{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261101-r03-open-cover-and-subcover-0057", "pilot_number": "R0031-16132", "number": 16132, "family_id": "qf-mod32-cover-minimal-subcover", "title": { "zh-Hant": "第 16132 題：開覆蓋與子覆蓋", "en": "Question 16132: Inclusion-Minimal Does Not Mean Minimum Cardinality" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261101-r03-open-cover-and-subcover-0057", "batch_id": "batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-open-cover-and-subcover" ], "title_zh": "開覆蓋與子覆蓋", "stem_zh": "完整有限拓撲、目標集合與原始cover family。操作=cover-minimal-subcover；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,removal_rows,inclusion_minimal,boundary] 欄序回答。", "answer": { "kind": "expression", "value": "([1], 是, [[1, [], 否]], 是, inclusion_minimal_not_minimum_cardinality)", "explanation_zh": "Wave5根節點核對=([1], 是, [[1, [], 否]], 是, inclusion_minimal_not_minimum_cardinality)；依完整明示資料重算；每個cover成員須開且聯集包含目標；subcover只能選原成員；一個有限subcover不證空間compact。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "開覆蓋與子覆蓋" ] }, "variant": { "family_id": "qf-mod32-cover-minimal-subcover", "seed": 202611013057, "parameters": { "n": 5, "topology": [ [], [ 0 ], [ 0, 2 ], [ 0, 1, 2 ], [ 0, 1, 2, 3 ], [ 0, 1, 2, 3, 4 ] ], "target": [ 0, 1, 2, 3 ], "cover": [ [ 0, 2 ], [ 0, 1, 2, 3 ], [ 0, 1, 2, 3, 4 ] ], "selected": [ 1 ], "domain": "Complete_finite_topological_cover", "operation": "cover-minimal-subcover", "semantic_template": "qf-mod32-cover-minimal-subcover/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式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-20261101-r03-open-cover-and-subcover-0057", "batch_id": "batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "1db08be4d812b070bbb8d6df5fece2deb0c27841d9ed9e95098ddff6e5c0bd3a", "family_id": "qf-mod32-cover-minimal-subcover", "seed": 202611013057, "semantic_template": "qf-mod32-cover-minimal-subcover/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}" }, "fields": { "title": "Inclusion-Minimal Does Not Mean Minimum Cardinality", "stem": "Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.", "answer": { "kind": "expression", "value": "([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)", "explanation": "Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)." }, "skills": [ "Inclusion-Minimal Does Not Mean Minimum Cardinality" ], "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": "完整有限拓撲、目標集合與原始cover family。操作=cover-minimal-subcover；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,removal_rows,inclusion_minimal,boundary] 欄序回答。", "en": "Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "same finite carrier Fin(5);complete original ordered topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]", "after": "unique empty and whole opens=是;all ordered open pairs=36;first union closure failure=null;first intersection closure failure=null", "relation": "implies", "rule_id": "cover_qualified_complete_topology_no_silent_metric", "conditions": [], "explanation": { "zh-Hant": "先核子集表示、空集與全集、全部兩開集的聯／交仍開。因拓撲族本身有限，任意聯集只含有限種不同開集，反覆二元聯封閉便涵蓋它；有限交亦由二元交與全集得到。未供度量，不引入距離。", "en": "Check canonical subsets,empty and whole sets,and union/intersection of every ordered pair.Since the topology family is finite,an arbitrary union involves only finitely many distinct opens and follows by repeated binary unions.Finite intersections follow similarly with the whole set.No metric is supplied or inferred." }, "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 cover family index=0;member=[0, 2];complete same topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]", "after": "original member open=是", "relation": "implies", "rule_id": "cover_every_original_index_open_including_unused_members", "conditions": [], "explanation": { "zh-Hant": "逐原始索引核開性，相同集合可以在不同索引重複；不能去重改題，也不能因它沒被選入就略過其非開性。", "en": "Check openness at every original index.Equal sets at different indices remain distinct indexed members.Do not deduplicate the input or ignore a nonopen member merely because it is unselected." }, "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": "original cover family index=1;member=[0, 1, 2, 3];complete same topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]", "after": "original member open=是", "relation": "implies", "rule_id": "cover_every_original_index_open_including_unused_members", "conditions": [], "explanation": { "zh-Hant": "逐原始索引核開性，相同集合可以在不同索引重複；不能去重改題，也不能因它沒被選入就略過其非開性。", "en": "Check openness at every original index.Equal sets at different indices remain distinct indexed members.Do not deduplicate the input or ignore a nonopen member merely because it is unselected." }, "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": "original cover family index=2;member=[0, 1, 2, 3, 4];complete same topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]", "after": "original member open=是", "relation": "implies", "rule_id": "cover_every_original_index_open_including_unused_members", "conditions": [], "explanation": { "zh-Hant": "逐原始索引核開性，相同集合可以在不同索引重複；不能去重改題，也不能因它沒被選入就略過其非開性。", "en": "Check openness at every original index.Equal sets at different indices remain distinct indexed members.Do not deduplicate the input or ignore a nonopen member merely because it is unselected." }, "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": "same target=[0, 1, 2, 3];union of ALL original members=[0, 1, 2, 3, 4];nonopen original indices=[]", "after": "covers target=是;whole original open cover valid=是", "relation": "implies", "rule_id": "cover_openness_and_target_coverage_separate_conjunction", "conditions": [], "explanation": { "zh-Hant": "開性與覆蓋目標是獨立義務，聯集可超過目標；只要包含即可，不要求聯集等於目標。兩者同時成立才核可原cover。", "en": "Openness and coverage are separate obligations.The union may exceed the target;containment,not equality,is required.Certify the original cover only when both hold." }, "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": "raw selected ORIGINAL indices=[1];normalized duplicate-free indices=[1];selected original sets=[[0, 1, 2, 3]]", "after": "selected union=[0, 1, 2, 3];selected all open=是;selected covers target=是;original family qualified=是;valid selected subcover=是", "relation": "implies", "rule_id": "cover_selected_original_indices_whole_family_premise_retained", "conditions": [], "explanation": { "zh-Hant": "只依原索引取成員，排序索引但拒重複；空選取在空目標可合格。此協定另要求整個原族先為有效開覆蓋，所以好子族不能抹去未選中壞原成員。", "en": "Select only original indexed members,sorting indices but rejecting duplicates.Empty selection can qualify for an empty target.This protocol additionally requires the whole original family to qualify;an otherwise good selected subfamily cannot erase an unselected bad original member." }, "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": "remove original selected index=1;remaining selected indices=[];remaining union=[];same target=[0, 1, 2, 3]", "after": "removal coverage row=[1, [], 否]", "relation": "implies", "rule_id": "cover_every_single_member_removal_target_test", "conditions": [], "explanation": { "zh-Hant": "每次只刪這個已選成員，重算剩餘聯集是否仍覆蓋原目標；保留原索引與移除順序，不用全圖最小化替代本題的包含極小測試。", "en": "Remove this one selected member and recompute coverage of the same target.Keep original indices and removal order.This is inclusion minimality,not a global minimum-cardinality search." }, "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": "valid selected subcover=是;all single-removal rows=[[1, [], 否]]", "after": "inclusion-minimal=是;inclusion_minimal_not_minimum_cardinality", "relation": "implies", "rule_id": "cover_inclusion_minimal_not_global_smallest_size", "conditions": [], "explanation": { "zh-Hant": "從有效subcover出發，任何單一刪除都失敗才包含極小；若真有較小包含子族，先刪一個不在其中的成員仍會覆蓋，故單刪測試足夠。但原族其他不可比較選取可能張數更少，不能據此一般宣稱最小張數。", "en": "For a valid subcover,every single deletion must fail.If a proper selected subfamily covered,removing an element outside it would still cover,so these deletion tests suffice for inclusion minimality.An incomparable choice from the original family may use fewer sets;this is not generally minimum cardinality." }, "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 ], true, [ [ 1, [], false ] ], true, "inclusion_minimal_not_minimum_cardinality" ] }, "conclusion": { "canonical_value": "([1], 是, [[1, [], 否]], 是, inclusion_minimal_not_minimum_cardinality)", "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-20261101-r03-open-cover-and-subcover-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations/questions.jsonl", "batch_sha256": "5f697bf41267aa623a8a11083ff731599a7d7be75b692fa2339606a98819c149", "original_line": 57, "original_line_sha256": "fef10734a85b1a23f395033fafe9a6ef4d0a8f3f1ccbef29add41a592a65a167", "question_compact_sha256": "1db08be4d812b070bbb8d6df5fece2deb0c27841d9ed9e95098ddff6e5c0bd3a", "english_row_compact_sha256": "1812b8128a8d37f444e4e051666e3ff4c4d1c6e75a109ef7c4087c453f8ae771", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations.en.jsonl" }, "mko_bindings": { "mko-open-cover-and-subcover": { "sha256": "bd5a852d24868cb59747e9030f8da58e604c78a87a2442558dc2643a01d8ecb1", "english_sha256": "a58afb586a37df65445b878eb9611ccd4f1ad7f03e2d7e5ef5d651f785f43d00", "original_path": "public/data/mko/open-cover-and-subcover.json", "english_path": "public/data/i18n/editions/mko-open-cover-and-subcover.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "633793a5f242dece0a240bd92f77a33c587d12463e0beaedd8f70a14c70cf9e7", "original_candidate_sha256": "fa60f5632f430023af6541cfa61a83bb2ea52c91cc7e0c15f0d74697764b5da8", "original_candidate_line_sha256": "fa60f5632f430023af6541cfa61a83bb2ea52c91cc7e0c15f0d74697764b5da8" }, "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-p32", "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": "55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369a", "question_first_witness_sha256": "754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bb" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 8, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "295c2040516962e624935575e57f0e48301f0bee179ad122bce672c4e1dda8ba", "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." } }