{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261109-r03-compact-subset-0085", "pilot_number": "R0042-21635", "number": 21635, "family_id": "qf-mod40-compact-nonproperty-boundary", "title": { "zh-Hant": "第 21635 題：緊緻子集", "en": "Question 21635: Compactness Does Not Supply General Ambient Closedness or Boundedness" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261109-r03-compact-subset-0085", "batch_id": "batch-20261014-advance-20261109-r03-compact-subset-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-compact-subset" ], "title_zh": "緊緻子集", "stem_zh": "完整有限subspace topology與relative open covers。操作=compact-nonproperty-boundary；資料={\"n\":5,\"parent_topology\":[[],[0],[0,1,2,3,4],[0,2],[0,2,3],[0,2,3,4]],\"subset\":[1,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([1, 2, 4], 是, 否, 否, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)", "explanation_zh": "Wave6第三層核對=([1, 2, 4], 是, 否, 否, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)；依完整明示資料獨立重算；cover必須覆蓋K且subcover只能選自original cover；single cover不證一般compactness；finite-space compactness不外推infinite spaces；不把compact普遍當closed-and-bounded，empty set convention明示。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "緊緻子集" ] }, "variant": { "family_id": "qf-mod40-compact-nonproperty-boundary", "seed": 202611093085, "parameters": { "n": 5, "parent_topology": [ [], [ 0 ], [ 0, 1, 2, 3, 4 ], [ 0, 2 ], [ 0, 2, 3 ], [ 0, 2, 3, 4 ] ], "subset": [ 1, 2, 4 ], "domain": "Complete_finite_compact_subspace", "operation": "compact-nonproperty-boundary", "semantic_template": "qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}" } }, "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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r03-compact-subset-0085", "batch_id": "batch-20261014-advance-20261109-r03-compact-subset-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "b4625a9cb8e1286813e37a767c950d56826e92be5aceec66946e31bc648a6ec4", "family_id": "qf-mod40-compact-nonproperty-boundary", "seed": 202611093085, "semantic_template": "qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}" }, "fields": { "title": "Compactness Does Not Supply General Ambient Closedness or Boundedness", "stem": "Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).", "answer": { "kind": "expression", "value": "([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)", "explanation": "Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)." }, "skills": [ "Compactness Does Not Supply General Ambient Closedness or Boundedness" ], "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subspace topology與relative open covers。操作=compact-nonproperty-boundary；資料={\"n\":5,\"parent_topology\":[[],[0],[0,1,2,3,4],[0,2],[0,2,3],[0,2,3,4]],\"subset\":[1,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=compact-nonproperty-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題歷史介面標記保留，本頁直接列來源有序欄位，不需JavaScript。tag不是額外證據，固定No保留其原聲稱範圍。", "en": "Keep the original historical interface marker while listing ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their 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 parent Fin(5);parent open index=0,U=[];complete topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]]", "after": "ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1, 2, 3, 4], []], [3, [0, 2], []], [4, [0, 2, 3], []], [5, [0, 2, 3, 4], []]];each belongs to the SAME parent family", "relation": "implies", "rule_id": "subtop_qualified_parent_all_pairs_union_intersection", "conditions": [], "explanation": { "zh-Hant": "先驗母拓撲含空集與全集，再核全部有序open pair的union／intersection閉合。有限family的任意union可去重成有限union，故二元閉合足夠；不能把不合格母資料當True。", "en": "First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected." }, "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 parent Fin(5);parent open index=1,U=[0];complete topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]]", "after": "ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1, 2, 3, 4], [0]], [3, [0, 2], [0]], [4, [0, 2, 3], [0]], [5, [0, 2, 3, 4], [0]]];each belongs to the SAME parent family", "relation": "implies", "rule_id": "subtop_qualified_parent_all_pairs_union_intersection", "conditions": [], "explanation": { "zh-Hant": "先驗母拓撲含空集與全集，再核全部有序open pair的union／intersection閉合。有限family的任意union可去重成有限union，故二元閉合足夠；不能把不合格母資料當True。", "en": "First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected." }, "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 parent Fin(5);parent open index=2,U=[0, 1, 2, 3, 4];complete topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]]", "after": "ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2, 3, 4], []], [1, [0, 1, 2, 3, 4], [0]], [2, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4]], [3, [0, 1, 2, 3, 4], [0, 2]], [4, [0, 1, 2, 3, 4], [0, 2, 3]], [5, [0, 1, 2, 3, 4], [0, 2, 3, 4]]];each belongs to the SAME parent family", "relation": "implies", "rule_id": "subtop_qualified_parent_all_pairs_union_intersection", "conditions": [], "explanation": { "zh-Hant": "先驗母拓撲含空集與全集，再核全部有序open pair的union／intersection閉合。有限family的任意union可去重成有限union，故二元閉合足夠；不能把不合格母資料當True。", "en": "First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected." }, "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 parent Fin(5);parent open index=3,U=[0, 2];complete topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]]", "after": "ALL [other_open_index,union,intersection] rows=[[0, [0, 2], []], [1, [0, 2], [0]], [2, [0, 1, 2, 3, 4], [0, 2]], [3, [0, 2], [0, 2]], [4, [0, 2, 3], [0, 2]], [5, [0, 2, 3, 4], [0, 2]]];each belongs to the SAME parent family", "relation": "implies", "rule_id": "subtop_qualified_parent_all_pairs_union_intersection", "conditions": [], "explanation": { "zh-Hant": "先驗母拓撲含空集與全集，再核全部有序open pair的union／intersection閉合。有限family的任意union可去重成有限union，故二元閉合足夠；不能把不合格母資料當True。", "en": "First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected." }, "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 parent Fin(5);parent open index=4,U=[0, 2, 3];complete topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]]", "after": "ALL [other_open_index,union,intersection] rows=[[0, [0, 2, 3], []], [1, [0, 2, 3], [0]], [2, [0, 1, 2, 3, 4], [0, 2, 3]], [3, [0, 2, 3], [0, 2]], [4, [0, 2, 3], [0, 2, 3]], [5, [0, 2, 3, 4], [0, 2, 3]]];each belongs to the SAME parent family", "relation": "implies", "rule_id": "subtop_qualified_parent_all_pairs_union_intersection", "conditions": [], "explanation": { "zh-Hant": "先驗母拓撲含空集與全集，再核全部有序open pair的union／intersection閉合。有限family的任意union可去重成有限union，故二元閉合足夠；不能把不合格母資料當True。", "en": "First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected." }, "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": "ORIGINAL parent Fin(5);parent open index=5,U=[0, 2, 3, 4];complete topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]]", "after": "ALL [other_open_index,union,intersection] rows=[[0, [0, 2, 3, 4], []], [1, [0, 2, 3, 4], [0]], [2, [0, 1, 2, 3, 4], [0, 2, 3, 4]], [3, [0, 2, 3, 4], [0, 2]], [4, [0, 2, 3, 4], [0, 2, 3]], [5, [0, 2, 3, 4], [0, 2, 3, 4]]];each belongs to the SAME parent family", "relation": "implies", "rule_id": "subtop_qualified_parent_all_pairs_union_intersection", "conditions": [], "explanation": { "zh-Hant": "先驗母拓撲含空集與全集，再核全部有序open pair的union／intersection閉合。有限family的任意union可去重成有限union，故二元閉合足夠；不能把不合格母資料當True。", "en": "First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected." }, "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": "ORIGINAL parent open index=0,U=[];ORIGINAL subset Y=[1, 2, 4]", "after": "actual U intersect Y=[];retain parent index even if another open has the same intersection", "relation": "implies", "rule_id": "subtop_every_parent_open_intersection_with_original_Y", "conditions": [], "explanation": { "zh-Hant": "對每個母open逐點交Y；相同交集只在誘導family去重，原母witness不能丟。U可含Y外的點，不強迫U subset Y。", "en": "Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "ORIGINAL parent open index=1,U=[0];ORIGINAL subset Y=[1, 2, 4]", "after": "actual U intersect Y=[];retain parent index even if another open has the same intersection", "relation": "implies", "rule_id": "subtop_every_parent_open_intersection_with_original_Y", "conditions": [], "explanation": { "zh-Hant": "對每個母open逐點交Y；相同交集只在誘導family去重，原母witness不能丟。U可含Y外的點，不強迫U subset Y。", "en": "Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "ORIGINAL parent open index=2,U=[0, 1, 2, 3, 4];ORIGINAL subset Y=[1, 2, 4]", "after": "actual U intersect Y=[1, 2, 4];retain parent index even if another open has the same intersection", "relation": "implies", "rule_id": "subtop_every_parent_open_intersection_with_original_Y", "conditions": [], "explanation": { "zh-Hant": "對每個母open逐點交Y；相同交集只在誘導family去重，原母witness不能丟。U可含Y外的點，不強迫U subset Y。", "en": "Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y." }, "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": "S011", "uses": [ "G001", "S010" ], "before": "ORIGINAL parent open index=3,U=[0, 2];ORIGINAL subset Y=[1, 2, 4]", "after": "actual U intersect Y=[2];retain parent index even if another open has the same intersection", "relation": "implies", "rule_id": "subtop_every_parent_open_intersection_with_original_Y", "conditions": [], "explanation": { "zh-Hant": "對每個母open逐點交Y；相同交集只在誘導family去重，原母witness不能丟。U可含Y外的點，不強迫U subset Y。", "en": "Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y." }, "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": "S012", "uses": [ "G001", "S011" ], "before": "ORIGINAL parent open index=4,U=[0, 2, 3];ORIGINAL subset Y=[1, 2, 4]", "after": "actual U intersect Y=[2];retain parent index even if another open has the same intersection", "relation": "implies", "rule_id": "subtop_every_parent_open_intersection_with_original_Y", "conditions": [], "explanation": { "zh-Hant": "對每個母open逐點交Y；相同交集只在誘導family去重，原母witness不能丟。U可含Y外的點，不強迫U subset Y。", "en": "Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y." }, "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": "S013", "uses": [ "G001", "S012" ], "before": "ORIGINAL parent open index=5,U=[0, 2, 3, 4];ORIGINAL subset Y=[1, 2, 4]", "after": "actual U intersect Y=[2, 4];retain parent index even if another open has the same intersection", "relation": "implies", "rule_id": "subtop_every_parent_open_intersection_with_original_Y", "conditions": [], "explanation": { "zh-Hant": "對每個母open逐點交Y；相同交集只在誘導family去重，原母witness不能丟。U可含Y外的點，不強迫U subset Y。", "en": "Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y." }, "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": "S014", "uses": [ "G001", "S013" ], "before": "subtop distinct relative open count := sum(1, 1, 1, 1)", "after": "subtop distinct relative open count = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。", "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 subtop distinct relative open count，不以小數近似替代。", "en": "Recompute subtop distinct relative open count with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1", "1", "1" ], "output": "4", "label": "subtop distinct relative open count" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "ORIGINAL Y=[1, 2, 4];ALL parent intersections=[[], [], [1, 2, 4], [2], [2], [2, 4]]", "after": "normalized relative topology=[[], [1, 2, 4], [2], [2, 4]];distinct count=4;empty and Y included;unions and finite intersections commute with intersect Y", "relation": "implies", "rule_id": "subtop_induced_axioms_and_empty_Y", "conditions": [], "explanation": { "zh-Hant": "任意union與有限intersection都可把交Y提出，再由母拓撲封閉得到相對open。Y空時所有交集都是空集，誘導拓撲是只含空集的一個family，不是零個open。", "en": "Intersection with Y commutes with arbitrary unions and finite intersections;parent closure gives the induced axioms.For empty Y every intersection is empty,so the topology has one open,not zero." }, "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": "S016", "uses": [ "G001", "S015" ], "before": "ORIGINAL K=[1, 2, 4];COMPLETE finite relative open family=[[], [1, 2, 4], [2], [2, 4]];distinct open count=4", "after": "for ANY relative-open cover of K,retain one used original index per DISTINCT open;at most this many opens form a finite subcover;empty K covered by the empty subfamily", "relation": "implies", "rule_id": "compact_finite_DISTINCT_open_family_general_cover_argument", "conditions": [], "explanation": { "zh-Hant": "這是全部cover的論證，不只一份測例。即使原cover有無窮重複索引，只有有限種不同open；每種選一個原索引即可。來源tag的finite指此distinct family，不宣稱所有indexed cover只有有限索引。", "en": "This argument covers every cover,not one sample.Even infinitely repeated indices use only finitely many distinct opens;choose one original index for each used open.The source finite tag concerns distinct opens,not the number of indices in every indexed cover." }, "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": "S017", "uses": [ "G001", "S016" ], "before": "ACTUAL finite K=[1, 2, 4] is compact in its relative topology", "after": "two fixed No markers WITHHOLD general ambient-closedness and boundedness claims;no metric or Hausdorff premise is supplied", "relation": "implies", "rule_id": "compact_general_closed_bounded_nonclaims_not_instance_negations", "conditions": [], "explanation": { "zh-Hant": "有限模型的compact性不自動給一般空間的closed-and-bounded等價；這兩個No不是宣稱此K不closed或不bounded，空K等特例仍可closed。", "en": "Finite-model compactness is not a general closed-and-bounded equivalence.The two No markers do not classify this K as nonclosed or unbounded;special cases such as empty K remain possible." }, "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, 2, 4 ], true, false, false, "compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure" ] }, "conclusion": { "canonical_value": "([1, 2, 4], 是, 否, 否, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)", "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-20261109-r03-compact-subset-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261109-r03-compact-subset-foundations/questions.jsonl", "batch_sha256": "eab9e79f2ff71ca90c747f44b7ec58eb5aff1593541da3d0672d1c83666f787a", "original_line": 85, "original_line_sha256": "a23609aa8a7276a7898eba1f60367fc75ce901594e0185a6a232248e343e3ff3", "question_compact_sha256": "b4625a9cb8e1286813e37a767c950d56826e92be5aceec66946e31bc648a6ec4", "english_row_compact_sha256": "f1f834d57548bb1b0e0270248f5680320635967147a58283863fb659e79e854a", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261109-r03-compact-subset-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261109-r03-compact-subset-foundations.en.jsonl" }, "mko_bindings": { "mko-compact-subset": { "sha256": "b4f23af78bd1197e0c8a2e10786c521b3e087c5ec8033bdec3c84135b9f380fd", "english_sha256": "d9367471b34cd6cfb707454aaba2ecd52662b39d867d6a3407ccd260560d1856", "original_path": "public/data/mko/compact-subset.json", "english_path": "public/data/i18n/editions/mko-compact-subset.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a84fbe07a4e46e8e47a8ad331f35fd3f81b821aef6cb27e7852de854aaa96b9a", "original_candidate_sha256": "a0d6486952490897c00b65d8d96552d9f7501fefafb79c1a4def660c630fcf34", "original_candidate_line_sha256": "a0d6486952490897c00b65d8d96552d9f7501fefafb79c1a4def660c630fcf34" }, "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-20261026-009-p43", "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": "dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1a", "question_first_witness_sha256": "9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 17, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "824c1904423be86c0f4bf35f63ef96d1cf50755d406bf7f059c0bc5a19667f86", "arithmetic_steps": 1 }, "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." } }