{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261107-r05-pigeonhole-principle-0052", "pilot_number": "R0039-20377", "number": 20377, "family_id": "qf-mod38-pigeon-premise-audit", "title": { "zh-Hant": "第 20377 題：鴿籠原理", "en": "Question 20377: The Size Premise Forces a Collision but Has No General Reverse Inference" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261107-r05-pigeonhole-principle-0052", "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-premise-audit；資料={\"domain_count\":6,\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(6, 4, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)", "explanation_zh": "Wave6根層核對=(6, 4, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)；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-premise-audit", "seed": 202611075052, "parameters": { "domain_count": 6, "codomain_count": 4, "domain": "Complete_finite_total_map_pigeonhole", "operation": "pigeon-premise-audit", "semantic_template": "qf-mod38-pigeon-premise-audit/{\"codomain_count\":4,\"domain_count\":6}" } }, "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-0052", "batch_id": "batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "784eda480c1c09ef42ba2fc31a659ee26c40be35bbf600421b8255e18e835e2a", "family_id": "qf-mod38-pigeon-premise-audit", "seed": 202611075052, "semantic_template": "qf-mod38-pigeon-premise-audit/{\"codomain_count\":4,\"domain_count\":6}" }, "fields": { "title": "The Size Premise Forces a Collision but Has No General Reverse Inference", "stem": "Use complete finite domain size=6≥0 and codomain size=4≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).", "answer": { "kind": "expression", "value": "(6, 4, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)", "explanation": "Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (6, 4, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)." }, "skills": [ "The Size Premise Forces a Collision but Has No General Reverse Inference" ], "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-premise-audit；資料={\"domain_count\":6,\"codomain_count\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use complete finite domain size=6≥0 and codomain size=4≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=pigeon-premise-audit;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary]", "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 finite domain size=6;nonempty codomain size=4;NO particular map table supplied", "after": "domain larger=是;size-forced collision=是;if a total map were injective,each fibre<=1 implies domain<=codomain", "relation": "implies", "rule_id": "pigeon_size_premise_all_maps_not_particular_sample", "conditions": [], "explanation": { "zh-Hant": "若source較多而每桶至多1個source，全部source最多codomain個，矛盾，所以任意total map都碰撞。若大小條件不成立，只是不由這個嚴格條件強迫碰撞，不是在檢一個未給的map。", "en": "If the source is larger but every fibre contains at most1 source,there can be at most codomain-many sources,a contradiction.So every total map collides.When the strict size premise fails,this premise does not force a collision;it is not an audit of an unspecified map." }, "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": "ACTUAL size comparison=是;no given total-map array", "after": "general reverse implication CLAIM=否;domain<=codomain does not in GENERAL force a given map injective;special empty/singleton cases are not denied", "relation": "implies", "rule_id": "pigeon_general_reverse_No_not_instance_injectivity_negation", "conditions": [], "explanation": { "zh-Hant": "一般逆推可由2點到2點的constant map反駁；這不是宣稱每個不滿足嚴格大小前提的map都碰撞。空／單點source另由定義確可單射，最後No不改寫這個特殊事實。", "en": "A constant map from2 points to2 points refutes the general reverse implication.This does not say every map failing the strict size premise collides.Empty/singleton cases may be injective by definition;the final No does not deny them." }, "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": [ 6, 4, true, true, false, "premise_guarantees_collision_but_converse_does_not_guarantee_injectivity" ] }, "conclusion": { "canonical_value": "(6, 4, 是, 是, 否, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)", "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-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": 52, "original_line_sha256": "b22ea7d7e2a882e238d267dfec4c7e657dd700b831147167e1a1e758644d54b6", "question_compact_sha256": "784eda480c1c09ef42ba2fc31a659ee26c40be35bbf600421b8255e18e835e2a", "english_row_compact_sha256": "d13e41a728f333bff4ed09261a615bc5a321e44e84982ccf79725041df628656", "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": "7d94f1423fb65e169827c1e654f10c83ae214b78be150c12e7bcfc8af4d1acff", "original_candidate_line_sha256": "7d94f1423fb65e169827c1e654f10c83ae214b78be150c12e7bcfc8af4d1acff" }, "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": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "188e323746f4276696fbb306390d56a4a05585807196f89c3f9b10f29b9ceb7c", "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." } }