{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0001","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":2}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([0, 1], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315001,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0002","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[1,2,1,2,1,2],\"k\":3}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1, 2], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([1, 2], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315002,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[1,2,1,2,1,2],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[1,2,1,2,1,2],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0003","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[2,3,2,3,2,3],\"k\":4}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([2, 3], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([2, 3], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315003,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[2,3,2,3,2,3],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[2,3,2,3,2,3],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0004","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[3,4,3,4,3,4],\"k\":2}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([3, 4], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([3, 4], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315004,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[3,4,3,4,3,4],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[3,4,3,4,3,4],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0005","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[4,5,4,5,4,5],\"k\":3}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([4, 5], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([4, 5], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315005,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[4,5,4,5,4,5],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[4,5,4,5,4,5],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0006","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[5,6,5,6,5,6],\"k\":4}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([5, 6], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([5, 6], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315006,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[5,6,5,6,5,6],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[5,6,5,6,5,6],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0007","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[6,7,6,7,6,7],\"k\":2}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([6, 7], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([6, 7], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315007,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[6,7,6,7,6,7],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[6,7,6,7,6,7],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0008","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[7,8,7,8,7,8],\"k\":3}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([7, 8], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([7, 8], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315008,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[7,8,7,8,7,8],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[7,8,7,8,7,8],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0009","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[8,9,8,9,8,9],\"k\":4}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([8, 9], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([8, 9], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315009,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[8,9,8,9,8,9],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[8,9,8,9,8,9],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0010","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[9,10,9,10,9,10],\"k\":2}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([9, 10], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([9, 10], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315010,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[9,10,9,10,9,10],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[9,10,9,10,9,10],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0011","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[10,11,10,11,10,11],\"k\":3}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([10, 11], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([10, 11], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315011,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[10,11,10,11,10,11],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[10,11,10,11,10,11],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0012","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[11,12,11,12,11,12],\"k\":4}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([11, 12], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([11, 12], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315012,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[11,12,11,12,11,12],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[11,12,11,12,11,12],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0013","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[12,13,12,13,12,13],\"k\":2}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([12, 13], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([12, 13], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315013,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[12,13,12,13,12,13],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[12,13,12,13,12,13],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0014","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[13,14,13,14,13,14],\"k\":3}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([13, 14], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([13, 14], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315014,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[13,14,13,14,13,14],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[13,14,13,14,13,14],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0015","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[14,15,14,15,14,15],\"k\":4}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([14, 15], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([14, 15], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315015,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[14,15,14,15,14,15],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[14,15,14,15,14,15],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0016","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[15,16,15,16,15,16],\"k\":2}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([15, 16], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([15, 16], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315016,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[15,16,15,16,15,16],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[15,16,15,16,15,16],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0017","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[16,17,16,17,16,17],\"k\":3}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([16, 17], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([16, 17], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315017,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[16,17,16,17,16,17],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[16,17,16,17,16,17],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0018","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[17,18,17,18,17,18],\"k\":4}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([17, 18], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([17, 18], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315018,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[17,18,17,18,17,18],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[17,18,17,18,17,18],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0019","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[18,19,18,19,18,19],\"k\":2}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([18, 19], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([18, 19], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315019,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[18,19,18,19,18,19],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[18,19,18,19,18,19],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0020","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[19,20,19,20,19,20],\"k\":3}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([19, 20], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([19, 20], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315020,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[19,20,19,20,19,20],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[19,20,19,20,19,20],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0021","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[20,21,20,21,20,21],\"k\":4}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([20, 21], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([20, 21], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315021,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[20,21,20,21,20,21],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[20,21,20,21,20,21],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0022","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[21,22,21,22,21,22],\"k\":2}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([21, 22], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([21, 22], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315022,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[21,22,21,22,21,22],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[21,22,21,22,21,22],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0023","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[22,23,22,23,22,23],\"k\":3}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([22, 23], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([22, 23], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315023,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[22,23,22,23,22,23],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[22,23,22,23,22,23],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0024","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[23,24,23,24,23,24],\"k\":4}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([23, 24], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([23, 24], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315024,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[23,24,23,24,23,24],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[23,24,23,24,23,24],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0025","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[24,25,24,25,24,25],\"k\":2}。請依 [used_colors,conflicts,edge_proper,at_most_k,proper_k_coloring,boundary] 欄序回答。","answer":{"kind":"expression","value":"([24, 25], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)","explanation_zh":"Wave4收束核對=([24, 25], [], 是, 是, 是, proper_k_coloring_uses_at_most_k_colors)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-proper-audit","seed":202610315025,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[24,25,24,25,24,25],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-proper-audit","semantic_template":"qf-mod31-coloring-proper-audit/{\"colors\":[24,25,24,25,24,25],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0026","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,0,1,0,1,0]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315026,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,0,1,0,1,0],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[0,0,1,0,1,0],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0027","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[1,1,2,1,2,1]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315027,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[1,1,2,1,2,1],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[1,1,2,1,2,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0028","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[2,2,3,2,3,2]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315028,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[2,2,3,2,3,2],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[2,2,3,2,3,2],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0029","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[3,3,4,3,4,3]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315029,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[3,3,4,3,4,3],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[3,3,4,3,4,3],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0030","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[4,4,5,4,5,4]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315030,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[4,4,5,4,5,4],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[4,4,5,4,5,4],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0031","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[5,5,6,5,6,5]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315031,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[5,5,6,5,6,5],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[5,5,6,5,6,5],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0032","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[6,6,7,6,7,6]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315032,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[6,6,7,6,7,6],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[6,6,7,6,7,6],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0033","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[7,7,8,7,8,7]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315033,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[7,7,8,7,8,7],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[7,7,8,7,8,7],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0043","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[17,17,18,17,18,17]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315043,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[17,17,18,17,18,17],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[17,17,18,17,18,17],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0045","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[19,19,20,19,20,19]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315045,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[19,19,20,19,20,19],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[19,19,20,19,20,19],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0046","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[20,20,21,20,21,20]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315046,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[20,20,21,20,21,20],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[20,20,21,20,21,20],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0047","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[21,21,22,21,22,21]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315047,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[21,21,22,21,22,21],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[21,21,22,21,22,21],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0048","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[22,22,23,22,23,22]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315048,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[22,22,23,22,23,22],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[22,22,23,22,23,22],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0049","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[23,23,24,23,24,23]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315049,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[23,23,24,23,24,23],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[23,23,24,23,24,23],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0050","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-conflict；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[24,24,25,24,25,24]}。請依 [conflicting_edges,count,proper] 欄序回答。","answer":{"kind":"expression","value":"([[0, 1]], 1, 否)","explanation_zh":"Wave4收束核對=([[0, 1]], 1, 否)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-conflict","seed":202610315050,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[24,24,25,24,25,24],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-conflict","semantic_template":"qf-mod31-coloring-conflict/{\"colors\":[24,24,25,24,25,24],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0051","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":2}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315051,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0052","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[1,2,1,2,1,2],\"k\":3}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315052,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[1,2,1,2,1,2],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[1,2,1,2,1,2],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0053","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[2,3,2,3,2,3],\"k\":4}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315053,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[2,3,2,3,2,3],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[2,3,2,3,2,3],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0054","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[3,4,3,4,3,4],\"k\":2}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315054,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[3,4,3,4,3,4],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[3,4,3,4,3,4],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0055","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[4,5,4,5,4,5],\"k\":3}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315055,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[4,5,4,5,4,5],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[4,5,4,5,4,5],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0056","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[5,6,5,6,5,6],\"k\":4}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315056,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[5,6,5,6,5,6],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[5,6,5,6,5,6],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0057","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[6,7,6,7,6,7],\"k\":2}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315057,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[6,7,6,7,6,7],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[6,7,6,7,6,7],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0058","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[7,8,7,8,7,8],\"k\":3}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315058,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[7,8,7,8,7,8],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[7,8,7,8,7,8],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0059","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[8,9,8,9,8,9],\"k\":4}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315059,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[8,9,8,9,8,9],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[8,9,8,9,8,9],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0060","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[9,10,9,10,9,10],\"k\":2}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315060,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[9,10,9,10,9,10],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[9,10,9,10,9,10],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0061","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[10,11,10,11,10,11],\"k\":3}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315061,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[10,11,10,11,10,11],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[10,11,10,11,10,11],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0062","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[11,12,11,12,11,12],\"k\":4}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315062,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[11,12,11,12,11,12],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[11,12,11,12,11,12],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0063","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[12,13,12,13,12,13],\"k\":2}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315063,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[12,13,12,13,12,13],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[12,13,12,13,12,13],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0064","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[13,14,13,14,13,14],\"k\":3}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315064,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[13,14,13,14,13,14],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[13,14,13,14,13,14],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0065","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[14,15,14,15,14,15],\"k\":4}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315065,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[14,15,14,15,14,15],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[14,15,14,15,14,15],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0066","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[15,16,15,16,15,16],\"k\":2}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315066,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[15,16,15,16,15,16],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[15,16,15,16,15,16],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0067","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[16,17,16,17,16,17],\"k\":3}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315067,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[16,17,16,17,16,17],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[16,17,16,17,16,17],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0068","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[17,18,17,18,17,18],\"k\":4}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315068,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[17,18,17,18,17,18],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[17,18,17,18,17,18],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0069","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[18,19,18,19,18,19],\"k\":2}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315069,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[18,19,18,19,18,19],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[18,19,18,19,18,19],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0070","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[19,20,19,20,19,20],\"k\":3}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315070,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[19,20,19,20,19,20],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[19,20,19,20,19,20],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0071","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[20,21,20,21,20,21],\"k\":4}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315071,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[20,21,20,21,20,21],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[20,21,20,21,20,21],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0072","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[21,22,21,22,21,22],\"k\":2}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315072,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[21,22,21,22,21,22],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[21,22,21,22,21,22],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0073","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[22,23,22,23,22,23],\"k\":3}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 3, 是, 是, chromatic_number_at_most_3, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315073,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[22,23,22,23,22,23],"k":3,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[22,23,22,23,22,23],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0074","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[23,24,23,24,23,24],\"k\":4}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 4, 是, 是, chromatic_number_at_most_4, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315074,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[23,24,23,24,23,24],"k":4,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[23,24,23,24,23,24],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0075","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[24,25,24,25,24,25],\"k\":2}。請依 [used_count,k,proper,certificate,conclusion,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)","explanation_zh":"Wave4收束核對=(2, 2, 是, 是, chromatic_number_at_most_2, upper_bound_not_exact_optimum)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-upper-bound","seed":202610315075,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[24,25,24,25,24,25],"k":2,"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-upper-bound","semantic_template":"qf-mod31-coloring-upper-bound/{\"colors\":[24,25,24,25,24,25],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"k\":2,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0076","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[10,40]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [10, 40, 10, 40, 10, 40], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [10, 40, 10, 40, 10, 40], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315076,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[10,40],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[10,40]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0077","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[11,41]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [11, 41, 11, 41, 11, 41], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [11, 41, 11, 41, 11, 41], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315077,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[11,41],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[11,41]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0078","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[12,42]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [12, 42, 12, 42, 12, 42], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [12, 42, 12, 42, 12, 42], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315078,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[12,42],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[12,42]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0079","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[13,43]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [13, 43, 13, 43, 13, 43], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [13, 43, 13, 43, 13, 43], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315079,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[13,43],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[13,43]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0080","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[14,44]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [14, 44, 14, 44, 14, 44], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [14, 44, 14, 44, 14, 44], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315080,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[14,44],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[14,44]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0081","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[15,45]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [15, 45, 15, 45, 15, 45], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [15, 45, 15, 45, 15, 45], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315081,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[15,45],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[15,45]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0082","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[16,46]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [16, 46, 16, 46, 16, 46], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [16, 46, 16, 46, 16, 46], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315082,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[16,46],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[16,46]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0083","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[17,47]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [17, 47, 17, 47, 17, 47], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [17, 47, 17, 47, 17, 47], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315083,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[17,47],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[17,47]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0084","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[18,48]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [18, 48, 18, 48, 18, 48], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [18, 48, 18, 48, 18, 48], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315084,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[18,48],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[18,48]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0085","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[19,49]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [19, 49, 19, 49, 19, 49], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [19, 49, 19, 49, 19, 49], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315085,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[19,49],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[19,49]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0086","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[20,50]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [20, 50, 20, 50, 20, 50], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [20, 50, 20, 50, 20, 50], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315086,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[20,50],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[20,50]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0087","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[21,51]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [21, 51, 21, 51, 21, 51], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [21, 51, 21, 51, 21, 51], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315087,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[21,51],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[21,51]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0088","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[22,52]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [22, 52, 22, 52, 22, 52], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [22, 52, 22, 52, 22, 52], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315088,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[22,52],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[22,52]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0089","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[23,53]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [23, 53, 23, 53, 23, 53], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [23, 53, 23, 53, 23, 53], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315089,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[23,53],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[23,53]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0090","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[24,54]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [24, 54, 24, 54, 24, 54], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [24, 54, 24, 54, 24, 54], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315090,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[24,54],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[24,54]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0091","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[25,55]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [25, 55, 25, 55, 25, 55], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [25, 55, 25, 55, 25, 55], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315091,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[25,55],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[25,55]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0092","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[26,56]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [26, 56, 26, 56, 26, 56], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [26, 56, 26, 56, 26, 56], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315092,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[26,56],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[26,56]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0093","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[27,57]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [27, 57, 27, 57, 27, 57], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [27, 57, 27, 57, 27, 57], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315093,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[27,57],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[27,57]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0094","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[28,58]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [28, 58, 28, 58, 28, 58], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [28, 58, 28, 58, 28, 58], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315094,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[28,58],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[28,58]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0095","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[29,59]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [29, 59, 29, 59, 29, 59], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [29, 59, 29, 59, 29, 59], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315095,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[29,59],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[29,59]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0096","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[30,60]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [30, 60, 30, 60, 30, 60], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [30, 60, 30, 60, 30, 60], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315096,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[30,60],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[30,60]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0097","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[31,61]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [31, 61, 31, 61, 31, 61], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [31, 61, 31, 61, 31, 61], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315097,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[31,61],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[31,61]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0098","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[32,62]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [32, 62, 32, 62, 32, 62], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [32, 62, 32, 62, 32, 62], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315098,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[32,62],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[32,62]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0099","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[33,63]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [33, 63, 33, 63, 33, 63], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [33, 63, 33, 63, 33, 63], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315099,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[33,63],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[33,63]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261031-r05-graph-coloring-0100","batch_id":"batch-20261014-advance-20261031-r05-graph-coloring-foundations","language":"zh-Hant","source_mko_ids":["mko-graph-coloring"],"title_zh":"圖著色","stem_zh":"完整有限無向簡單圖與每頂點顏色。操作=coloring-relabel-invariance；資料={\"n\":6,\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"relabel\":[34,64]}。請依 [colors,recolored,proper_before,proper_after,same_conflicts,boundary] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [34, 64, 34, 64, 34, 64], 是, 是, 是, color_names_do_not_change_vertex_partition)","explanation_zh":"Wave4收束核對=([0, 1, 0, 1, 0, 1], [34, 64, 34, 64, 34, 64], 是, 是, 是, color_names_do_not_change_vertex_partition)；依完整明示資料重算；只要求相鄰端點異色；非相鄰點可同色；一個proper coloring只給上界且greedy不自動最優。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["圖著色"]},"variant":{"family_id":"qf-mod31-coloring-relabel-invariance","seed":202610315100,"parameters":{"n":6,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"relabel":[34,64],"domain":"Complete_finite_simple_graph_vertex_coloring","operation":"coloring-relabel-invariance","semantic_template":"qf-mod31-coloring-relabel-invariance/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]],\"n\":6,\"relabel\":[34,64]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}