{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261102-r05-compact-topological-space-0080", "pilot_number": "R0032-16855", "number": 16855, "family_id": "qf-mod33-compact-single-cover-boundary", "title": { "zh-Hant": "第 16855 題：緊緻拓撲空間", "en": "Question 16855: Keep One-Cover Evidence Separate from the Finite Universal Theorem" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261102-r05-compact-topological-space-0080", "batch_id": "batch-20261014-advance-20261102-r05-compact-topological-space-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-compact-topological-space" ], "title_zh": "緊緻拓撲空間", "stem_zh": "完整有限carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。", "answer": { "kind": "expression", "value": "([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)", "explanation_zh": "Wave5第二層核對=([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)；依完整明示資料重算；每個有限拓撲空間由有限性全稱定理compact；單一cover有有限subcover不證一般或無限空間compact。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "緊緻拓撲空間" ] }, "variant": { "family_id": "qf-mod33-compact-single-cover-boundary", "seed": 202611025080, "parameters": { "n": 5, "topology": [ [], [ 0 ], [ 0, 1 ], [ 0, 1, 4 ], [ 0, 1, 2, 4 ], [ 0, 1, 2, 3, 4 ] ], "target": [ 0, 1, 2, 4 ], "cover": [ [ 0, 1 ], [ 0, 1, 2, 4 ], [ 0, 1, 2, 3, 4 ] ], "selected": [ 1 ], "domain": "Complete_finite_topology_compactness", "operation": "compact-single-cover-boundary", "semantic_template": "qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[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、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、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-20261102-r05-compact-topological-space-0080", "batch_id": "batch-20261014-advance-20261102-r05-compact-topological-space-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "19c6fb6c1e73899fc747f14e47702acadaaab5342ac15f65cd91b4f17a788345", "family_id": "qf-mod33-compact-single-cover-boundary", "seed": 202611025080, "semantic_template": "qf-mod33-compact-single-cover-boundary/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}" }, "fields": { "title": "Keep One-Cover Evidence Separate from the Finite Universal Theorem", "stem": "Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Target=[0, 1, 2, 4], original ordered cover=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original member must be open and the original union must contain the target. A subcover selects only original family indexes, starting at 0, sorted increasingly with duplicate indexes forbidden. This supplied-cover audit is separate from the universal finite-space compactness theorem. selected=[1]. Give (selected_indexes,valid_subcover?,proves_general_compactness?,one_supplied_cover_does_not_prove_general_compactness).", "answer": { "kind": "expression", "value": "([1], Yes, No, one_supplied_cover_does_not_prove_general_compactness)", "explanation": "The last No withholds a general compactness conclusion from this one supplied-cover audit. It does not classify this finite space as noncompact: a separate finiteness argument already supplies its universal compactness certificate. Infinite spaces cannot be classified from finitely sampled covers alone. The exact result, retaining the requested order, is ([1], Yes, No, one_supplied_cover_does_not_prove_general_compactness)." }, "skills": [ "Keep One-Cover Evidence Separate from the Finite Universal Theorem" ], "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carrier與完整拓撲或明示supplied cover。操作=compact-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_general_compactness,boundary] 欄序回答。", "en": "Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Target=[0, 1, 2, 4], original ordered cover=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original member must be open and the original union must contain the target. A subcover selects only original family indexes, starting at 0, sorted increasingly with duplicate indexes forbidden. This supplied-cover audit is separate from the universal finite-space compactness theorem. selected=[1]. Give (selected_indexes,valid_subcover?,proves_general_compactness?,one_supplied_cover_does_not_prove_general_compactness)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "same complete carrier Fin(5);original ordered complete topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]", "after": "unique empty/whole opens=是;ordered open pairs checked=36;first union failure=null;first intersection failure=null", "relation": "implies", "rule_id": "topology_complete_finite_parent_all_pairs", "conditions": [], "explanation": { "zh-Hant": "個別子集須照原載體遞增且無重複，拓撲族須有唯一空集、全集並核全部二元聯／交。因不同開集有限，任意聯只涉及有限種開集，二元封閉加空集便足夠；不另供度量。保留拓撲原順序。", "en": "Each subset is increasing and duplicate-free.The topology has unique empty/whole opens and closure under every binary union/intersection.Finitely many distinct opens reduce arbitrary unions to finite ones,handled by binary closure and the empty set.No metric is supplied.Keep original topology order." }, "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 supplied cover index=0;member=[0, 1];same full topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]", "after": "original member open=是", "relation": "implies", "rule_id": "compact_every_original_supplied_member_open", "conditions": [], "explanation": { "zh-Hant": "逐原族成員核開性，保留重複集合的不同索引；未選中的非開成員也不能抹去。這只是供給族的合格性，不是反駁空間緊緻。", "en": "Audit every original member's openness,retaining equal sets at distinct indices.An unselected nonopen original member is not erased.This qualifies the supplied family,not the space's compactness." }, "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 supplied cover index=1;member=[0, 1, 2, 4];same full topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]", "after": "original member open=是", "relation": "implies", "rule_id": "compact_every_original_supplied_member_open", "conditions": [], "explanation": { "zh-Hant": "逐原族成員核開性，保留重複集合的不同索引；未選中的非開成員也不能抹去。這只是供給族的合格性，不是反駁空間緊緻。", "en": "Audit every original member's openness,retaining equal sets at distinct indices.An unselected nonopen original member is not erased.This qualifies the supplied family,not the space's compactness." }, "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 supplied cover index=2;member=[0, 1, 2, 3, 4];same full topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]", "after": "original member open=是", "relation": "implies", "rule_id": "compact_every_original_supplied_member_open", "conditions": [], "explanation": { "zh-Hant": "逐原族成員核開性，保留重複集合的不同索引；未選中的非開成員也不能抹去。這只是供給族的合格性，不是反駁空間緊緻。", "en": "Audit every original member's openness,retaining equal sets at distinct indices.An unselected nonopen original member is not erased.This qualifies the supplied family,not the space's compactness." }, "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 supplied target=[0, 1, 2, 4];whole original family union=[0, 1, 2, 3, 4];nonopen indices=[]", "after": "covers target=是;original supplied cover valid=是", "relation": "implies", "rule_id": "compact_supplied_openness_and_coverage_not_universal_theorem", "conditions": [], "explanation": { "zh-Hant": "覆蓋只要求聯集包含這個目標，可超過目標；全體開性與覆蓋須同時成立。供給族失敗不會否定有限空間的全稱緊緻定理。", "en": "Coverage means containment of the supplied target;the union may be larger.Both all-member openness and coverage are required.A failed supplied family does not refute the finite-space universal theorem." }, "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 selected indices=[1];sorted duplicate-free indices=[1];original indexed members=[[0, 1, 2, 4]]", "after": "selected union=[0, 1, 2, 4];selected all open=是;selected covers target=是;whole original family qualified=是;valid subcover=是", "relation": "implies", "rule_id": "compact_selected_original_subcover_whole_family_premise", "conditions": [], "explanation": { "zh-Hant": "只依原索引選取，不製造新開集；原族仍須先有效，好的選取不能修好未選原成員的壞資料。空目標可由空選取覆蓋。", "en": "Select only original indexed members,not invented opens.The original family must first qualify;a good selection does not repair a bad unselected original member.Empty selection can cover an empty target." }, "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": "this one supplied cover;selected original indices=[1];valid subcover=是", "after": "proves general compactness from one audit=否;one_supplied_cover_does_not_prove_general_compactness", "relation": "implies", "rule_id": "compact_single_instance_not_all_covers_or_noncompactness", "conditions": [], "explanation": { "zh-Hant": "No只保留單次證據不能負擔一般全稱的界線，不把這個已由有限性另有證書的空間說成不緊緻，也不據有限測試分類無限空間。", "en": "The No limits the inference from one audit to a universal property.It does not declare this finite space noncompact;finiteness supplies a separate certificate.Nor does it classify an infinite space from finite tests." }, "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, false, "one_supplied_cover_does_not_prove_general_compactness" ] }, "conclusion": { "canonical_value": "([1], 是, 否, one_supplied_cover_does_not_prove_general_compactness)", "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-20261102-r05-compact-topological-space-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261102-r05-compact-topological-space-foundations/questions.jsonl", "batch_sha256": "01b7b492658c47414f5f927eb995ff63a21e4da461bafc0cf2fb9f40397a1db2", "original_line": 80, "original_line_sha256": "6e45cbe2db70017567ad52761625d4bcea946efba54b3dc9842e261b60665f82", "question_compact_sha256": "19c6fb6c1e73899fc747f14e47702acadaaab5342ac15f65cd91b4f17a788345", "english_row_compact_sha256": "1af3a538dd91a3bbd0a2b5d4c7491d625aa3020e8d3db34f6cc04a330488f8b7", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261102-r05-compact-topological-space-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261102-r05-compact-topological-space-foundations.en.jsonl" }, "mko_bindings": { "mko-compact-topological-space": { "sha256": "13f302a171c38a633a5a145da984410cab9bef79a09c4aef91bdaf9e016226e6", "english_sha256": "cb6c5eefd1b67e492a6fcd011cf21c954ed3dcfdf9a15d5bbc46db1460c8e204", "original_path": "public/data/mko/compact-topological-space.json", "english_path": "public/data/i18n/editions/mko-compact-topological-space.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "fe4c48735753bdd58c10db9b2ca3a2a3baeb31d0424c3fc09dea62317185ddae", "original_candidate_sha256": "fd8d03560221c6923a3615429a38af00a15e9a6c3263a839d27d99e459a9cfa3", "original_candidate_line_sha256": "fd8d03560221c6923a3615429a38af00a15e9a6c3263a839d27d99e459a9cfa3" }, "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-p33", "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": "d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e", "question_first_witness_sha256": "790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 7, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "2fc702d08e75a1e0bd370b82c26e920365ccb83b01bc128aeb87309ef965d128", "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." } }