{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261030-r04-quotient-set-0047", "pilot_number": "R0029-15472", "number": 15472, "family_id": "qf-mod30-quotient-projection", "title": { "zh-Hant": "第 15472 題：商集", "en": "Question 15472: Canonical Projection Returns a Class, Not a Chosen Label" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261030-r04-quotient-set-0047", "batch_id": "batch-20261014-advance-20261030-r04-quotient-set-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-quotient-set" ], "title_zh": "商集", "stem_zh": "同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":3}。請依 [element,class] 欄序回答。", "answer": { "kind": "expression", "value": "(3, [0, 3])", "explanation_zh": "Wave4第四層核對=(3, [0, 3])；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；商集不是代表元集合且classes覆蓋互斥；商運算須另證代表元無關。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "商集" ] }, "variant": { "family_id": "qf-mod30-quotient-projection", "seed": 202610304047, "parameters": { "n": 6, "pairs": [ [ 0, 0 ], [ 0, 3 ], [ 1, 1 ], [ 2, 2 ], [ 2, 4 ], [ 2, 5 ], [ 3, 0 ], [ 3, 3 ], [ 4, 2 ], [ 4, 4 ], [ 4, 5 ], [ 5, 2 ], [ 5, 4 ], [ 5, 5 ] ], "element": 3, "domain": "Complete_finite_quotient_set", "operation": "quotient-projection", "semantic_template": "qf-mod30-quotient-projection/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}" } }, "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-r04-quotient-set-0047", "batch_id": "batch-20261014-advance-20261030-r04-quotient-set-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "388c466e54fe0d4f7e73de0c2be4c8cf31dbd36143187815d084602b878237c8", "family_id": "qf-mod30-quotient-projection", "seed": 202610304047, "semantic_template": "qf-mod30-quotient-projection/{\"element\":3,\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]]}" }, "fields": { "title": "Canonical Projection Returns a Class, Not a Chosen Label", "stem": "Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=3. Give (element,class).", "answer": { "kind": "expression", "value": "(3, [0, 3])", "explanation": "Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (3, [0, 3])." }, "skills": [ "Canonical Projection Returns a Class, Not a Chosen Label" ], "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": "同一完整等價關係的互異classes。操作=quotient-projection；資料={\"n\":6,\"pairs\":[[0,0],[0,3],[1,1],[2,2],[2,4],[2,5],[3,0],[3,3],[4,2],[4,4],[4,5],[5,2],[5,4],[5,5]],\"element\":3}。請依 [element,class] 欄序回答。", "en": "Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. element=3. Give (element,class)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "complete carrier Fin(6);original relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]", "after": "set of pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]];reflexive/symmetric/transitive=[是, 是, 是]", "relation": "implies", "rule_id": "quotient_complete_equivalence_not_connectedness_only", "conditions": [], "explanation": { "zh-Hant": "把原pairs按集合語義讀，重複列不改關係；先核自反、對稱、傳遞，不能把非等價關係的連通分量當原關係之商。", "en": "Read pairs with set semantics;duplicate entries do not change the relation.Check reflexivity,symmetry and transitivity before forming the quotient.Connected components of a non-equivalence relation are not its quotient classes." }, "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": "least uncovered carrier label=0;all labels related from it=[0, 3]", "after": "next canonical full class=[0, 3];covered labels=[0, 3]", "relation": "implies", "rule_id": "quotient_full_classes_in_least_representative_order", "conditions": [], "explanation": { "zh-Hant": "類內照原載體遞增列全體元素，類與類依最小未覆蓋元素排序；商元素是整個類，不是只有代表的標籤。", "en": "List every element in the class in carrier order,and order classes by the least not-yet-covered element.A quotient element is the whole class,not merely its representative label." }, "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": "least uncovered carrier label=1;all labels related from it=[1]", "after": "next canonical full class=[1];covered labels=[0, 1, 3]", "relation": "implies", "rule_id": "quotient_full_classes_in_least_representative_order", "conditions": [], "explanation": { "zh-Hant": "類內照原載體遞增列全體元素，類與類依最小未覆蓋元素排序；商元素是整個類，不是只有代表的標籤。", "en": "List every element in the class in carrier order,and order classes by the least not-yet-covered element.A quotient element is the whole class,not merely its representative label." }, "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": "least uncovered carrier label=2;all labels related from it=[2, 4, 5]", "after": "next canonical full class=[2, 4, 5];covered labels=[0, 1, 2, 3, 4, 5]", "relation": "implies", "rule_id": "quotient_full_classes_in_least_representative_order", "conditions": [], "explanation": { "zh-Hant": "類內照原載體遞增列全體元素，類與類依最小未覆蓋元素排序；商元素是整個類，不是只有代表的標籤。", "en": "List every element in the class in carrier order,and order classes by the least not-yet-covered element.A quotient element is the whole class,not merely its representative label." }, "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": "all complete canonical classes=[[0, 3], [1], [2, 4, 5]];original carrier=[0, 1, 2, 3, 4, 5]", "after": "cover whole carrier=是;pairwise disjoint=是;class count=3", "relation": "implies", "rule_id": "quotient_partition_cover_and_disjoint_not_count_only", "conditions": [], "explanation": { "zh-Hant": "等價類覆蓋原載體且兩兩不交，各元素恰屬一類；空載體的空族亦保留空覆蓋與不交，不憑類數猜partition。", "en": "Equivalence classes cover the original carrier and are pairwise disjoint,so each element lies in exactly one class.The empty family on an empty carrier also satisfies empty coverage and disjointness;class counts alone are not the partition test." }, "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": "original element=3;canonical quotient elements=[[0, 3], [1], [2, 4, 5]]", "after": "projection sends3 to whole class=[0, 3]", "relation": "implies", "rule_id": "quotient_projection_element_to_full_class", "conditions": [], "explanation": { "zh-Hant": "自然投影送元素到其全等價類，不是送到類的最小元素，也未給商上的加法乘法。", "en": "The natural projection sends an element to its entire equivalence class,not its least element.No quotient addition or multiplication has been defined." }, "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": [ 3, [ 0, 3 ] ] }, "conclusion": { "canonical_value": "(3, [0, 3])", "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-r04-quotient-set-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261030-r04-quotient-set-foundations/questions.jsonl", "batch_sha256": "43bb6f0acd8d0dad5a31f92053946dc624118bf67dbf9ccb67760346ccf89c8f", "original_line": 47, "original_line_sha256": "d206137c4290d36b52baec2b9353109b6e70cecd1c9f34b4c33ab41613804de7", "question_compact_sha256": "388c466e54fe0d4f7e73de0c2be4c8cf31dbd36143187815d084602b878237c8", "english_row_compact_sha256": "ef36c23e5f498b37fda93e893eb3f76cecd22acbced592779ddaefce166df25b", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261030-r04-quotient-set-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261030-r04-quotient-set-foundations.en.jsonl" }, "mko_bindings": { "mko-quotient-set": { "sha256": "3741dd3a04bde75f8d82e8fd31849761617fb2410c548387ba4434f502159236", "english_sha256": "dae34ce1e56ae7006833ff81818110a2d41c8df3ecf627d111fe3e8120eb954f", "original_path": "public/data/mko/quotient-set.json", "english_path": "public/data/i18n/editions/mko-quotient-set.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "54de60dca082018923bf089f2b1eb0a6666cc401916115af792056c8bae935db", "original_candidate_sha256": "50489e5bf358f98d545bb0e35ee732b29651ff81234d58b61bb08c6b85e8a590", "original_candidate_line_sha256": "50489e5bf358f98d545bb0e35ee732b29651ff81234d58b61bb08c6b85e8a590" }, "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-p30", "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": "8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725", "question_first_witness_sha256": "c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 6, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "bc46f5d5d0f19f58b96baeac7e0c2c6d3aa4b04a4f0058a77c77b492cef4c6e7", "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." } }