{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0001","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062001,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0002","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062002,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0003","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062003,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0004","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062004,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0005","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[2,4],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062005,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[2,4],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0006","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062006,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0007","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062007,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0008","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062008,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0009","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062009,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0010","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062010,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0011","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062011,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0012","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062012,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0013","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062013,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0014","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062014,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0015","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062015,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0016","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062016,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0017","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,5],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062017,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,5],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0018","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062018,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0019","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062019,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0020","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062020,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0021","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062021,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0022","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062022,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0023","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062023,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0024","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062024,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0025","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-clique-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"vertices\":[0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 是, 3, candidate_clique_only_lower_bound_on_omega)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-clique-audit","seed":202611062025,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],"vertices":[0,1,2],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-clique-audit","semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0026","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062026,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0027","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062027,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0028","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062028,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0029","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062029,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0030","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[2,4],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062030,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[2,4],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0031","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062031,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0032","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062032,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0033","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062033,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0034","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062034,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0035","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062035,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0036","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062036,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0037","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062037,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0038","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062038,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0039","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062039,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0040","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062040,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0041","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062041,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0042","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,5],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062042,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,5],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0043","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062043,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0044","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062044,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0045","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062045,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0046","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062046,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0047","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062047,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0048","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062048,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0049","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062049,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0050","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-independent-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"vertices\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation_zh":"Wave5封閉層核對=([3, 5], 是, 是, 2, candidate_independent_set_only_lower_bound_on_alpha)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-independent-audit","seed":202611062050,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],"vertices":[3,5],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-independent-audit","semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[3,5]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0051","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062051,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0052","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062052,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0053","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062053,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0054","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062054,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0055","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[2,4],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062055,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[2,4],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[2,4],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0056","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062056,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0057","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062057,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0058","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062058,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0059","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062059,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0060","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062060,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0061","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062061,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0062","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062062,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0063","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062063,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0064","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [1, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062064,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0065","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062065,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0066","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062066,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0067","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,5],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062067,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,5],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0068","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062068,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0069","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062069,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0070","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062070,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0071","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062071,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0072","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062072,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0073","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [0, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062073,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0074","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062074,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0075","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-exact-extrema；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)","explanation_zh":"Wave5封閉層核對=([0, 1, 2], 3, [2, 3, 5], 3, 是, exhaustive_all_vertex_subsets)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-exact-extrema","seed":202611062075,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],"domain":"Complete_finite_simple_graph_extrema","operation":"clique-exact-extrema","semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"n\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0076","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062076,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0077","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062077,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0078","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062078,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0079","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062079,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0080","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[2,4],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062080,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[2,4],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0081","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062081,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0082","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062082,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0083","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062083,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0084","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062084,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0085","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062085,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0086","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062086,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0087","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062087,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0088","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062088,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0089","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062089,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0090","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062090,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0091","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062091,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0092","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,5],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062092,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,5],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0093","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062093,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0094","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062094,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0095","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062095,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0096","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062096,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0097","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062097,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0098","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062098,"parameters":{"n":6,"edges":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0099","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062099,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0100","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","language":"zh-Hant","source_mko_ids":["mko-clique-and-independent-set"],"title_zh":"Clique與獨立集","stem_zh":"完整有限簡單圖與全部候選頂點對。操作=clique-maximal-vs-maximum；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"vertices\":[3,4],\"kind\":\"clique\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation_zh":"Wave5封閉層核對=(clique, [3, 4], 是, 是, 3, 否, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)；依完整明示資料獨立重算；clique逐對相鄰且independent set逐對不相鄰；單一候選只給extremum下界；maximal不等於maximum；maximum須全域上界或exhaustive證書且clique只給chi下界。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Clique與獨立集"]},"variant":{"family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062100,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],"vertices":[3,4],"kind":"clique","domain":"Complete_finite_simple_graph_extrema","operation":"clique-maximal-vs-maximum","semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}