{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0001","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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-audit","seed":202611013001,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0002","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[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]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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-audit","seed":202611013002,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[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]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0003","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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-audit","seed":202611013003,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0004","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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-audit","seed":202611013004,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0005","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-audit；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]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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-audit","seed":202611013005,"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]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0006","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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-audit","seed":202611013006,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0007","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-audit；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]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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-audit","seed":202611013007,"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]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0008","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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-audit","seed":202611013008,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0009","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[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]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013009,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[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]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0010","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013010,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0011","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013011,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0012","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013012,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0013","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013013,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0014","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013014,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0015","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013015,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0016","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013016,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0017","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013017,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0018","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013018,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0019","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013019,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0020","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013020,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0021","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013021,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0022","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013022,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0023","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013023,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0024","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-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013024,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0025","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-audit；資料={\"n\":5,\"topology\":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]]}。請依 [invalid_open_indexes,covered_union,covers_target,valid_open_cover] 欄序回答。","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], 是, 是)","explanation_zh":"Wave5根節點核對=([], [0, 1, 2, 3, 4], 是, 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-audit","seed":202611013025,"parameters":{"n":5,"topology":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"domain":"Complete_finite_topological_cover","operation":"cover-audit","semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0026","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)；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-subcover-audit","seed":202611013026,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,1],[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],[0,1],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0027","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[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,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 4]], [0, 1, 2, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 4]], [0, 1, 2, 4], 是)；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-subcover-audit","seed":202611013027,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[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_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"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],[0,1,2],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0028","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)；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-subcover-audit","seed":202611013028,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,1],[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],[0,1,3],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0029","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 3, 4]], [0, 1, 3, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 3, 4]], [0, 1, 3, 4], 是)；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-subcover-audit","seed":202611013029,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0030","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-subcover-audit；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,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 4]], [0, 1, 2, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 4]], [0, 1, 2, 4], 是)；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-subcover-audit","seed":202611013030,"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_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0031","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 3, 4]], [0, 1, 3, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 3, 4]], [0, 1, 3, 4], 是)；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-subcover-audit","seed":202611013031,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0032","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-subcover-audit；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,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)；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-subcover-audit","seed":202611013032,"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-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0033","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 4]], [0, 1, 2, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 4]], [0, 1, 2, 4], 是)；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-subcover-audit","seed":202611013033,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,2],[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],[0,1,2],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0034","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[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,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013034,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[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-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"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],[0,2,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0035","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 2, 3, 4]], [0, 2, 3, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 2, 3, 4]], [0, 2, 3, 4], 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013035,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0036","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 4]], [0, 1, 2, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 4]], [0, 1, 2, 4], 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013036,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,2],[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],[0,2,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0037","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 2, 3, 4]], [0, 2, 3, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 2, 3, 4]], [0, 2, 3, 4], 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013037,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0038","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013038,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,3],[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],[0,1,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0039","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 3, 4]], [0, 1, 3, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 3, 4]], [0, 1, 3, 4], 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013039,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0040","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013040,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,3],[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],[0,2,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0041","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 2, 3, 4]], [0, 2, 3, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 2, 3, 4]], [0, 2, 3, 4], 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013041,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0042","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 3, 4]], [0, 1, 3, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 3, 4]], [0, 1, 3, 4], 是)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013042,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0043","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 2, 3, 4]], [0, 2, 3, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 2, 3, 4]], [0, 2, 3, 4], 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013043,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0044","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 4]], [0, 1, 2, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 4]], [0, 1, 2, 4], 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013044,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,4],[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],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0045","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 3, 4]], [0, 1, 3, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 3, 4]], [0, 1, 3, 4], 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013045,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0046","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 4]], [0, 1, 2, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 4]], [0, 1, 2, 4], 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013046,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,4],[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],[0,2,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0047","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 2, 3, 4]], [0, 2, 3, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 2, 3, 4]], [0, 2, 3, 4], 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013047,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0048","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 3, 4]], [0, 1, 3, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 3, 4]], [0, 1, 3, 4], 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013048,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0049","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 2, 3, 4]], [0, 2, 3, 4], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 2, 3, 4]], [0, 2, 3, 4], 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013049,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0050","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-subcover-audit；資料={\"n\":5,\"topology\":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,selected_sets,covered_union,valid_subcover] 欄序回答。","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)","explanation_zh":"Wave5根節點核對=([1], [[0, 1, 2, 3]], [0, 1, 2, 3], 是)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-subcover-audit","seed":202611013050,"parameters":{"n":5,"topology":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-subcover-audit","semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,1],[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],[0,1],[1]]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0051","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],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[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":202611013051,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[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\":[[0,1],[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],[0,1],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0052","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],[0,1],[0,1,2],[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,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":202611013052,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[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_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"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],[0,1,2],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0053","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],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[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":202611013053,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[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\":[[0,1],[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],[0,1,3],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0054","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],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[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":202611013054,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0055","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],[0,1],[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,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":202611013055,"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_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0056","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],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[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":202611013056,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[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、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"}}
{"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0058","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],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[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":202611013058,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[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],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0059","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],[0,2,3],[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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013059,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[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],[0,2,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0060","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],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013060,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[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],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0061","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],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013061,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[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],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0062","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],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013062,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[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],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0063","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],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013063,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[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\":[[0,3],[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],[0,1,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0064","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],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013064,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0065","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],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013065,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[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\":[[0,3],[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],[0,2,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0066","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],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013066,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0067","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],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013067,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0068","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],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013068,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0069","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],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013069,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,4],[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],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0070","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],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013070,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0071","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],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013071,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,4],[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],[0,2,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0072","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],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013072,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0073","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],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013073,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0074","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],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013074,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-minimal-subcover","semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0075","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\":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-minimal-subcover","seed":202611013075,"parameters":{"n":5,"topology":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[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\":[[0,1],[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],[0,1],[1]]}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0076","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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-single-cover-boundary","seed":202611013076,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,1],[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],[0,1],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0077","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,2],[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_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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-single-cover-boundary","seed":202611013077,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,2],[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_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-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],[0,1,2],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0078","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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-single-cover-boundary","seed":202611013078,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,1],[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],[0,1,3],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0079","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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-single-cover-boundary","seed":202611013079,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0080","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-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_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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-single-cover-boundary","seed":202611013080,"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_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0081","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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-single-cover-boundary","seed":202611013081,"parameters":{"n":5,"topology":[[],[0],[0,1],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,1],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[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、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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0082","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-single-cover-boundary；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,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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-single-cover-boundary","seed":202611013082,"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-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0083","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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-single-cover-boundary","seed":202611013083,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,1,2],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,2],[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],[0,1,2],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0084","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[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,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013084,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[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-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"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],[0,2,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0085","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013085,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0086","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013086,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,2],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,2],[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],[0,2,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0087","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013087,"parameters":{"n":5,"topology":[[],[0],[0,2],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,2],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0088","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013088,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,3],[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],[0,1,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0089","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013089,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,1,3],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0090","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013090,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,3],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,3],[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],[0,2,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0091","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013091,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,2,3],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0092","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":2,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013092,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,3],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0093","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013093,"parameters":{"n":5,"topology":[[],[0],[0,3],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,3],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0094","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013094,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,4],[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],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0095","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013095,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,1,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0096","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],\"target\":[0,1,2,4],\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013096,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,1,2,4],[0,1,2,3,4]],"target":[0,1,2,4],"cover":[[0,4],[0,1,2,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,4],[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],[0,2,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0097","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013097,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,2,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0098","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],\"target\":[0,1,3,4],\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013098,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,1,3,4],[0,1,2,3,4]],"target":[0,1,3,4],"cover":[[0,4],[0,1,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0099","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],\"target\":[0,2,3,4],\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013099,"parameters":{"n":5,"topology":[[],[0],[0,4],[0,3,4],[0,2,3,4],[0,1,2,3,4]],"target":[0,2,3,4],"cover":[[0,4],[0,2,3,4],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0100","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-single-cover-boundary；資料={\"n\":5,\"topology\":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],\"target\":[0,1,2,3],\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"selected\":[1]}。請依 [selected_indexes,valid_subcover,proves_compactness,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1], 是, 否, one_finite_subcover_does_not_prove_compactness)","explanation_zh":"Wave5根節點核對=([1], 是, 否, one_finite_subcover_does_not_prove_compactness)；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":3,"skills":["開覆蓋與子覆蓋"]},"variant":{"family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013100,"parameters":{"n":5,"topology":[[],[1],[0,1],[0,1,2],[0,1,2,3],[0,1,2,3,4]],"target":[0,1,2,3],"cover":[[0,1],[0,1,2,3],[0,1,2,3,4]],"selected":[1],"domain":"Complete_finite_topological_cover","operation":"cover-single-cover-boundary","semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,1],[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],[0,1],[1]]}"}},"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"}}