{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r07-surjective-0031",
  "pilot_number": "R0014-07706",
  "number": 7706,
  "family_id": "qf-density16-surj-changed-codomain-qualification",
  "title": {
    "zh-Hant": "第 7706 題：新陪域的全表資格",
    "en": "Question 7706: Changed Codomain Must Contain All Original Outputs"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r07-surjective-0031",
    "batch_id": "batch-20261012-advance-20261016-r07-surjective-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-surjective"
    ],
    "title_zh": "新陪域的全表資格",
    "stem_zh": "Fin型別從0起，完整總表；Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]，images=[0, 0, 0, 0, 1]。候選新陪域原序codomain=[2, 1, 0]，可用原Fin(m)外的相異整數標籤。求 (原images, 排序新陪域, 全原輸出合新陪域資格, 合格時缺目標或不合陪域, 滿射資格或不求值)。",
    "answer": {
      "kind": "expression",
      "value": "([0, 0, 0, 0, 1], [0, 1, 2], 是, [2], 否)",
      "explanation_zh": "有限結構核對=([0, 0, 0, 0, 1], [0, 1, 2], 是, [2], 否)。先核每個仍存在的原輸出；全部落在新陪域才列新缺目標，不丟掉輸出來洗白。有限精確核對不是一般形式證明。先資格後求值，空函數條件依宣告陪域。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "新陪域的全表資格"
      ]
    },
    "variant": {
      "family_id": "qf-density16-surj-changed-codomain-qualification",
      "seed": 202610167031,
      "parameters": {
        "n": 5,
        "m": 3,
        "images": [
          0,
          0,
          0,
          0,
          1
        ],
        "codomain": [
          2,
          1,
          0
        ],
        "domain": "Fin_total_table",
        "operation": "surj-changed-codomain-qualification",
        "semantic_template": "qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[0,0,0,0,1],\"m\":3,\"n\":5}"
      }
    },
    "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": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261016-r07-surjective-0031",
    "batch_id": "batch-20261012-advance-20261016-r07-surjective-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "b7ce9e0c5bd6548974bcff7ca3906606d3a3f5c7ded59e234e1dba471b7ae3c3",
      "family_id": "qf-density16-surj-changed-codomain-qualification",
      "seed": 202610167031,
      "semantic_template": "qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[0,0,0,0,1],\"m\":3,\"n\":5}"
    },
    "fields": {
      "title": "Changed Codomain Must Contain All Original Outputs",
      "stem": "Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).",
      "answer": {
        "kind": "expression",
        "value": "([0, 0, 0, 0, 1], [0, 1, 2], Yes, [2], No)",
        "explanation": "Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [0, 1, 2], Yes, [2], No)."
      },
      "skills": [
        "Changed Codomain Must Contain All Original Outputs"
      ],
      "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": "Fin型別從0起，完整總表；Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]，images=[0, 0, 0, 0, 1]。候選新陪域原序codomain=[2, 1, 0]，可用原Fin(m)外的相異整數標籤。求 (原images, 排序新陪域, 全原輸出合新陪域資格, 合格時缺目標或不合陪域, 滿射資格或不求值)。",
        "en": "Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "n=5;m=3;images=[0, 0, 0, 0, 1]",
      "after": "all original inputs and outputs form a typed total table;labels start at0",
      "relation": "implies",
      "rule_id": "finite_surjection_typed_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核整個原表為Fin(n)到Fin(m)的總函數，包含不會被候選查詢的格；目標資格通過後才談其原像與滿射。",
        "en": "First validate the entire original table as a total function from Fin(n) to Fin(m),including cells not used by a query.Only then evaluate preimages for eligible targets and surjectivity."
      },
      "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": "original images=[0, 0, 0, 0, 1];candidate codomain=[0, 1, 2]",
      "after": "all original outputs eligible=是",
      "relation": "implies",
      "rule_id": "changed_codomain_full_range_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "新陪域可以是任意相異整數標籤，但必須包含每個原輸出；不可丟棄域外輸出來偽造函數資格。",
        "en": "The new codomain may use arbitrary distinct integer labels,but must contain every original output.Do not discard outside outputs to manufacture function eligibility."
      },
      "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": "phase=changed codomain;target=0;source labels=[0, 1, 2, 3, 4];images on labels=[0, 0, 0, 0, 1]",
      "after": "fiber=[0, 1, 2, 3];missing=否",
      "relation": "implies",
      "rule_id": "surjection_complete_target_fiber",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對這個已合型目標掃描全部保留原輸入，列出所有原像而不是只找第一個；沒有原像才是缺目標。",
        "en": "For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty."
      },
      "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": "phase=changed codomain;target=1;source labels=[0, 1, 2, 3, 4];images on labels=[0, 0, 0, 0, 1]",
      "after": "fiber=[4];missing=否",
      "relation": "implies",
      "rule_id": "surjection_complete_target_fiber",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對這個已合型目標掃描全部保留原輸入，列出所有原像而不是只找第一個；沒有原像才是缺目標。",
        "en": "For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty."
      },
      "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": "phase=changed codomain;target=2;source labels=[0, 1, 2, 3, 4];images on labels=[0, 0, 0, 0, 1]",
      "after": "fiber=[];missing=是",
      "relation": "implies",
      "rule_id": "surjection_complete_target_fiber",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對這個已合型目標掃描全部保留原輸入，列出所有原像而不是只找第一個；沒有原像才是缺目標。",
        "en": "For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty."
      },
      "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": "phase=changed codomain;targets=[0, 1, 2];fibers=[[0, 1, 2, 3], [4], []]",
      "after": "missing=[2];surjective=否",
      "relation": "implies",
      "rule_id": "surjection_all_declared_targets",
      "conditions": [],
      "explanation": {
        "zh-Hant": "滿射相對於宣告陪域：每個目標都須有原像。空陪域沒有待查目標，條件真；陪域非空且域空則每個目標都缺。",
        "en": "Surjectivity is relative to the declared codomain:every target needs a preimage.An empty codomain has no targets to test and satisfies the condition;an empty domain with a nonempty codomain misses every target."
      },
      "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": "qualified=是;missing/status=[2]",
      "after": "surjectivity/status=否",
      "relation": "implies",
      "rule_id": "changed_codomain_guarded_result",
      "conditions": [],
      "explanation": {
        "zh-Hant": "不合新陪域時沒有這個指定型別的函數，因此滿射欄是不求值而不是假；合格時才使用新陪域全部目標。",
        "en": "If an output is outside the new codomain,the specified function type is not valid,so surjectivity is unevaluated rather than False.When eligible,use all targets of the new 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": [
      [
        0,
        0,
        0,
        0,
        1
      ],
      [
        0,
        1,
        2
      ],
      true,
      [
        2
      ],
      "否"
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 0, 0, 0, 1], [0, 1, 2], 是, [2], 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-20261012-advance-20261016-r07-surjective-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261016-r07-surjective-foundations/questions.jsonl",
      "batch_sha256": "2e96f82e444d7357027861ae1bf02bfabdf513a5219df4e4a643472e67389b30",
      "original_line": 31,
      "original_line_sha256": "3e080f1fd9f63245c59c942edb69aa8d200bfaf21e4605a1609dd3a28dbfbdd2",
      "question_compact_sha256": "b7ce9e0c5bd6548974bcff7ca3906606d3a3f5c7ded59e234e1dba471b7ae3c3",
      "english_row_compact_sha256": "a2274324f714b2316b9e6365a3bef9fa910e5eb55f0b5111cd60cd9072e83698",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261016-r07-surjective-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261016-r07-surjective-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-surjective": {
        "sha256": "9e23f524f4cd5d7a32f8fed322c2a1717b66148676cc519dc1288a1346303feb",
        "english_sha256": "dc93237b8c19a6b091095797c7bb5b1ffc1e339f1c4437a9a8fecc626775520a",
        "original_path": "public/data/mko/surjective.json",
        "english_path": "public/data/i18n/editions/mko-surjective.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "d7860245c70062705fa4858c13821db6ff286678dec63f22464364f5d24cfcc1",
    "original_candidate_sha256": "94a90b97b7eb0bb94854c7ea7037ed01067c3e8d652685c9e20a4325e0702def",
    "original_candidate_line_sha256": "94a90b97b7eb0bb94854c7ea7037ed01067c3e8d652685c9e20a4325e0702def"
  },
  "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-p15",
    "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": "23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77",
      "question_first_witness_sha256": "17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "8958f34f75d5e2285b9f22b119c5c8664d4ae17937423ca8368789874f0e5f70",
      "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."
  }
}