{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261030-r05-cardinality-comparison-0087", "pilot_number": "R0028-14862", "number": 14862, "family_id": "qf-mod30-cardinality-pigeonhole", "title": { "zh-Hant": "第 14862 題：集合基數比較", "en": "Question 14862: A Larger Finite Source Rules Out Every Injection" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261030-r05-cardinality-comparison-0087", "batch_id": "batch-20261014-advance-20261030-r05-cardinality-comparison-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-cardinality-comparison" ], "title_zh": "集合基數比較", "stem_zh": "mapping綁精確有限source與target。操作=cardinality-pigeonhole；資料={\"source\":[110,111,112,113,114],\"target\":[210,211]}。請依 [source_size,target_size,larger,conclusion] 欄序回答。", "answer": { "kind": "expression", "value": "(5, 2, 是, no_injection_finite_pigeonhole)", "explanation_zh": "Wave4第四層核對=(5, 2, 是, no_injection_finite_pigeonhole)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；injective surjective bijective責任分開；surjection逆向比較限有限集合且不外推無限。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "集合基數比較" ] }, "variant": { "family_id": "qf-mod30-cardinality-pigeonhole", "seed": 202610305087, "parameters": { "source": [ 110, 111, 112, 113, 114 ], "target": [ 210, 211 ], "domain": "Exact_finite_cardinality_mapping", "operation": "cardinality-pigeonhole", "semantic_template": "qf-mod30-cardinality-pigeonhole/{\"source\":[110,111,112,113,114],\"target\":[210,211]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_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、一般域／Hilbert／martingale／無限基數／一般收斂、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-20261030-r05-cardinality-comparison-0087", "batch_id": "batch-20261014-advance-20261030-r05-cardinality-comparison-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "751ae8653c7b253568937c8ff3db53c2ef9b46c6562916859bf78cdfa26d1e00", "family_id": "qf-mod30-cardinality-pigeonhole", "seed": 202610305087, "semantic_template": "qf-mod30-cardinality-pigeonhole/{\"source\":[110,111,112,113,114],\"target\":[210,211]}" }, "fields": { "title": "A Larger Finite Source Rules Out Every Injection", "stem": "Use the complete finite source=[110, 111, 112, 113, 114] and target=[210, 211] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.", "answer": { "kind": "expression", "value": "(5, 2, Yes, no_injection_finite_pigeonhole)", "explanation": "Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole)." }, "skills": [ "A Larger Finite Source Rules Out Every Injection" ], "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": "mapping綁精確有限source與target。操作=cardinality-pigeonhole；資料={\"source\":[110,111,112,113,114],\"target\":[210,211]}。請依 [source_size,target_size,larger,conclusion] 欄序回答。", "en": "Use the complete finite source=[110, 111, 112, 113, 114] and target=[210, 211] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "raw source list=[110, 111, 112, 113, 114];raw target list=[210, 211]", "after": "distinct finite source=[110, 111, 112, 113, 114];target=[210, 211];sizes=[5, 2]", "relation": "implies", "rule_id": "pigeonhole_set_counts_not_list_multiplicity", "conditions": [], "explanation": { "zh-Hant": "本族比較集合大小，重複出現不增加元素數；尚未給任何特定mapping，不捏造碰撞見證的元素對。", "en": "This family compares set cardinalities,so repetitions do not add elements.No particular mapping is supplied,and no specific collision pair is invented." }, "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": "finite sizes source=5;target=2", "after": "source larger=是;conclusion=no_injection_finite_pigeonhole", "relation": "implies", "rule_id": "finite_pigeonhole_all_injections_not_one_map_failure", "conditions": [], "explanation": { "zh-Hant": "若單射，來源的不同元素有同樣多不同像，須容於目標，所以ns≤nt；ns>nt矛盾，排除所有單射。No只說此大小不可行證據未成立，不是某個未給映射失敗。", "en": "An injection has as many distinct images as source elements,all within the target,hence ns≤nt.A larger source contradicts this and excludes every injection.No means this size-based impossibility certificate does not apply,not that an unspecified map failed." }, "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, 2, true, "no_injection_finite_pigeonhole" ] }, "conclusion": { "canonical_value": "(5, 2, 是, no_injection_finite_pigeonhole)", "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-20261030-r05-cardinality-comparison-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261030-r05-cardinality-comparison-foundations/questions.jsonl", "batch_sha256": "601304f1f2812ae71284fc9774945ccecf61443483ed83e7496eeb8e794c970f", "original_line": 87, "original_line_sha256": "45e2cf1f01256fed016bd8439d67d60ba3e8b0494f74e8768c612246d48e0170", "question_compact_sha256": "751ae8653c7b253568937c8ff3db53c2ef9b46c6562916859bf78cdfa26d1e00", "english_row_compact_sha256": "e422bf87867ef5088ccbaeded1feffd0e0f7f43ad4975eda107ce9fd140748b9", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261030-r05-cardinality-comparison-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261030-r05-cardinality-comparison-foundations.en.jsonl" }, "mko_bindings": { "mko-cardinality-comparison": { "sha256": "50efb74b57a576c8cec0b8230fc37bf7cd9c7ecd6303efe09810085d9a570227", "english_sha256": "d5d927d20ab09e9e7df6d3a33e4ed265b08e1db823fee7f5cb2a61a667172b4a", "original_path": "public/data/mko/cardinality-comparison.json", "english_path": "public/data/i18n/editions/mko-cardinality-comparison.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "466f72389eb454e24322e59df2871ec206c351545a4a7b072d39fb5f6afc2697", "original_candidate_sha256": "85c5304d30ce67dcd51de7fdf1ff5a3210d8a8851e3ace15be64f915b1f315e4", "original_candidate_line_sha256": "85c5304d30ce67dcd51de7fdf1ff5a3210d8a8851e3ace15be64f915b1f315e4" }, "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-20261026-007-p29", "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": "bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209ae", "question_first_witness_sha256": "61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 2, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "605f003672fd9af614f0d21f31482ee559786b4fb654864ab44c9bac474803fc", "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." } }