression",
      "value": "([0], [1, 2, 3, 4, 5, 6, 7], 否)",
      "explanation_zh": "值域與未用陪域=([0], [1, 2, 3, 4, 5, 6, 7], 否)。先去重所有輸出，再枚舉陪域未用項；全定義不保證滿射，非部分程式函數。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "值域與未用陪域"
      ]
    },
    "variant": {
      "family_id": "qf-density12-mapping-range-codomain-defect",
      "seed": 202610127029,
      "parameters": {
        "n": 1,
        "m": 8,
        "images": [
          0
        ],
        "operation": "density12_mapping_range_codomain_defect",
        "domain": "finite_total_mapping_protocol",
        "semantic_template": "qf-density12-mapping-range-codomain-defect/[8,[0]]"
      }
    },
    "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": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261012-r07-function-mapping-0029",
    "batch_id": "batch-20261008-advance-20261012-r07-function-mapping-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ffeabb8bf09256e22a65fc357a4ca98c127b66b78e66eeef4fc89e753d612060",
      "family_id": "qf-density12-mapping-range-codomain-defect",
      "seed": 202610127029,
      "semantic_template": "qf-density12-mapping-range-codomain-defect/[8,[0]]"
    },
    "fields": {
      "title": "Actual Range and Unused Codomain",
      "stem": "The complete total map f:Fin(1)→Fin(8) has images=[0] indexed from 0; Fin(1)={0,…,0}. Every table entry must lie in the target type. Give (increasing actual range, increasing unused codomain elements, does the range equal the entire codomain?).",
      "answer": {
        "kind": "expression",
        "value": "([0], [1, 2, 3, 4, 5, 6, 7], No)",
        "explanation": "Deduplicate all outputs, then enumerate unused codomain elements; total definition alone does not imply surjectivity. The exact result, retaining the requested order, is ([0], [1, 2, 3, 4, 5, 6, 7], No)."
      },
      "skills": [
        "Actual Range and Unused Codomain"
      ],
      "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": "完整有限總映射 f:Fin(1)→Fin(8)，索引從0；images=[0]。依序列 (遞增實際值域, 遞增未使用的Fin(m)陪域元素, 值域是否等於整個陪域)。",
        "en": "The complete total map f:Fin(1)→Fin(8) has images=[0] indexed from 0; Fin(1)={0,…,0}. Every table entry must lie in the target type. Give (increasing actual range, increasing unused codomain elements, does the range equal the entire codomain?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "f:Fin(1)→Fin(8);source indices start at0",
      "after": "complete total map,not a partial program function",
      "relation": "implies",
      "rule_id": "total_finite_mapping_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個合法來源元素恰有一個陪域內值；Fin(0)是空型別，域外查詢不合型別，不是合法輸入的部分函數空洞。",
        "en": "Every legal source element has one codomain value.Fin(0) is empty;an outside query is ill-typed,not a hole of a partial function at a legal 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": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "f input=0∈Fin(1)",
      "after": "f(0)=0∈Fin(8)",
      "relation": "implies",
      "rule_id": "complete_mapping_entry",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整核對每個表項的來源及陪域資格，未查詢的項也不能非法。",
        "en": "Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal."
      },
      "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": "codomain element=0",
      "after": "preimages=[0];used=是",
      "relation": "implies",
      "rule_id": "codomain_fiber_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個陪域目標都核對，未被命中仍是陪域元素但不在實際值域。",
        "en": "Audit every codomain target;an unhit target remains in the codomain but not in the actual range."
      },
      "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": "codomain element=1",
      "after": "preimages=[];used=否",
      "relation": "implies",
      "rule_id": "codomain_fiber_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個陪域目標都核對，未被命中仍是陪域元素但不在實際值域。",
        "en": "Audit every codomain target;an unhit target remains in the codomain but not in the actual range."
      },
      "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": "codomain element=2",
      "after": "preimages=[];used=否",
      "relation": "implies",
      "rule_id": "codomain_fiber_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個陪域目標都核對，未被命中仍是陪域元素但不在實際值域。",
        "en": "Audit every codomain target;an unhit target remains in the codomain but not in the actual range."
      },
      "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": "codomain element=3",
      "after": "preimages=[];used=否",
      "relation": "implies",
      "rule_id": "codomain_fiber_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個陪域目標都核對，未被命中仍是陪域元素但不在實際值域。",
        "en": "Audit every codomain target;an unhit target remains in the codomain but not in the actual range."
      },
      "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": "codomain element=4",
      "after": "preimages=[];used=否",
      "relation": "implies",
      "rule_id": "codomain_fiber_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個陪域目標都核對，未被命中仍是陪域元素但不在實際值域。",
        "en": "Audit every codomain target;an unhit target remains in the codomain but not in the actual range."
      },
      "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": "codomain element=5",
      "after": "preimages=[];used=否",
      "relation": "implies",
      "rule_id": "codomain_fiber_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個陪域目標都核對，未被命中仍是陪域元素但不在實際值域。",
        "en": "Audit every codomain target;an unhit target remains in the codomain but not in the actual range."
      },
      "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": "codomain element=6",
      "after": "preimages=[];used=否",
      "relation": "implies",
      "rule_id": "codomain_fiber_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個陪域目標都核對，未被命中仍是陪域元素但不在實際值域。",
        "en": "Audit every codomain target;an unhit target remains in the codomain but not in the actual range."
      },
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "codomain element=7",
      "after": "preimages=[];used=否",
      "relation": "implies",
      "rule_id": "codomain_fiber_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個陪域目標都核對，未被命中仍是陪域元素但不在實際值域。",
        "en": "Audit every codomain target;an unhit target remains in the codomain but not in the actual range."
      },
      "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": "actual range=[0];unused codomain=[1, 2, 3, 4, 5, 6, 7]",
      "after": "range equals codomain=否",
      "relation": "implies",
      "rule_id": "range_vs_declared_codomain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "總函數不保證滿射；只有全部陪域都有原像才值域等於陪域。Fin(0)到Fin(0)的空映射在此判準下通過。",
        "en": "Totality does not imply surjectivity;the range equals the codomain only if every target has a preimage.The empty map Fin(0)→Fin(0) satisfies this criterion."
      },
      "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
      ],
      [
        1,
        2,
        3,
        4,
        5,
        6,
        7
      ],
      false
    ],
    "codomain_fibers": [
      [
        0,
        [
          0
        ]
      ],
      [
        1,
        []
      ],
      [
        2,
        []
      ],
      [
        3,
        []
      ],
      [
        4,
        []
      ],
      [
        5,
        []
      ],
      [
        6,
        []
      ],
      [
        7,
        []
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([0], [1, 2, 3, 4, 5, 6, 7], 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-20261008-advance-20261012-r07-function-mapping-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261012-r07-function-mapping-foundations/questions.jsonl",
      "batch_sha256": "37b7f1f791b76c79f8565271927c13bc33d4c471e6c21c12196d8dfc284d099f",
      "original_line": 29,
      "original_line_sha256": "e6936ee4f2126f79989c7a353dc73854aa3d8844e5f96ac2f169b5cf955ac020",
      "question_compact_sha256": "ffeabb8bf09256e22a65fc357a4ca98c127b66b78e66eeef4fc89e753d612060",
      "english_row_compact_sha256": "1abe601502075d8c58632036495bd7bd918a1b96e19f1bd10bea37576cc12d57",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261012-r07-function-mapping-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261012-r07-function-mapping-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-function-mapping": {
        "sha256": "f103408037c3680d770316d998ec3b92d19797f2865d8c94fa74efaf38bbc2b0",
        "english_sha256": "d94512a955a174d5185733591c5b2d72ff09a0546f48e27a9024138e9275b64b",
        "original_path": "public/data/mko/function-mapping.json",
        "english_path": "public/data/i18n/editions/mko-function-mapping.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "aeab777a3897ae2c01315b86b5e9b9a43ffb59b075efae44fd95d447123d32f1",
    "original_candidate_sha256": "a9f6ee1869cf901602bba4bc2b3dc2bc60b74e3eeba5a44cafa4fa51c41b34f1",
    "original_candidate_line_sha256": "a9f6ee1869cf901602bba4bc2b3dc2bc60b74e3eeba5a44cafa4fa51c41b34f1"
  },
  "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-p11",
    "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": "0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f",
      "question_first_witness_sha256": "0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc"
    },
    "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": "c8e24941148c436f9438cd515ed8c313dfaaceac6a6606659929af5ae5ef4431",
      "arithmetic_steps": 0
    },
    "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."
  }
}