{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261016-r07-surjective-0053",
  "pilot_number": "R0014-07728",
  "number": 7728,
  "family_id": "qf-density16-surj-delete-input-missing",
  "title": {
    "zh-Hant": "第 7728 題：刪一輸入的缺目標變化",
    "en": "Question 7728: Source Deletion with Declared Codomain Kept"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261016-r07-surjective-0053",
    "batch_id": "batch-20261012-advance-20261016-r07-surjective-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-surjective"
    ],
    "title_zh": "刪一輸入的缺目標變化",
    "stem_zh": "Fin型別從0起，完整總表；Fin(3)=[0, 1, 2]→Fin(3)=[0, 1, 2]，images=[0, 1, 1]。刪原輸入at=2，原陪域保留。求 (原images, 原缺目標, 保留原輸入標籤, 剩餘images, 後缺目標, 後滿射資格)。",
    "answer": {
      "kind": "expression",
      "value": "([0, 1, 1], [2], [0, 1], [0, 1], [2], 否)",
      "explanation_zh": "有限結構核對=([0, 1, 1], [2], [0, 1], [0, 1], [2], 否)。保留原陪域，刪除該來源後重新收集輸出；移去唯一原像會新增缺目標。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": [
        "刪一輸入的缺目標變化"
      ]
    },
    "variant": {
      "family_id": "qf-density16-surj-delete-input-missing",
      "seed": 202610167053,
      "parameters": {
        "n": 3,
        "m": 3,
        "images": [
          0,
          1,
          1
        ],
        "at": 2,
        "domain": "Fin_total_table",
        "operation": "surj-delete-input-missing",
        "semantic_template": "qf-density16-surj-delete-input-missing/{\"at\":2,\"images\":[0,1,1],\"m\":3,\"n\":3}"
      }
    },
    "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-0053",
    "batch_id": "batch-20261012-advance-20261016-r07-surjective-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "9b71589c650a45606e9f143b04ae55105afccd45faa392aa58051aafb0cf949f",
      "family_id": "qf-density16-surj-delete-input-missing",
      "seed": 202610167053,
      "semantic_template": "qf-density16-surj-delete-input-missing/{\"at\":2,\"images\":[0,1,1],\"m\":3,\"n\":3}"
    },
    "fields": {
      "title": "Source Deletion with Declared Codomain Kept",
      "stem": "Finite types start at 0: Fin(3)={0,…,2}→Fin(3), complete images=[0, 1, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).",
      "answer": {
        "kind": "expression",
        "value": "([0, 1, 1], [2], [0, 1], [0, 1], [2], No)",
        "explanation": "Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1], [2], [0, 1], [0, 1], [2], No)."
      },
      "skills": [
        "Source Deletion with Declared Codomain Kept"
      ],
      "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(3)=[0, 1, 2]→Fin(3)=[0, 1, 2]，images=[0, 1, 1]。刪原輸入at=2，原陪域保留。求 (原images, 原缺目標, 保留原輸入標籤, 剩餘images, 後缺目標, 後滿射資格)。",
        "en": "Finite types start at 0: Fin(3)={0,…,2}→Fin(3), complete images=[0, 1, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "n=3;m=3;images=[0, 1, 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": "phase=original;target=0;source labels=[0, 1, 2];images on labels=[0, 1, 1]",
      "after": "fiber=[0];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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "phase=original;target=1;source labels=[0, 1, 2];images on labels=[0, 1, 1]",
      "after": "fiber=[1, 2];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=original;target=2;source labels=[0, 1, 2];images on labels=[0, 1, 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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "phase=original;targets=[0, 1, 2];fibers=[[0], [1, 2], []]",
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "deleted original input=2;old labels=[0, 1, 2]",
      "after": "kept labels=[0, 1];remaining images=[0, 1];codomain Fin(3)",
      "relation": "implies",
      "rule_id": "delete_input_keeps_original_labels_and_codomain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "刪除指定原輸入，不改原陪域；保留標籤不重新編號。剩餘images是按保留標籤順序列出的值，不是新的標籤宣告。",
        "en": "Delete the specified original input without changing the codomain or renumbering surviving labels.The remaining images list is ordered by retained labels,not a declaration of relabeled 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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "phase=after deletion;target=0;source labels=[0, 1];images on labels=[0, 1]",
      "after": "fiber=[0];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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "phase=after deletion;target=1;source labels=[0, 1];images on labels=[0, 1]",
      "after": "fiber=[1];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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "phase=after deletion;target=2;source labels=[0, 1];images on labels=[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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "phase=after deletion;targets=[0, 1, 2];fibers=[[0], [1], []]",
      "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
    }
  ],
  "math_model": {
    "result": [
      [
        0,
        1,
        1
      ],
      [
        2
      ],
      [
        0,
        1
      ],
      [
        0,
        1
      ],
      [
        2
      ],
      false
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 1, 1], [2], [0, 1], [0, 1], [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": 53,
      "original_line_sha256": "a8ff607f7b53bffd4bfe91fc465eb9137597841821cdb58d6fb54dd2dbf7574a",
      "question_compact_sha256": "9b71589c650a45606e9f143b04ae55105afccd45faa392aa58051aafb0cf949f",
      "english_row_compact_sha256": "f567e18a86913030f721220c6a21727dd76935fe4d71475d4956692e3b757113",
      "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": "ab25e7af3dd590d9465448f0aebf633f80511b795ef2b783201e12b285a15c4d",
    "original_candidate_line_sha256": "ab25e7af3dd590d9465448f0aebf633f80511b795ef2b783201e12b285a15c4d"
  },
  "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": 10,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "0aa3b01a5fba8a2c66a2b28f085768561190b5d773abfe44358ae7ec3d20fc8f",
      "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."
  }
}