{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0001","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"candidate\":[\"v1\",\"v2\",\"v3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3], [[v0, v1], [v1, v2], [v2, v3]], [v1, v2, v3], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3], [[v0, v1], [v1, v2], [v2, v3]], [v1, v2, v3], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147001,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"]],"candidate":["v1","v2","v3"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0002","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"candidate\":[\"v0\",\"v2\",\"v3\",\"v4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], [[v0, v1], [v1, v2], [v2, v3], [v3, v4]], [v0, v2, v3, v4], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4], [[v0, v1], [v1, v2], [v2, v3], [v3, v4]], [v0, v2, v3, v4], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147002,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"]],"candidate":["v0","v2","v3","v4"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0003","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"candidate\":[\"v0\",\"v1\",\"v3\",\"v4\",\"v5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v3, v4, v5], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v3, v4, v5], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147003,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"]],"candidate":["v0","v1","v3","v4","v5"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0004","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v4\",\"v5\",\"v6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [v0, v1, v2, v4, v5, v6], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [v0, v1, v2, v4, v5, v6], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147004,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"]],"candidate":["v0","v1","v2","v4","v5","v6"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0005","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v5\",\"v6\",\"v7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v5, v6, v7], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v5, v6, v7], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147005,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"]],"candidate":["v0","v1","v2","v3","v5","v6","v7"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0006","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147006,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"]],"candidate":["v0","v1","v2","v3","v4","v6","v7","v8"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0007","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v7\",\"v8\",\"v9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [[[v0, v1], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147007,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"]],"candidate":["v0","v1","v2","v3","v4","v5","v7","v8","v9"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0008","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v8\",\"v9\",\"v10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [[[v0, v1], 是], [[v1, v2], 是], [[v10, v9], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [[[v0, v1], 是], [[v1, v2], 是], [[v10, v9], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147008,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v8","v9","v10"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0009","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v11\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v9\",\"v10\",\"v11\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v9], [[[v0, v1], 是], [[v1, v2], 是], [[v10, v11], 是], [[v10, v9], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v9], [[[v0, v1], 是], [[v1, v2], 是], [[v10, v11], 是], [[v10, v9], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147009,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v11"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v9","v10","v11"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0010","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v0\"]],\"candidate\":[\"v0\",\"v2\",\"v3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], [v0, v2, v3], [[[v0, v1], 是], [[v0, v3], 是], [[v1, v2], 是], [[v2, v3], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], [v0, v2, v3], [[[v0, v1], 是], [[v0, v3], 是], [[v1, v2], 是], [[v2, v3], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147010,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v0"]],"candidate":["v0","v2","v3"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0011","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v0\"]],\"candidate\":[\"v1\",\"v2\",\"v3\",\"v4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], [v1, v2, v3, v4], [[[v0, v1], 是], [[v0, v4], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], [v1, v2, v3, v4], [[[v0, v1], 是], [[v0, v4], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147011,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v0"]],"candidate":["v1","v2","v3","v4"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0012","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v2, v3, v4], [[[v0, v1], 是], [[v0, v5], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v2, v3, v4], [[[v0, v1], 是], [[v0, v5], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147012,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v0"]],"candidate":["v0","v1","v2","v3","v4"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0013","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [v0, v1, v2, v3, v4, v6], [[[v0, v1], 是], [[v0, v6], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [v0, v1, v2, v3, v4, v6], [[[v0, v1], 是], [[v0, v6], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147013,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v0"]],"candidate":["v0","v1","v2","v3","v4","v6"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v6\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0014","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v4, v6, v7], [[[v0, v1], 是], [[v0, v7], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v4, v6, v7], [[[v0, v1], 是], [[v0, v7], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147014,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v0"]],"candidate":["v0","v1","v2","v3","v4","v6","v7"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v7\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0015","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], 是], [[v0, v8], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], 是], [[v0, v8], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147015,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v0"]],"candidate":["v0","v1","v2","v3","v4","v6","v7","v8"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v8\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0016","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\",\"v9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], 是], [[v0, v9], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], 是], [[v0, v9], 是], [[v1, v2], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147016,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v0"]],"candidate":["v0","v1","v2","v3","v4","v6","v7","v8","v9"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v9\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0017","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], 是], [[v0, v10], 是], [[v1, v2], 是], [[v10, v9], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], 是], [[v0, v10], 是], [[v1, v2], 是], [[v10, v9], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147017,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v0"]],"candidate":["v0","v1","v2","v3","v4","v6","v7","v8","v9","v10"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v10\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0018","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v11\"],[\"v11\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], 是], [[v0, v11], 是], [[v1, v2], 是], [[v10, v11], 是], [[v10, v9], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], 是], [[v0, v11], 是], [[v1, v2], 是], [[v10, v11], 是], [[v10, v9], 是], [[v2, v3], 是], [[v3, v4], 是], [[v4, v5], 是], [[v5, v6], 是], [[v6, v7], 是], [[v7, v8], 是], [[v8, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147018,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v11"],["v11","v0"]],"candidate":["v0","v1","v2","v3","v4","v6","v7","v8","v9","v10","v11"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v11\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0019","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"candidate\":[\"v0\",\"v1\",\"v3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3], [[v0, v1], [v0, v2], [v0, v3]], [v0, v1, v3], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3], [[v0, v1], [v0, v2], [v0, v3]], [v0, v1, v3], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147019,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"]],"candidate":["v0","v1","v3"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0020","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4]], [v0, v1, v2, v3], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是], [[v0, v4], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4]], [v0, v1, v2, v3], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是], [[v0, v4], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147020,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"]],"candidate":["v0","v1","v2","v3"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0021","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"candidate\":[\"v0\",\"v1\",\"v3\",\"v4\",\"v5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]], [v0, v1, v3, v4, v5], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是], [[v0, v4], 是], [[v0, v5], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]], [v0, v1, v3, v4, v5], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是], [[v0, v4], 是], [[v0, v5], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147021,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"]],"candidate":["v0","v1","v3","v4","v5"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0022","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"candidate\":[\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]], [v1, v2, v3, v4, v5, v6], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是], [[v0, v4], 是], [[v0, v5], 是], [[v0, v6], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]], [v1, v2, v3, v4, v5, v6], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是], [[v0, v4], 是], [[v0, v5], 是], [[v0, v6], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147022,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"]],"candidate":["v1","v2","v3","v4","v5","v6"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0023","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [v0, v1, v2, v3, v4, v5, v7], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是], [[v0, v4], 是], [[v0, v5], 是], [[v0, v6], 是], [[v0, v7], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [v0, v1, v2, v3, v4, v5, v7], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是], [[v0, v4], 是], [[v0, v5], 是], [[v0, v6], 是], [[v0, v7], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147023,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"]],"candidate":["v0","v1","v2","v3","v4","v5","v7"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0024","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是], [[v0, v4], 是], [[v0, v5], 是], [[v0, v6], 是], [[v0, v7], 是], [[v0, v8], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是], [[v0, v4], 是], [[v0, v5], 是], [[v0, v6], 是], [[v0, v7], 是], [[v0, v8], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147024,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"]],"candidate":["v0","v1","v2","v3","v4","v6","v7","v8"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0025","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-complete-edge-audit；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是], [[v0, v4], 是], [[v0, v5], 是], [[v0, v6], 是], [[v0, v7], 是], [[v0, v8], 是], [[v0, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation_zh":"Wave6第八層核對=([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [[[v0, v1], 是], [[v0, v2], 是], [[v0, v3], 是], [[v0, v4], 是], [[v0, v5], 是], [[v0, v6], 是], [[v0, v7], 是], [[v0, v8], 是], [[v0, v9], 是]], 是, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147025,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"],["v0","v9"]],"candidate":["v0","v1","v2","v3","v5","v6","v7","v8","v9"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-complete-edge-audit","semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0026","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147026,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0027","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147027,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0028","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147028,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0029","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147029,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0030","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147030,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0031","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147031,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0032","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147032,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0033","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v1, v2], 否], [[v10, v9], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v1, v2], 否], [[v10, v9], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147033,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0034","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v11\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v1, v2], 否], [[v10, v11], 否], [[v10, v9], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v1, v2], 否], [[v10, v11], 否], [[v10, v9], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147034,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v11"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0035","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v0\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v3], 否], [[v1, v2], 否], [[v2, v3], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v3], 否], [[v1, v2], 否], [[v2, v3], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147035,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v0"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0036","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v0\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v4], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v4], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147036,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v0"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0037","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v0\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v5], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v5], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147037,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v0"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0038","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v0\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v6], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v6], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147038,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v0"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v6\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0039","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v0\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v7], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v7], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147039,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v0"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v7\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0040","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v0\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v8], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v8], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147040,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v0"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v8\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0041","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v0\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v9], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v9], 否], [[v1, v2], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147041,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v0"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v9\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0042","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v0\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v10], 否], [[v1, v2], 否], [[v10, v9], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v10], 否], [[v1, v2], 否], [[v10, v9], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147042,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v0"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v10\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0043","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"],[\"v10\",\"v11\"],[\"v11\",\"v0\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v11], 否], [[v1, v2], 否], [[v10, v11], 否], [[v10, v9], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v11], 否], [[v1, v2], 否], [[v10, v11], 否], [[v10, v9], 否], [[v2, v3], 否], [[v3, v4], 否], [[v4, v5], 否], [[v5, v6], 否], [[v6, v7], 否], [[v7, v8], 否], [[v8, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147043,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"],["v10","v11"],["v11","v0"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v11\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0044","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147044,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0045","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否], [[v0, v4], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否], [[v0, v4], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147045,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0046","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否], [[v0, v4], 否], [[v0, v5], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否], [[v0, v4], 否], [[v0, v5], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147046,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0047","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否], [[v0, v4], 否], [[v0, v5], 否], [[v0, v6], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否], [[v0, v4], 否], [[v0, v5], 否], [[v0, v6], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147047,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0048","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否], [[v0, v4], 否], [[v0, v5], 否], [[v0, v6], 否], [[v0, v7], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否], [[v0, v4], 否], [[v0, v5], 否], [[v0, v6], 否], [[v0, v7], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147048,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0049","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否], [[v0, v4], 否], [[v0, v5], 否], [[v0, v6], 否], [[v0, v7], 否], [[v0, v8], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否], [[v0, v4], 否], [[v0, v5], 否], [[v0, v6], 否], [[v0, v7], 否], [[v0, v8], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147049,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0050","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-uncovered-edge-witness；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"candidate\":[]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否], [[v0, v4], 否], [[v0, v5], 否], [[v0, v6], 否], [[v0, v7], 否], [[v0, v8], 否], [[v0, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation_zh":"Wave6第八層核對=([], [[[v0, v1], 否], [[v0, v2], 否], [[v0, v3], 否], [[v0, v4], 否], [[v0, v5], 否], [[v0, v6], 否], [[v0, v7], 否], [[v0, v8], 否], [[v0, v9], 否]], [v0, v1], 否, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147050,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"],["v0","v9"]],"candidate":[],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-uncovered-edge-witness","semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0051","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\"],\"edges\":[],\"candidate\":[],\"superset\":[\"v1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [v0, v1], [], 是, [v1], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(是, 是, [v0, v1], [], 是, [v1], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147051,"parameters":{"vertices":["v0","v1"],"edges":[],"candidate":[],"superset":["v1"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[],\"edges\":[],\"superset\":[\"v1\"],\"vertices\":[\"v0\",\"v1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0052","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"]],\"candidate\":[\"v0\",\"v1\",\"v2\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v3], [v0, v1, v2], 是, [v0, v1, v2, v3], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v3], [v0, v1, v2], 是, [v0, v1, v2, v3], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147052,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"]],"candidate":["v0","v1","v2"],"superset":["v0","v1","v2","v3"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0053","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v4], [v0, v1, v2, v3], 是, [v0, v1, v2, v3, v4], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v4], [v0, v1, v2, v3], 是, [v0, v1, v2, v3, v4], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147053,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"]],"candidate":["v0","v1","v2","v3"],"superset":["v0","v1","v2","v3","v4"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0054","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v5], [v0, v1, v2, v3, v4], 是, [v0, v1, v2, v3, v4, v5], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v5], [v0, v1, v2, v3, v4], 是, [v0, v1, v2, v3, v4, v5], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147054,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"]],"candidate":["v0","v1","v2","v3","v4"],"superset":["v0","v1","v2","v3","v4","v5"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0055","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v6], [v0, v1, v2, v3, v4, v5], 是, [v0, v1, v2, v3, v4, v5, v6], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v6], [v0, v1, v2, v3, v4, v5], 是, [v0, v1, v2, v3, v4, v5, v6], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147055,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"]],"candidate":["v0","v1","v2","v3","v4","v5"],"superset":["v0","v1","v2","v3","v4","v5","v6"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0056","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v7], [v0, v1, v2, v3, v4, v5, v6], 是, [v0, v1, v2, v3, v4, v5, v6, v7], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v7], [v0, v1, v2, v3, v4, v5, v6], 是, [v0, v1, v2, v3, v4, v5, v6, v7], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147056,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"]],"candidate":["v0","v1","v2","v3","v4","v5","v6"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0057","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v8], [v0, v1, v2, v3, v4, v5, v6, v7], 是, [v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v8], [v0, v1, v2, v3, v4, v5, v6, v7], 是, [v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147057,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0058","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147058,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0059","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147059,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"superset\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0060","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"],[\"v9\",\"v10\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147060,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v6"],["v6","v7"],["v7","v8"],["v8","v9"],["v9","v10"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"superset\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0061","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\"],\"edges\":[[\"v0\",\"v1\"]],\"candidate\":[\"v0\",\"v1\"],\"superset\":[\"v0\",\"v1\",\"v2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v2], [v0, v1], 是, [v0, v1, v2], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v2], [v0, v1], 是, [v0, v1, v2], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147061,"parameters":{"vertices":["v0","v1","v2"],"edges":[["v0","v1"]],"candidate":["v0","v1"],"superset":["v0","v1","v2"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\"],\"edges\":[[\"v0\",\"v1\"]],\"superset\":[\"v0\",\"v1\",\"v2\"],\"vertices\":[\"v0\",\"v1\",\"v2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0062","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"]],\"candidate\":[\"v0\",\"v1\",\"v2\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v3], [v0, v1, v2], 是, [v0, v1, v2, v3], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v3], [v0, v1, v2], 是, [v0, v1, v2, v3], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147062,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v0","v2"]],"candidate":["v0","v1","v2"],"superset":["v0","v1","v2","v3"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0063","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v4], [v0, v1, v2, v3], 是, [v0, v1, v2, v3, v4], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v4], [v0, v1, v2, v3], 是, [v0, v1, v2, v3, v4], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147063,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"]],"candidate":["v0","v1","v2","v3"],"superset":["v0","v1","v2","v3","v4"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0064","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v5], [v0, v1, v2, v3, v4], 是, [v0, v1, v2, v3, v4, v5], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v5], [v0, v1, v2, v3, v4], 是, [v0, v1, v2, v3, v4, v5], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147064,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"]],"candidate":["v0","v1","v2","v3","v4"],"superset":["v0","v1","v2","v3","v4","v5"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0065","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v6], [v0, v1, v2, v3, v4, v5], 是, [v0, v1, v2, v3, v4, v5, v6], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v6], [v0, v1, v2, v3, v4, v5], 是, [v0, v1, v2, v3, v4, v5, v6], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147065,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"]],"candidate":["v0","v1","v2","v3","v4","v5"],"superset":["v0","v1","v2","v3","v4","v5","v6"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0066","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v7], [v0, v1, v2, v3, v4, v5, v6], 是, [v0, v1, v2, v3, v4, v5, v6, v7], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v7], [v0, v1, v2, v3, v4, v5, v6], 是, [v0, v1, v2, v3, v4, v5, v6, v7], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147066,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"]],"candidate":["v0","v1","v2","v3","v4","v5","v6"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0067","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v8], [v0, v1, v2, v3, v4, v5, v6, v7], 是, [v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v8], [v0, v1, v2, v3, v4, v5, v6, v7], 是, [v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147067,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0068","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8], 是, [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147068,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0069","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], 是, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147069,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"],["v0","v9"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"superset\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0070","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"],[\"v0\",\"v10\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\",\"v10\",\"v11\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], 是, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147070,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"edges":[["v0","v1"],["v0","v2"],["v0","v3"],["v0","v4"],["v0","v5"],["v0","v6"],["v0","v7"],["v0","v8"],["v0","v9"],["v0","v10"]],"candidate":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10"],"superset":["v0","v1","v2","v3","v4","v5","v6","v7","v8","v9","v10","v11"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v10\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"superset\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0071","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\"],\"edges\":[],\"candidate\":[\"v0\",\"v1\"],\"superset\":[\"v0\",\"v1\",\"v2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [v0, v1, v2], [v0, v1], 是, [v0, v1, v2], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(是, 是, [v0, v1, v2], [v0, v1], 是, [v0, v1, v2], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147071,"parameters":{"vertices":["v0","v1","v2"],"edges":[],"candidate":["v0","v1"],"superset":["v0","v1","v2"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\"],\"edges\":[],\"superset\":[\"v0\",\"v1\",\"v2\"],\"vertices\":[\"v0\",\"v1\",\"v2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0072","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v3], [v0, v1, v2], 是, [v0, v1, v2, v3], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v3], [v0, v1, v2], 是, [v0, v1, v2, v3], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147072,"parameters":{"vertices":["v0","v1","v2","v3"],"edges":[["v0","v1"],["v1","v2"],["v2","v0"]],"candidate":["v0","v1","v2"],"superset":["v0","v1","v2","v3"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v1\",\"v2\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0073","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v4], [v0, v1, v2, v3], 是, [v0, v1, v2, v3, v4], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v4], [v0, v1, v2, v3], 是, [v0, v1, v2, v3, v4], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147073,"parameters":{"vertices":["v0","v1","v2","v3","v4"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v0"]],"candidate":["v0","v1","v2","v3"],"superset":["v0","v1","v2","v3","v4"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0074","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v5], [v0, v1, v2, v3, v4], 是, [v0, v1, v2, v3, v4, v5], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v5], [v0, v1, v2, v3, v4], 是, [v0, v1, v2, v3, v4, v5], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147074,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v0"]],"candidate":["v0","v1","v2","v3","v4"],"superset":["v0","v1","v2","v3","v4","v5"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0075","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-empty-isolated-superset；資料={\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v0\"]],\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(否, null, [v6], [v0, v1, v2, v3, v4, v5], 是, [v0, v1, v2, v3, v4, v5, v6], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation_zh":"Wave6第八層核對=(否, null, [v6], [v0, v1, v2, v3, v4, v5], 是, [v0, v1, v2, v3, v4, v5, v6], 是, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147075,"parameters":{"vertices":["v0","v1","v2","v3","v4","v5","v6"],"edges":[["v0","v1"],["v1","v2"],["v2","v3"],["v3","v4"],["v4","v5"],["v5","v0"]],"candidate":["v0","v1","v2","v3","v4","v5"],"superset":["v0","v1","v2","v3","v4","v5","v6"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-empty-isolated-superset","semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0076","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"]],\"candidate\":[\"l0_0\",\"l0_1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1], [[l0_0, [l0_1], 否], [l0_1, [l0_0], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1], [[l0_0, [l0_1], 否], [l0_1, [l0_0], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147076,"parameters":{"vertices":["c0","l0_0","l0_1"],"edges":[["c0","l0_0"],["c0","l0_1"]],"candidate":["l0_0","l0_1"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0077","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2], [[l0_0, [l0_1, l0_2], 否], [l0_1, [l0_0, l0_2], 否], [l0_2, [l0_0, l0_1], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2], [[l0_0, [l0_1, l0_2], 否], [l0_1, [l0_0, l0_2], 否], [l0_2, [l0_0, l0_1], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147077,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"]],"candidate":["l0_0","l0_1","l0_2"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0078","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3], [[l0_0, [l0_1, l0_2, l0_3], 否], [l0_1, [l0_0, l0_2, l0_3], 否], [l0_2, [l0_0, l0_1, l0_3], 否], [l0_3, [l0_0, l0_1, l0_2], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l0_3], [[l0_0, [l0_1, l0_2, l0_3], 否], [l0_1, [l0_0, l0_2, l0_3], 否], [l0_2, [l0_0, l0_1, l0_3], 否], [l0_3, [l0_0, l0_1, l0_2], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147078,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"]],"candidate":["l0_0","l0_1","l0_2","l0_3"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0079","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4], [[l0_0, [l0_1, l0_2, l0_3, l0_4], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l0_3, l0_4], [[l0_0, [l0_1, l0_2, l0_3, l0_4], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147079,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0080","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147080,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4","l0_5"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"],["c0","l0_5"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4","l0_5"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0081","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147081,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"],["c0","l0_5"],["c0","l0_6"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0082","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7], 否], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7], 否], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147082,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"],["c0","l0_5"],["c0","l0_6"],["c0","l0_7"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0083","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8], 否], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8], 否], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8], 否], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8], 否], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147083,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7","l0_8"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"],["c0","l0_5"],["c0","l0_6"],["c0","l0_7"],["c0","l0_8"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7","l0_8"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0084","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"],[\"c0\",\"l0_9\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], 否], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], 否], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], 否], [l0_9, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], 否], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], 否], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], 否], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], 否], [l0_9, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147084,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7","l0_8","l0_9"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"],["c0","l0_5"],["c0","l0_6"],["c0","l0_7"],["c0","l0_8"],["c0","l0_9"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7","l0_8","l0_9"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"],[\"c0\",\"l0_9\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0085","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\",\"l0_10\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"],[\"c0\",\"l0_9\"],[\"c0\",\"l0_10\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\",\"l0_10\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_1, [l0_0, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_10, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_2, [l0_0, l0_1, l0_10, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_3, [l0_0, l0_1, l0_10, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_4, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_5, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], 否], [l0_6, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], 否], [l0_7, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], 否], [l0_8, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], 否], [l0_9, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_1, [l0_0, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_10, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_2, [l0_0, l0_1, l0_10, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_3, [l0_0, l0_1, l0_10, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_4, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], 否], [l0_5, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], 否], [l0_6, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], 否], [l0_7, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], 否], [l0_8, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], 否], [l0_9, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], 否]], 是, 1, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147085,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7","l0_8","l0_9","l0_10"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c0","l0_4"],["c0","l0_5"],["c0","l0_6"],["c0","l0_7"],["c0","l0_8"],["c0","l0_9"],["c0","l0_10"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l0_4","l0_5","l0_6","l0_7","l0_8","l0_9","l0_10"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_10\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_10\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"],[\"c0\",\"l0_9\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_10\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0086","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1], [[l0_0, [l0_1, l1_0, l1_1], 否], [l0_1, [l0_0, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l1_1], 否], [l1_1, [l0_0, l0_1, l1_0], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l1_0, l1_1], [[l0_0, [l0_1, l1_0, l1_1], 否], [l0_1, [l0_0, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l1_1], 否], [l1_1, [l0_0, l0_1, l1_0], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147086,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"]],"candidate":["l0_0","l0_1","l1_0","l1_1"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0087","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l1_0, l1_1, l1_2], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l1_0, l1_1, l1_2], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147087,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1","l1_2"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"]],"candidate":["l0_0","l0_1","l1_0","l1_1","l1_2"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0088","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147088,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1","l1_2","l1_3"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"]],"candidate":["l0_0","l0_1","l1_0","l1_1","l1_2","l1_3"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0089","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147089,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1","l1_2","l1_3","l1_4"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"]],"candidate":["l0_0","l0_1","l1_0","l1_1","l1_2","l1_3","l1_4"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0090","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147090,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"],["c1","l1_5"]],"candidate":["l0_0","l0_1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0091","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], 否], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], 否], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147091,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5","l1_6"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"],["c1","l1_5"],["c1","l1_6"]],"candidate":["l0_0","l0_1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5","l1_6"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0092","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\",\"l1_7\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"],[\"c1\",\"l1_7\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\",\"l1_7\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6, l1_7], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6, l1_7], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6, l1_7], 否], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_7], 否], [l1_7, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6, l1_7], 否], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6, l1_7], 否], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6, l1_7], 否], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6, l1_7], 否], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_7], 否], [l1_7, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147092,"parameters":{"vertices":["c0","l0_0","l0_1","c1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5","l1_6","l1_7"],"edges":[["c0","l0_0"],["c0","l0_1"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"],["c1","l1_5"],["c1","l1_6"],["c1","l1_7"]],"candidate":["l0_0","l0_1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5","l1_6","l1_7"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\",\"l1_7\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"],[\"c1\",\"l1_7\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\",\"l1_7\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0093","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"c1\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l1_0, l1_1], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l1_0, l1_1], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147093,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0094","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147094,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1","l1_2"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1","l1_2"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0095","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147095,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1","l1_2","l1_3"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1","l1_2","l1_3"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0096","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], 否], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], 否], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147096,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1","l1_2","l1_3","l1_4"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1","l1_2","l1_3","l1_4"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0097","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5], 否], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5], 否], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5], 否], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5], 否], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147097,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"],["c1","l1_5"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0098","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], 否], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], 否], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], 否], [l1_6, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], 否], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], 否], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], 否], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], 否], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], 否], [l1_6, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147098,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","c1","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5","l1_6"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"],["c1","l1_3"],["c1","l1_4"],["c1","l1_5"],["c1","l1_6"]],"candidate":["l0_0","l0_1","l0_2","l1_0","l1_1","l1_2","l1_3","l1_4","l1_5","l1_6"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0099","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"c1\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1], 否], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1], 否], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1], 否], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1], 否], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1], 否], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1], 否], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1], 否], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1], 否], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1], 否], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147099,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","c1","l1_0","l1_1"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c1","l1_0"],["c1","l1_1"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l1_0","l1_1"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r07-vertex-cover-0100","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","language":"zh-Hant","source_mko_ids":["mko-vertex-cover"],"title_zh":"頂點覆蓋","stem_zh":"有限simple undirected graph的全部edges與candidate subset完整核對。操作=vcover-minimal-vs-minimum-boundary；資料={\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"c1\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\",\"l1_2\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], 否], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], 否], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], 否], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], 否], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], 否], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], 否], [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation_zh":"Wave6第八層核對=([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], 否], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], 否], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], 否], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], 否], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], 否], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], 否], [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], 否]], 是, 2, 否, 否, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)；依完整明示資料獨立重算；uncovered edge是反例；empty-edge、isolated vertex與cover superset邊界分開；inclusion-minimal不是minimum；不混用edge cover、independent set或directed cover。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["頂點覆蓋"]},"variant":{"family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147100,"parameters":{"vertices":["c0","l0_0","l0_1","l0_2","l0_3","c1","l1_0","l1_1","l1_2"],"edges":[["c0","l0_0"],["c0","l0_1"],["c0","l0_2"],["c0","l0_3"],["c1","l1_0"],["c1","l1_1"],["c1","l1_2"]],"candidate":["l0_0","l0_1","l0_2","l0_3","l1_0","l1_1","l1_2"],"domain":"Complete_finite_simple_graph_vertex_cover","operation":"vcover-minimal-vs-minimum-boundary","semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\",\"l1_2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}