{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261107-r05-pigeonhole-principle-0002",
  "pilot_number": "R0039-20302",
  "number": 20302,
  "family_id": "qf-mod38-pigeon-collision-witness",
  "title": {
    "zh-Hant": "第 20302 題：鴿籠原理",
    "en": "Question 20302: A Complete Larger-Domain Map Has Two Distinct Sources with One Image"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261107-r05-pigeonhole-principle-0002",
    "batch_id": "batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-pigeonhole-principle"
    ],
    "title_zh": "鴿籠原理",
    "stem_zh": "完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[1,3,1,3,1],\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(5, 4, [0, 2, 1], 是, complete_total_map_collision)",
      "explanation_zh": "Wave6根層核對=(5, 4, [0, 2, 1], 是, complete_total_map_collision)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "鴿籠原理"
      ]
    },
    "variant": {
      "family_id": "qf-mod38-pigeon-collision-witness",
      "seed": 202611075002,
      "parameters": {
        "mapping": [
          1,
          3,
          1,
          3,
          1
        ],
        "codomain_count": 4,
        "domain": "Complete_finite_total_map_pigeonhole",
        "operation": "pigeon-collision-witness",
        "semantic_template": "qf-mod38-pigeon-collision-witness/{\"codomain_count\":4,\"mapping\":[1,3,1,3,1]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261107-r05-pigeonhole-principle-0002",
    "batch_id": "batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1b3900c045e990ed67fd09d59b90a360a41299ae6a4f82036ae445b410f5f075",
      "family_id": "qf-mod38-pigeon-collision-witness",
      "seed": 202611075002,
      "semantic_template": "qf-mod38-pigeon-collision-witness/{\"codomain_count\":4,\"mapping\":[1,3,1,3,1]}"
    },
    "fields": {
      "title": "A Complete Larger-Domain Map Has Two Distinct Sources with One Image",
      "stem": "Use the complete total mapping=[1, 3, 1, 3, 1] from source indices Fin(5)={0,…,4} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(5, 4, [0, 2, 1], Yes, complete_total_map_collision)",
        "explanation": "Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (5, 4, [0, 2, 1], Yes, complete_total_map_collision)."
      },
      "skills": [
        "A Complete Larger-Domain Map Has Two Distinct Sources with One Image"
      ],
      "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": "完整有限集合大小與total map。操作=pigeon-collision-witness；資料={\"mapping\":[1,3,1,3,1],\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the complete total mapping=[1, 3, 1, 3, 1] from source indices Fin(5)={0,…,4} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=pigeon-collision-witness;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[domain_count, codomain_count, collision, has_collision, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。欄位標籤不是額外數學證據，No保持原題的聲稱範圍。",
        "en": "Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope."
      },
      "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 complete source-indexed total map=[1, 3, 1, 3, 1];source Fin(5);ACTUAL target Fin(4)",
      "after": "every source has ONE in-range image;all target labels0..3 retained even if unused;empty source allowed",
      "relation": "implies",
      "rule_id": "pigeon_complete_total_map_not_sample_or_bin_relabeling",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原source索引順序與target標籤保留，不以首次遇見次序重新編號桶。完整total map才可算全部fibre與碰撞；空domain也仍映入非空codomain。",
        "en": "Retain original source order and target labels rather than relabeling bins by first appearance.Complete total-map data support full fibres and collisions,including an empty source into a nonempty 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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "next LEX ordered original source pair=[0, 1] with left<right;original images=[1, 3]",
      "after": "same target image=否;every earlier LEFT-then-RIGHT pair already checked unequal",
      "relation": "implies",
      "rule_id": "pigeon_first_collision_lex_pair_not_first_repeat_stream",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐left再逐right核，第一組相同才是來源首LEX pair；不能改成依right時間首次遇到重複，兩者的witness可能不同。",
        "en": "Check left first and then right;the first equal pair is the source's first lexicographic pair,which may differ from the earliest repeat encountered by increasing right endpoint."
      },
      "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": "next LEX ordered original source pair=[0, 2] with left<right;original images=[1, 1]",
      "after": "same target image=是;every earlier LEFT-then-RIGHT pair already checked unequal",
      "relation": "implies",
      "rule_id": "pigeon_first_collision_lex_pair_not_first_repeat_stream",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐left再逐right核，第一組相同才是來源首LEX pair；不能改成依right時間首次遇到重複，兩者的witness可能不同。",
        "en": "Check left first and then right;the first equal pair is the source's first lexicographic pair,which may differ from the earliest repeat encountered by increasing right endpoint."
      },
      "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": "ORIGINAL full domain size=5>target size=4;actual first collision=[0, 2, 1]",
      "after": "two DISTINCT original sources have the SAME original image;exact instance witness and finite size theorem agree",
      "relation": "implies",
      "rule_id": "pigeon_actual_distinct_source_collision_not_equal_indices",
      "conditions": [],
      "explanation": {
        "zh-Hant": "witness的兩個source索引必不同；這裡既有完整total map的真碰撞，也有任何m>n total map不可單射的大小論證，不從部分取樣推全圖。",
        "en": "The two source indices in the witness are distinct.This is an actual collision in a complete total map,consistent with the size argument for every map with m>n,not a global inference from partial samples."
      },
      "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": [
      5,
      4,
      [
        0,
        2,
        1
      ],
      true,
      "complete_total_map_collision"
    ]
  },
  "conclusion": {
    "canonical_value": "(5, 4, [0, 2, 1], 是, complete_total_map_collision)",
    "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": {
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    "question_binding": {
      "batch_id": "batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations/questions.jsonl",
      "batch_sha256": "d58393244c048d683f7cf78cef4425c148092c211893c3c037d0c067040e52fb",
      "original_line": 2,
      "original_line_sha256": "732cd29cdca44e6126fc576d40fc59a2fc3900dda09c626f3702bd51d94c2fb9",
      "question_compact_sha256": "1b3900c045e990ed67fd09d59b90a360a41299ae6a4f82036ae445b410f5f075",
      "english_row_compact_sha256": "1bcab06fcaf6dfe4484fb5d8f99415afde36a4b8c278c885a6125c8f8f318537",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-pigeonhole-principle": {
        "sha256": "3a161ec4318e1b1c3993a14395e4e5710b5529f42a9311aa9b041380c9b9eaba",
        "english_sha256": "4ea6bc9d5425a0918302e7a898231e46d17e569bb3d5b6c5661e5c57bd32235b",
        "original_path": "public/data/mko/pigeonhole-principle.json",
        "english_path": "public/data/i18n/editions/mko-pigeonhole-principle.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "521929dfe9b0919224c38312ad78e5fa56457fc886faaa12b137edd83962251e",
    "original_candidate_sha256": "1f5b0c1951badb02e761429932def3fed0dd94294a18dafe27ccae03dcf670d6",
    "original_candidate_line_sha256": "1f5b0c1951badb02e761429932def3fed0dd94294a18dafe27ccae03dcf670d6"
  },
  "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-20261026-008-p40",
    "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": "6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034",
      "question_first_witness_sha256": "cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 5,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "5fb77341d63f9ab5d781751cba0ba3025c047005e92130eedae91dd0a1a4e5d2",
      "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."
  }
}