{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261004-advance-20261005-r03-bijective-0061",
  "pilot_number": "R0004-02861",
  "number": 2861,
  "family_id": "qf-bijective-fiber-profile",
  "title": {
    "zh-Hant": "第 2861 題：完整陪域纖維概況",
    "en": "Question 2861: Complete Codomain Fiber Profile"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261004-advance-20261005-r03-bijective-0061",
    "batch_id": "batch-20261004-advance-20261005-r03-bijective-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-bijective"
    ],
    "title_zh": "完整陪域纖維概況",
    "stem_zh": "明示總函數 f:A→B，A={-3, -2, -1, 0}、B={1, 3, 5, 7, 9}，函數表為 {(-3, 1), (-2, 3), (-1, 5), (0, 7)}。按目標數值順序列出全部 (b,|f⁻¹({b})|)，未命中的目標也須列零。",
    "answer": {
      "kind": "short_text",
      "value": "{(1, 1), (3, 1), (5, 1), (7, 1), (9, 0)}",
      "explanation_zh": "纖維概況={(1, 1), (3, 1), (5, 1), (7, 1), (9, 0)}。每個 B 中目標都出現一次；概況包含零纖維，並非只把已命中的值列出。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "完整陪域纖維概況"
      ]
    },
    "variant": {
      "family_id": "qf-bijective-fiber-profile",
      "seed": 202610053061,
      "parameters": {
        "A": [
          -3,
          -2,
          -1,
          0
        ],
        "B": [
          1,
          3,
          5,
          7,
          9
        ],
        "pairs": [
          [
            -3,
            1
          ],
          [
            -2,
            3
          ],
          [
            -1,
            5
          ],
          [
            0,
            7
          ]
        ],
        "operation": "fiber_profile",
        "semantic_template": "qf-bijective-fiber-profile/{\"A\":[-3,-2,-1,0],\"B\":[1,3,5,7,9],\"operation\":\"fiber_profile\",\"pairs\":[[-3,1],[-2,3],[-1,5],[0,7]]}"
      }
    },
    "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": "Codex independent producer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-04",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261004-advance-20261005-r03-bijective-0061",
    "batch_id": "batch-20261004-advance-20261005-r03-bijective-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "808de77851100e2be0b1abf957d8a44898b8dc4b4932d10a576bbe1b3105e61f",
      "family_id": "qf-bijective-fiber-profile",
      "seed": 202610053061,
      "semantic_template": "qf-bijective-fiber-profile/{\"A\":[-3,-2,-1,0],\"B\":[1,3,5,7,9],\"operation\":\"fiber_profile\",\"pairs\":[[-3,1],[-2,3],[-1,5],[0,7]]}"
    },
    "fields": {
      "title": "Complete Codomain Fiber Profile",
      "stem": "The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.",
      "answer": {
        "kind": "short_text",
        "value": "{(1, 1), (3, 1), (5, 1), (7, 1), (9, 0)}",
        "explanation": "Count the permitted preimages of every element of B to obtain {(1, 1), (3, 1), (5, 1), (7, 1), (9, 0)}. The profile includes zero-size fibers, not only attained targets."
      },
      "skills": [
        "Complete Codomain Fiber Profile"
      ],
      "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": "明示總函數 f:A→B，A={-3, -2, -1, 0}、B={1, 3, 5, 7, 9}，函數表為 {(-3, 1), (-2, 3), (-1, 5), (0, 7)}。按目標數值順序列出全部 (b,|f⁻¹({b})|)，未命中的目標也須列零。",
        "en": "The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "A=[-3, -2, -1, 0]; B=[1, 3, 5, 7, 9]; table=[[-3, 1], [-2, 3], [-1, 5], [0, 7]]",
      "after": "legal total function on the declared A→B",
      "relation": "implies",
      "rule_id": "declared_total_function",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先確認每個A元素恰有一個表值，表中沒有論域外輸入或陪域外輸出；單射與滿射都依這個陪域判定。",
        "en": "First check exactly one table value for every A-element,and no input outside A or output outside B.Injectivity and surjectivity refer to this declared 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
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "input -3",
      "after": "f(-3)=1∈B; exactly one listed value",
      "relation": "implies",
      "rule_id": "total_function_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一核對總函數的定義域列，不只比較集合大小。",
        "en": "Audit each domain row of the total function,not just set cardinalities."
      },
      "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": "input -2",
      "after": "f(-2)=3∈B; exactly one listed value",
      "relation": "implies",
      "rule_id": "total_function_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一核對總函數的定義域列，不只比較集合大小。",
        "en": "Audit each domain row of the total function,not just set cardinalities."
      },
      "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": "input -1",
      "after": "f(-1)=5∈B; exactly one listed value",
      "relation": "implies",
      "rule_id": "total_function_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一核對總函數的定義域列，不只比較集合大小。",
        "en": "Audit each domain row of the total function,not just set cardinalities."
      },
      "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": "input 0",
      "after": "f(0)=7∈B; exactly one listed value",
      "relation": "implies",
      "rule_id": "total_function_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一核對總函數的定義域列，不只比較集合大小。",
        "en": "Audit each domain row of the total function,not just set cardinalities."
      },
      "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": "target 1; all source rows examined",
      "after": "fiber=[-3]; size=1",
      "relation": "implies",
      "rule_id": "complete_codomain_fiber",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出此目標的所有原像；沒有命中的陪域元素仍列空纖維及0。",
        "en": "List every preimage of this target;an unhit codomain element still has an empty fiber of size0."
      },
      "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": "target 3; all source rows examined",
      "after": "fiber=[-2]; size=1",
      "relation": "implies",
      "rule_id": "complete_codomain_fiber",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出此目標的所有原像；沒有命中的陪域元素仍列空纖維及0。",
        "en": "List every preimage of this target;an unhit codomain element still has an empty fiber of size0."
      },
      "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": "target 5; all source rows examined",
      "after": "fiber=[-1]; size=1",
      "relation": "implies",
      "rule_id": "complete_codomain_fiber",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出此目標的所有原像；沒有命中的陪域元素仍列空纖維及0。",
        "en": "List every preimage of this target;an unhit codomain element still has an empty fiber of size0."
      },
      "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": "target 7; all source rows examined",
      "after": "fiber=[0]; size=1",
      "relation": "implies",
      "rule_id": "complete_codomain_fiber",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出此目標的所有原像；沒有命中的陪域元素仍列空纖維及0。",
        "en": "List every preimage of this target;an unhit codomain element still has an empty fiber of size0."
      },
      "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": "target 9; all source rows examined",
      "after": "fiber=[]; size=0",
      "relation": "implies",
      "rule_id": "complete_codomain_fiber",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整列出此目標的所有原像；沒有命中的陪域元素仍列空纖維及0。",
        "en": "List every preimage of this target;an unhit codomain element still has an empty fiber of size0."
      },
      "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": "[]",
      "after": "all fibers have size≤1: 是",
      "relation": "implies",
      "rule_id": "injectivity_fiber_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "單射表示兩個不同輸入不能有相同輸出；纖維至多一個元素正是此條件。空定義域時條件真。",
        "en": "Injectivity forbids distinct inputs with the same output,exactly the condition that every fiber has size at most1.It holds vacuously for an empty domain."
      },
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "missing codomain targets=[9]",
      "after": "all codomain fibers nonempty: 否",
      "relation": "implies",
      "rule_id": "surjectivity_fiber_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "滿射要求每個陪域元素都有原像；陪域為空時此全稱條件真，但總函數前提仍不可省略。",
        "en": "Surjectivity requires a preimage for every codomain element.The universal condition is true for an empty codomain,but the total-function premise is still required."
      },
      "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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "injective=是;surjective=否",
      "after": "requested result={(1, 1), (3, 1), (5, 1), (7, 1), (9, 0)}",
      "relation": "implies",
      "rule_id": "bijection_requested_classification",
      "conditions": [],
      "explanation": {
        "zh-Hant": "雙射是單射與滿射的合取；纖維概況按全部陪域數值排序，並保留零纖維。",
        "en": "A bijection is both injective and surjective;the fiber profile is numerically ordered over the entire codomain,including zero fibers."
      },
      "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": [
      [
        1,
        1
      ],
      [
        3,
        1
      ],
      [
        5,
        1
      ],
      [
        7,
        1
      ],
      [
        9,
        0
      ]
    ],
    "fibers": [
      [
        1,
        [
          -3
        ]
      ],
      [
        3,
        [
          -2
        ]
      ],
      [
        5,
        [
          -1
        ]
      ],
      [
        7,
        [
          0
        ]
      ],
      [
        9,
        []
      ]
    ],
    "injective": true,
    "surjective": false,
    "claim_scope": "declared_total_function"
  },
  "conclusion": {
    "canonical_value": "{(1, 1), (3, 1), (5, 1), (7, 1), (9, 0)}",
    "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-20261004-advance-20261005-r03-bijective-foundations",
      "batch_path": "public/data/questions/batches/batch-20261004-advance-20261005-r03-bijective-foundations/questions.jsonl",
      "batch_sha256": "f4a605cc0336af363b5bdd962e1eedcd3f9df30552bca14e1231a63e56c4783e",
      "original_line": 61,
      "original_line_sha256": "19411685cc8073599c8a523ff790141ef45f96ad87ce1cf20d595d32d18dd072",
      "question_compact_sha256": "808de77851100e2be0b1abf957d8a44898b8dc4b4932d10a576bbe1b3105e61f",
      "english_row_compact_sha256": "64a2313f7dbb061ffda6dc86b6c3ebe5035c32b1586d85e2cbbfcc0eb6beea57",
      "english_manifest_path": "public/data/i18n/editions/batch-20261004-advance-20261005-r03-bijective-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261004-advance-20261005-r03-bijective-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-bijective": {
        "sha256": "3f8aacffa7cd2d06121165a4baccd488e1af92185dc745b6ee2cd5733c59c73c",
        "english_sha256": "6263c10144a28f0b03f69a5c27ee3b32c7eaf33f0446b48924c55a2d093b71e9",
        "original_path": "public/data/mko/bijective.json",
        "english_path": "public/data/i18n/editions/mko-bijective.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b254245aa10315455f756d4620a6ac0243e0146bccea5d89e872f1359bb32a1d",
    "original_candidate_sha256": "852585e1fefd5a1988893197ea8da2a9770bef9bde8fedc75e42a3fabcd82915",
    "original_candidate_line_sha256": "852585e1fefd5a1988893197ea8da2a9770bef9bde8fedc75e42a3fabcd82915"
  },
  "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-20261026-004-p5",
    "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": "312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56",
      "question_first_witness_sha256": "831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 13,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "f696beae7def8136caf100ca94c227b6c5abe3afb83117d91fde40545aaaa378",
      "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."
  }
}