{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0001","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061001,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0002","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061002,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,2,0,1,2],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0003","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061003,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0004","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061004,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,2,1,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0005","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061005,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0006","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 2, 0, 1, 2], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061006,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"colors":[0,1,2,0,1,2],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0007","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061007,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0008","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,3,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061008,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"colors":[0,1,2,3,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0009","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061009,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0010","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061010,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,2,0,2,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0011","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061011,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0012","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061012,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,2,1,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0013","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061013,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0014","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061014,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,2,0,2,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0015","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061015,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0016","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,3,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061016,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,2,3,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0017","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061017,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0018","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,2,0,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061018,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,2,0,2,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0019","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061019,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0020","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,2,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 2, 1, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061020,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,2,1,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0021","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061021,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0022","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,2,0,2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 2, 0, 2, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061022,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,2,0,2,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0023","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 2, 0, 1], [0, 1, 2], 3, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061023,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0024","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,2,3,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 2, 3, 0, 1], [0, 1, 2, 3], 4, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061024,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,2,3,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0025","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-proper-audit；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)","explanation_zh":"Wave5封閉層核對=([0, 1, 0, 1, 0, 1], [0, 1], 2, 是, proper_coloring_only)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-proper-audit","seed":202611061025,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-proper-audit","semantic_template":"qf-mod37-chromatic-proper-audit/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0026","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(3, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061026,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0027","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,1,2],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061027,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,2,0,1,2],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0028","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(3, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061028,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0029","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,1,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061029,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,2,1,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0030","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061030,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0031","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,1,2],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061031,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"colors":[0,1,2,0,1,2],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0032","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061032,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0033","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,3,0,1],\"k\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(5, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061033,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"colors":[0,1,2,3,0,1],"k":5,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":5,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0034","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(3, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061034,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0035","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,2,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061035,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,2,0,2,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0036","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(3, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061036,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0037","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,1,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061037,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,2,1,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0038","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061038,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0039","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,2,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061039,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,2,0,2,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0040","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061040,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0041","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,3,0,1],\"k\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(5, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061041,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,2,3,0,1],"k":5,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":5,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0042","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(3, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061042,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0043","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,2,0,2,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061043,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,2,0,2,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0044","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(3, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061044,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0045","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,2,1,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061045,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,2,1,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0046","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061046,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0047","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,2,0,2,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061047,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,2,0,2,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0048","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(4, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061048,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0049","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,2,3,0,1],\"k\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(5, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061049,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,2,3,0,1],"k":5,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":5,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0050","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-upper-bound；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 是, 是, chi_at_most_k_not_equality)","explanation_zh":"Wave5封閉層核對=(3, 是, 是, 是, chi_at_most_k_not_equality)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-upper-bound","seed":202611061050,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-upper-bound","semantic_template":"qf-mod37-chromatic-upper-bound/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0051","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061051,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0052","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0, 1, 2], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 2, 0, 1, 2], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061052,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0053","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061053,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0054","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 2, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061054,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0055","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 0, 2, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 0, 2, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061055,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0056","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0, 1, 2], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 2, 0, 1, 2], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061056,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0057","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 0, 2, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 0, 2, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061057,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0058","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(4, [0, 1, 2, 3, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061058,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0059","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061059,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0060","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0, 2, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 2, 0, 2, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061060,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0061","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061061,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0062","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 2, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061062,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0063","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 0, 2, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 0, 2, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061063,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0064","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0, 2, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 2, 0, 2, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061064,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0065","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 0, 2, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 0, 2, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061065,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0066","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(4, [0, 1, 2, 3, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061066,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0067","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061067,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0068","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0, 2, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 2, 0, 2, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061068,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0069","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061069,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0070","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 2, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061070,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0071","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 0, 2, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 0, 2, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061071,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0072","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0, 2, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 2, 0, 2, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061072,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0073","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [0, 1, 0, 2, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(3, [0, 1, 0, 2, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061073,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0074","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(4, [0, 1, 2, 3, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061074,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0075","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-exact-number；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)","explanation_zh":"Wave5封閉層核對=(2, [0, 1, 0, 1, 0, 1], 是, exhaustive_lower_and_constructive_upper_certificates)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-exact-number","seed":202611061075,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-exact-number","semantic_template":"qf-mod37-chromatic-exact-number/{\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0076","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061076,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0077","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,1,2],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061077,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,2,0,1,2],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0078","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061078,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0079","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,1,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061079,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],"colors":[0,1,2,1,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0080","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061080,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0081","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,1,2],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061081,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"colors":[0,1,2,0,1,2],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,1,2],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0082","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061082,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0083","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,3,0,1],\"k\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 4, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(5, 是, 4, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061083,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],"colors":[0,1,2,3,0,1],"k":5,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[3,4],[4,5]],\"k\":5,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0084","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061084,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0085","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,2,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061085,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,2,0,2,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0086","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061086,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0087","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,1,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061087,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,2,1,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0088","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061088,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0089","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,0,2,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061089,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,2,0,2,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0090","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061090,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0091","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"colors\":[0,1,2,3,0,1],\"k\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 4, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(5, 是, 4, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061091,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],"colors":[0,1,2,3,0,1],"k":5,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[1,4],[2,3],[3,4],[4,5]],\"k\":5,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0092","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061092,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0093","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,2,0,2,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061093,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,2,0,2,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0094","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061094,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0095","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,2,1,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061095,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,2,1,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,1,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0096","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061096,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0097","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,2,0,2,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061097,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,2,0,2,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,0,2,1],\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0098","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,2,0,1],\"k\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(4, 是, 3, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061098,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,2,0,1],"k":4,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,2,0,1],\"edges\":[[0,1],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":4,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0099","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,2,3,0,1],\"k\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 是, 4, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(5, 是, 4, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061099,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,2,3,0,1],"k":5,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,2,3,0,1],\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,3],[2,5],[3,4],[4,5]],\"k\":5,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r01-chromatic-number-0100","batch_id":"batch-20261014-advance-20261106-r01-chromatic-number-foundations","language":"zh-Hant","source_mko_ids":["mko-chromatic-number"],"title_zh":"色數","stem_zh":"完整有限簡單圖、proper vertex coloring與exact最小性證書。操作=chromatic-minimum-boundary；資料={\"n\":6,\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],\"colors\":[0,1,0,1,0,1],\"k\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)","explanation_zh":"Wave5封閉層核對=(3, 是, 2, 否, k_colorable_means_at_most_k_and_greedy_not_optimality_proof)；依完整明示資料獨立重算；proper k-coloring只證chi小於等於k；等號還須排除k減1 coloring且greedy不證最優；空頂點圖chi等於0、非空無邊圖chi等於1；k-colorable表示至多k色。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["色數"]},"variant":{"family_id":"qf-mod37-chromatic-minimum-boundary","seed":202611061100,"parameters":{"n":6,"edges":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],"colors":[0,1,0,1,0,1],"k":3,"domain":"Complete_finite_simple_graph_coloring","operation":"chromatic-minimum-boundary","semantic_template":"qf-mod37-chromatic-minimum-boundary/{\"colors\":[0,1,0,1,0,1],\"edges\":[[0,1],[0,5],[1,2],[1,4],[2,3],[2,5],[3,4],[4,5]],\"k\":3,\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}