{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r06-injective-0059",
  "pilot_number": "R0013-07434",
  "number": 7434,
  "family_id": "qf-density16-inj-append-output-qualification",
  "title": {
    "zh-Hant": "第 7434 題：附加輸出的陪域資格",
    "en": "Question 7434: Append Candidate Guard before New Table and Collisions"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r06-injective-0059",
    "batch_id": "batch-20261012-advance-20261016-r06-injective-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-injective"
    ],
    "title_zh": "附加輸出的陪域資格",
    "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=[2, 0, 0, 0, 1]。在新輸入標籤n附加候選value=-1。求 (原images, 新值陪域資格, 新images或不合陪域, 新碰撞或不求值, 新單射資格或不求值)。合格附加可使新域有9標籤。",
    "answer": {
      "kind": "expression",
      "value": "([2, 0, 0, 0, 1], 否, 不合陪域, 不求值, 不求值)",
      "explanation_zh": "有限結構核對=([2, 0, 0, 0, 1], 否, 不合陪域, 不求值, 不求值)。候選先落在原陪域才附加新格並枚舉碰撞；域外不造新表或性質。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": 2,
      "skills": [
        "附加輸出的陪域資格"
      ]
    },
    "variant": {
      "family_id": "qf-density16-inj-append-output-qualification",
      "seed": 202610166059,
      "parameters": {
        "n": 5,
        "m": 3,
        "images": [
          2,
          0,
          0,
          0,
          1
        ],
        "value": -1,
        "domain": "Fin_total_table",
        "operation": "inj-append-output-qualification",
        "semantic_template": "qf-density16-inj-append-output-qualification/{\"images\":[2,0,0,0,1],\"m\":3,\"n\":5,\"value\":-1}"
      }
    },
    "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-r06-injective-0059",
    "batch_id": "batch-20261012-advance-20261016-r06-injective-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "290581d5df58f1f203bcdec61789ddd8eee265d94fa3fea9138bcc1e6c30f163",
      "family_id": "qf-density16-inj-append-output-qualification",
      "seed": 202610166059,
      "semantic_template": "qf-density16-inj-append-output-qualification/{\"images\":[2,0,0,0,1],\"m\":3,\"n\":5,\"value\":-1}"
    },
    "fields": {
      "title": "Append Candidate Guard before New Table and Collisions",
      "stem": "Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.",
      "answer": {
        "kind": "expression",
        "value": "([2, 0, 0, 0, 1], No, outside-codomain, not-evaluated, not-evaluated)",
        "explanation": "Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], No, outside-codomain, not-evaluated, not-evaluated)."
      },
      "skills": [
        "Append Candidate Guard before New Table and Collisions"
      ],
      "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=[2, 0, 0, 0, 1]。在新輸入標籤n附加候選value=-1。求 (原images, 新值陪域資格, 新images或不合陪域, 新碰撞或不求值, 新單射資格或不求值)。合格附加可使新域有9標籤。",
        "en": "Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source index=0∈Fin(5)",
      "after": "image[0]=2∈Fin(3)",
      "relation": "implies",
      "rule_id": "injective_complete_original_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。",
        "en": "First type-check every cell of the complete original table,not only cells queried later."
      },
      "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": "source index=1∈Fin(5)",
      "after": "image[1]=0∈Fin(3)",
      "relation": "implies",
      "rule_id": "injective_complete_original_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。",
        "en": "First type-check every cell of the complete original table,not only cells queried later."
      },
      "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": "source index=2∈Fin(5)",
      "after": "image[2]=0∈Fin(3)",
      "relation": "implies",
      "rule_id": "injective_complete_original_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。",
        "en": "First type-check every cell of the complete original table,not only cells queried later."
      },
      "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": "source index=3∈Fin(5)",
      "after": "image[3]=0∈Fin(3)",
      "relation": "implies",
      "rule_id": "injective_complete_original_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。",
        "en": "First type-check every cell of the complete original table,not only cells queried later."
      },
      "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": "source index=4∈Fin(5)",
      "after": "image[4]=1∈Fin(3)",
      "relation": "implies",
      "rule_id": "injective_complete_original_table_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。",
        "en": "First type-check every cell of the complete original table,not only cells queried later."
      },
      "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": "Fin(5)→Fin(3);original images=[2, 0, 0, 0, 1]",
      "after": "injectivity forbids equal outputs from distinct original input labels",
      "relation": "implies",
      "rule_id": "finite_injectivity_label_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "單射只禁止不同輸入同輸出；空函數可空真，不能混同滿射。所有原表格先合法，後續限制或刪除保留原始標籤。",
        "en": "Injectivity forbids equal outputs for distinct inputs;the empty function is vacuously injective,which is not surjectivity.Type-check the whole original table and retain original labels under restriction or deletion."
      },
      "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": "append candidate value=-1;old codomain Fin(3)",
      "after": "qualified=否",
      "relation": "implies",
      "rule_id": "append_output_codomain_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先要求新值落在原陪域；不合陪域時不造新表，也不把未求值的單射性填成假。",
        "en": "Require the new value to lie in the original codomain before constructing a table;an outside value leaves injectivity unevaluated,not False."
      },
      "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,
        0,
        0,
        0,
        1
      ],
      false,
      "不合陪域",
      "不求值",
      "不求值"
    ]
  },
  "conclusion": {
    "canonical_value": "([2, 0, 0, 0, 1], 否, 不合陪域, 不求值, 不求值)",
    "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-r06-injective-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261016-r06-injective-foundations/questions.jsonl",
      "batch_sha256": "65bff135724c246bb730628d91519bc7486fb8a7f68f57e348e6019c9edaa9e8",
      "original_line": 59,
      "original_line_sha256": "6cfe5228a2a896017f6abdaed5b477f36171500356b0f6cae2bb77d1f7b70331",
      "question_compact_sha256": "290581d5df58f1f203bcdec61789ddd8eee265d94fa3fea9138bcc1e6c30f163",
      "english_row_compact_sha256": "bbdcf74768271614782921f5610b913aa324affb30568d80ab78ef01fb4d43e7",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261016-r06-injective-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261016-r06-injective-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-injective": {
        "sha256": "3075724e6925d827d24d227d1a1ae7ef1d5fb63e41ca8cbf87adcb12c8c76941",
        "english_sha256": "9fe4b6d40c7d49c6b6b3428ebeba9f5e119a339a59cb8c0838fc01d5e17f69bc",
        "original_path": "public/data/mko/injective.json",
        "english_path": "public/data/i18n/editions/mko-injective.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "944e5a6e7ddc77eb28f9280c1b90da6c7083069b4790cbeb7488d009328efce8",
    "original_candidate_sha256": "f28679b3bc66cb87089ac6854cbf8a8ae4961fbaf4b03bbe90c1b2f79f353038",
    "original_candidate_line_sha256": "f28679b3bc66cb87089ac6854cbf8a8ae4961fbaf4b03bbe90c1b2f79f353038"
  },
  "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-p14",
    "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": "e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59",
      "question_first_witness_sha256": "0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299"
    },
    "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": "a2120abe29ad77da4c64481690ccf03f765f300f16e568a8868580cd33ab8e4d",
      "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."
  }
}