{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0001","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(0)=∅，無向邊列項=[]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(0, [], [], 是, 否, 否, 否)","explanation_zh":"離散結構核對=(0, [], [], 是, 否, 否, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176001,"parameters":{"n":0,"edges":[],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0002","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(1)=[0]，無向邊列項=[]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(1, [], [[0]], 是, 否, 是, 是)","explanation_zh":"離散結構核對=(1, [], [[0]], 是, 否, 是, 是)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176002,"parameters":{"n":1,"edges":[],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0003","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[1, 2], [1, 3], [2, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(4, [[1, 2], [1, 3], [2, 3]], [[0], [1, 2, 3]], 否, 是, 是, 否)","explanation_zh":"離散結構核對=(4, [[1, 2], [1, 3], [2, 3]], [[0], [1, 2, 3]], 否, 是, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176003,"parameters":{"n":4,"edges":[[1,2],[1,3],[2,3]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[1,2],[1,3],[2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0004","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[0, 1], [2, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(4, [[0, 1], [2, 3]], [[0, 1], [2, 3]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(4, [[0, 1], [2, 3]], [[0, 1], [2, 3]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176004,"parameters":{"n":4,"edges":[[0,1],[2,3]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0005","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[0, 1], [1, 2], [2, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(4, [[0, 1], [1, 2], [2, 3]], [[0, 1, 2, 3]], 是, 否, 是, 是)","explanation_zh":"離散結構核對=(4, [[0, 1], [1, 2], [2, 3]], [[0, 1, 2, 3]], 是, 否, 是, 是)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176005,"parameters":{"n":4,"edges":[[0,1],[1,2],[2,3]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[1,2],[2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0006","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[0, 1], [0, 2], [0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(4, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3]], 是, 否, 是, 是)","explanation_zh":"離散結構核對=(4, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3]], 是, 否, 是, 是)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176006,"parameters":{"n":4,"edges":[[0,1],[0,2],[0,3]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0007","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 1]], [[0, 1], [2], [3], [4]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 1]], [[0, 1], [2], [3], [4]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176007,"parameters":{"n":5,"edges":[[0,1]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0008","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 2]], [[0, 2], [1], [3], [4]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 2]], [[0, 2], [1], [3], [4]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176008,"parameters":{"n":5,"edges":[[0,2]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0009","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2]], [[0, 1, 2], [3], [4]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2]], [[0, 1, 2], [3], [4]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176009,"parameters":{"n":5,"edges":[[0,1],[0,2]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0010","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 3]], [[0, 3], [1], [2], [4]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 3]], [[0, 3], [1], [2], [4]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176010,"parameters":{"n":5,"edges":[[0,3]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0011","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 3]], [[0, 1, 3], [2], [4]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 3]], [[0, 1, 3], [2], [4]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176011,"parameters":{"n":5,"edges":[[0,1],[0,3]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0012","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 3]], [[0, 2, 3], [1], [4]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 2], [0, 3]], [[0, 2, 3], [1], [4]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176012,"parameters":{"n":5,"edges":[[0,2],[0,3]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,2],[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0013","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 3]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3], [4]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3], [4]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176013,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,3]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0014","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 4]], [[0, 4], [1], [2], [3]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 4]], [[0, 4], [1], [2], [3]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176014,"parameters":{"n":5,"edges":[[0,4]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0015","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 4]], [[0, 1, 4], [2], [3]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 4]], [[0, 1, 4], [2], [3]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176015,"parameters":{"n":5,"edges":[[0,1],[0,4]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0016","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 4]], [[0, 2, 4], [1], [3]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 2], [0, 4]], [[0, 2, 4], [1], [3]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176016,"parameters":{"n":5,"edges":[[0,2],[0,4]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,2],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0017","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 4]], [[0, 1, 2, 4], [3]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2], [0, 4]], [[0, 1, 2, 4], [3]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176017,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,4]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,2],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0018","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 3], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 3], [0, 4]], [[0, 3, 4], [1], [2]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 3], [0, 4]], [[0, 3, 4], [1], [2]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176018,"parameters":{"n":5,"edges":[[0,3],[0,4]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0019","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 3], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 3], [0, 4]], [[0, 1, 3, 4], [2]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 3], [0, 4]], [[0, 1, 3, 4], [2]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176019,"parameters":{"n":5,"edges":[[0,1],[0,3],[0,4]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0020","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [0, 3], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 3], [0, 4]], [[0, 2, 3, 4], [1]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 2], [0, 3], [0, 4]], [[0, 2, 3, 4], [1]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176020,"parameters":{"n":5,"edges":[[0,2],[0,3],[0,4]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,2],[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0021","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [0, 3], [0, 4]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[0, 1, 2, 3, 4]], 是, 否, 是, 是)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[0, 1, 2, 3, 4]], 是, 否, 是, 是)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176021,"parameters":{"n":5,"edges":[[0,1],[0,2],[0,3],[0,4]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0022","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[1, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[1, 2]], [[0], [1, 2], [3], [4]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[1, 2]], [[0], [1, 2], [3], [4]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176022,"parameters":{"n":5,"edges":[[1,2]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0023","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2]], [[0, 1, 2], [3], [4]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 1], [1, 2]], [[0, 1, 2], [3], [4]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176023,"parameters":{"n":5,"edges":[[0,1],[1,2]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0024","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 2], [1, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 2], [1, 2]], [[0, 1, 2], [3], [4]], 否, 否, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 2], [1, 2]], [[0, 1, 2], [3], [4]], 否, 否, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176024,"parameters":{"n":5,"edges":[[0,2],[1,2]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,2],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0025","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"全部分量與非空樹資格","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(5)=[0, 1, 2, 3, 4]，無向邊列項=[[0, 1], [0, 2], [1, 2]]；各邊兩端皆合法且相異。答 (n, 正規邊集, 依最小頂點排序的連通分量, 任兩頂點可連接資格, 有環資格, 非空資格, 樹資格)。空圖的任兩頂點條件為真，但空圖非樹。","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2]], [[0, 1, 2], [3], [4]], 否, 是, 是, 否)","explanation_zh":"離散結構核對=(5, [[0, 1], [0, 2], [1, 2]], [[0, 1, 2], [3], [4]], 否, 是, 是, 否)；DFS走遍所有分量並偵測非父邊；非空、單分量、無環共同決定樹。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全部分量與非空樹資格"]},"variant":{"family_id":"qf-density17-tree-component-qualification","seed":202610176025,"parameters":{"n":5,"edges":[[0,1],[0,2],[1,2]],"domain":"Fin_tree_qualification","operation":"tree-component-qualification","semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,2],[1,2]],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0026","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(0)=∅，無向邊列項=[]；各邊兩端皆合法且相異。start=0、end=-1。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(0, [], 否, [否, 否], 不成樹, 不求值)","explanation_zh":"離散結構核對=(0, [], 否, [否, 否], 不成樹, 不求值)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176026,"parameters":{"n":0,"edges":[],"start":0,"end":-1,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[],\"end\":\"untyped\",\"n\":0,\"start\":\"untyped\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0027","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(1)=[0]，無向邊列項=[]；各邊兩端皆合法且相異。start=0、end=0。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(1, [], 是, [是, 是], [0], 0)","explanation_zh":"離散結構核對=(1, [], 是, [是, 是], [0], 0)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176027,"parameters":{"n":1,"edges":[],"start":0,"end":0,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[],\"end\":0,\"n\":1,\"start\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0028","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[1, 2], [1, 3], [2, 3]]；各邊兩端皆合法且相異。start=0、end=2。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(4, [[1, 2], [1, 3], [2, 3]], 否, [是, 是], 不成樹, 不求值)","explanation_zh":"離散結構核對=(4, [[1, 2], [1, 3], [2, 3]], 否, [是, 是], 不成樹, 不求值)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176028,"parameters":{"n":4,"edges":[[1,2],[1,3],[2,3]],"start":0,"end":2,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[1,2],[1,3],[2,3]],\"end\":2,\"n\":4,\"start\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0029","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(3)=[0, 1, 2]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。start=-1、end=2。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], 是, [否, 是], 不合頂點, 不求值)","explanation_zh":"離散結構核對=(3, [[0, 1], [1, 2]], 是, [否, 是], 不合頂點, 不求值)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176029,"parameters":{"n":3,"edges":[[0,1],[1,2]],"start":-1,"end":2,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[1,2]],\"end\":2,\"n\":3,\"start\":\"untyped\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0030","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(3)=[0, 1, 2]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。start=1、end=1。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], 是, [是, 是], [1], 0)","explanation_zh":"離散結構核對=(3, [[0, 1], [1, 2]], 是, [是, 是], [1], 0)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176030,"parameters":{"n":3,"edges":[[0,1],[1,2]],"start":1,"end":1,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[1,2]],\"end\":1,\"n\":3,\"start\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0031","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[0, 1], [0, 2], [0, 3]]；各邊兩端皆合法且相異。start=1、end=3。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(4, [[0, 1], [0, 2], [0, 3]], 是, [是, 是], [1, 0, 3], 2)","explanation_zh":"離散結構核對=(4, [[0, 1], [0, 2], [0, 3]], 是, [是, 是], [1, 0, 3], 2)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176031,"parameters":{"n":4,"edges":[[0,1],[0,2],[0,3]],"start":1,"end":3,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,2],[0,3]],\"end\":3,\"n\":4,\"start\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0032","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]；各邊兩端皆合法且相異。start=0、end=3。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], 是, [是, 是], [0, 3], 1)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], 是, [是, 是], [0, 3], 1)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176032,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,4],[0,5]],"start":0,"end":3,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5]],\"end\":3,\"n\":6,\"start\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0033","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]]；各邊兩端皆合法且相異。start=1、end=4。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], 是, [是, 是], [1, 0, 4], 2)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], 是, [是, 是], [1, 0, 4], 2)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176033,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,4],[0,5],[1,2]],"start":1,"end":4,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,3],[0,4],[0,5],[1,2]],\"end\":4,\"n\":6,\"start\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0034","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]]；各邊兩端皆合法且相異。start=2、end=5。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], 是, [是, 是], [2, 0, 5], 2)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], 是, [是, 是], [2, 0, 5], 2)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176034,"parameters":{"n":6,"edges":[[0,2],[0,3],[0,4],[0,5],[1,2]],"start":2,"end":5,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2]],\"end\":5,\"n\":6,\"start\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0035","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]]；各邊兩端皆合法且相異。start=3、end=0。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], 是, [是, 是], [3, 0], 1)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], 是, [是, 是], [3, 0], 1)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176035,"parameters":{"n":6,"edges":[[0,2],[0,3],[0,4],[0,5],[1,3]],"start":3,"end":0,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,3]],\"end\":0,\"n\":6,\"start\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0036","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]]；各邊兩端皆合法且相異。start=4、end=1。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], 是, [是, 是], [4, 1], 1)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], 是, [是, 是], [4, 1], 1)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176036,"parameters":{"n":6,"edges":[[0,2],[0,3],[0,4],[0,5],[1,4]],"start":4,"end":1,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,4]],\"end\":1,\"n\":6,\"start\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0037","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]；各邊兩端皆合法且相異。start=5、end=2。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], 是, [是, 是], [5, 0, 2], 2)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], 是, [是, 是], [5, 0, 2], 2)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176037,"parameters":{"n":6,"edges":[[0,2],[0,3],[0,4],[0,5],[1,5]],"start":5,"end":2,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,5]],\"end\":2,\"n\":6,\"start\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0038","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]]；各邊兩端皆合法且相異。start=0、end=3。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], 是, [是, 是], [0, 1, 3], 2)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], 是, [是, 是], [0, 1, 3], 2)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176038,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,4],[0,5],[1,3]],"start":0,"end":3,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3]],\"end\":3,\"n\":6,\"start\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0039","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]]；各邊兩端皆合法且相異。start=1、end=4。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], 是, [是, 是], [1, 0, 4], 2)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], 是, [是, 是], [1, 0, 4], 2)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176039,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,2],[1,3]],"start":1,"end":4,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,4],[0,5],[1,2],[1,3]],\"end\":4,\"n\":6,\"start\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0040","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]；各邊兩端皆合法且相異。start=2、end=5。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], 是, [是, 是], [2, 1, 0, 5], 3)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], 是, [是, 是], [2, 1, 0, 5], 3)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176040,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,2],[2,3]],"start":2,"end":5,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,4],[0,5],[1,2],[2,3]],\"end\":5,\"n\":6,\"start\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0041","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]]；各邊兩端皆合法且相異。start=3、end=0。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], 是, [是, 是], [3, 1, 0], 2)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], 是, [是, 是], [3, 1, 0], 2)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176041,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,3],[2,3]],"start":3,"end":0,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,3]],\"end\":0,\"n\":6,\"start\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0042","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]]；各邊兩端皆合法且相異。start=4、end=1。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], 是, [是, 是], [4, 0, 1], 2)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], 是, [是, 是], [4, 0, 1], 2)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176042,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,3],[2,4]],"start":4,"end":1,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,4]],\"end\":1,\"n\":6,\"start\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0043","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]]；各邊兩端皆合法且相異。start=5、end=2。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], 是, [是, 是], [5, 2], 1)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], 是, [是, 是], [5, 2], 1)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176043,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,3],[2,5]],"start":5,"end":2,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,5]],\"end\":2,\"n\":6,\"start\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0044","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]]；各邊兩端皆合法且相異。start=0、end=3。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], 是, [是, 是], [0, 2, 3], 2)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], 是, [是, 是], [0, 2, 3], 2)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176044,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,4],[0,5],[2,3]],"start":0,"end":3,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,2],[0,4],[0,5],[2,3]],\"end\":3,\"n\":6,\"start\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0045","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]]；各邊兩端皆合法且相異。start=1、end=4。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], 是, [是, 是], [1, 2, 0, 4], 3)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], 是, [是, 是], [1, 2, 0, 4], 3)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176045,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,2],[1,3]],"start":1,"end":4,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,4],[0,5],[1,2],[1,3]],\"end\":4,\"n\":6,\"start\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0046","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]]；各邊兩端皆合法且相異。start=2、end=5。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], 是, [是, 是], [2, 0, 5], 2)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], 是, [是, 是], [2, 0, 5], 2)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176046,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,2],[2,3]],"start":2,"end":5,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,4],[0,5],[1,2],[2,3]],\"end\":5,\"n\":6,\"start\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0047","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]]；各邊兩端皆合法且相異。start=3、end=0。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], 是, [是, 是], [3, 2, 0], 2)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], 是, [是, 是], [3, 2, 0], 2)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176047,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,3],[2,3]],"start":3,"end":0,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,4],[0,5],[1,3],[2,3]],\"end\":0,\"n\":6,\"start\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0048","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]]；各邊兩端皆合法且相異。start=4、end=1。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], 是, [是, 是], [4, 1], 1)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], 是, [是, 是], [4, 1], 1)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176048,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,4],[2,3]],"start":4,"end":1,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,4],[0,5],[1,4],[2,3]],\"end\":1,\"n\":6,\"start\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0049","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]]；各邊兩端皆合法且相異。start=5、end=2。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], 是, [是, 是], [5, 0, 2], 2)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], 是, [是, 是], [5, 0, 2], 2)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176049,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,5],[2,3]],"start":5,"end":2,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,4],[0,5],[1,5],[2,3]],\"end\":2,\"n\":6,\"start\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0050","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"樹與端點門檻後的唯一道路","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 3], [0, 4], [0, 5], [2, 3]]；各邊兩端皆合法且相異。start=0、end=3。答 (n, 正規邊集, 原圖樹資格, 兩端點資格, 道路或不成樹或不合頂點, 邊數或不求值)。原圖非樹優先答不成樹；合法自身道路僅列[start]。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 3], [0, 4], [0, 5], [2, 3]], 是, [是, 是], [0, 3], 1)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 3], [0, 4], [0, 5], [2, 3]], 是, [是, 是], [0, 3], 1)；先驗整張原圖與端點，再用不重複頂點的DFS尋路。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["樹與端點門檻後的唯一道路"]},"variant":{"family_id":"qf-density17-tree-unique-path-gate","seed":202610176050,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,4],[0,5],[2,3]],"start":0,"end":3,"domain":"Fin_tree_qualification","operation":"tree-unique-path-gate","semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,3],[0,4],[0,5],[2,3]],\"end\":3,\"n\":6,\"start\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0051","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(0)=∅，無向邊列項=[]；各邊兩端皆合法且相異。候選刪除at=0。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(0, [], 否, 否, 不合頂點, 否, 不適用, 不適用, 不求值)","explanation_zh":"離散結構核對=(0, [], 否, 否, 不合頂點, 否, 不適用, 不適用, 不求值)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176051,"parameters":{"n":0,"edges":[],"at":0,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":\"untyped\",\"edges\":[],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0052","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(1)=[0]，無向邊列項=[]；各邊兩端皆合法且相異。候選刪除at=0。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(1, [], 是, 是, 0, 否, 不適用, 不適用, 不求值)","explanation_zh":"離散結構核對=(1, [], 是, 是, 0, 否, 不適用, 不適用, 不求值)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176052,"parameters":{"n":1,"edges":[],"at":0,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":0,\"edges\":[],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0053","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(2)=[0, 1]，無向邊列項=[[0, 1]]；各邊兩端皆合法且相異。候選刪除at=1。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(2, [[0, 1]], 是, 是, 1, 是, [0], [], 是)","explanation_zh":"離散結構核對=(2, [[0, 1]], 是, 是, 1, 是, [0], [], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176053,"parameters":{"n":2,"edges":[[0,1]],"at":1,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":1,\"edges\":[[0,1]],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0054","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[1, 2], [1, 3], [2, 3]]；各邊兩端皆合法且相異。候選刪除at=0。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(4, [[1, 2], [1, 3], [2, 3]], 否, 是, 0, 否, 不適用, 不適用, 不求值)","explanation_zh":"離散結構核對=(4, [[1, 2], [1, 3], [2, 3]], 否, 是, 0, 否, 不適用, 不適用, 不求值)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176054,"parameters":{"n":4,"edges":[[1,2],[1,3],[2,3]],"at":0,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":0,\"edges\":[[1,2],[1,3],[2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0055","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(3)=[0, 1, 2]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。候選刪除at=1。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], 是, 是, 2, 否, 不適用, 不適用, 不求值)","explanation_zh":"離散結構核對=(3, [[0, 1], [1, 2]], 是, 是, 2, 否, 不適用, 不適用, 不求值)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176055,"parameters":{"n":3,"edges":[[0,1],[1,2]],"at":1,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":1,\"edges\":[[0,1],[1,2]],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0056","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(3)=[0, 1, 2]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。候選刪除at=3。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], 是, 否, 不合頂點, 否, 不適用, 不適用, 不求值)","explanation_zh":"離散結構核對=(3, [[0, 1], [1, 2]], 是, 否, 不合頂點, 否, 不適用, 不適用, 不求值)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176056,"parameters":{"n":3,"edges":[[0,1],[1,2]],"at":3,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":\"untyped\",\"edges\":[[0,1],[1,2]],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0057","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[0, 1], [0, 2], [0, 3]]；各邊兩端皆合法且相異。候選刪除at=2。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(4, [[0, 1], [0, 2], [0, 3]], 是, 是, 1, 是, [0, 1, 3], [[0, 1], [0, 3]], 是)","explanation_zh":"離散結構核對=(4, [[0, 1], [0, 2], [0, 3]], 是, 是, 1, 是, [0, 1, 3], [[0, 1], [0, 3]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176057,"parameters":{"n":4,"edges":[[0,1],[0,2],[0,3]],"at":2,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":2,\"edges\":[[0,1],[0,2],[0,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0058","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 3], [0, 4]], 是)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 3], [0, 4]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176058,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,4],[0,5]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0059","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 3], [0, 4], [1, 2]], 是)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 3], [0, 4], [1, 2]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176059,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,4],[0,5],[1,2]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,3],[0,4],[0,5],[1,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0060","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 2]], 是)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 2]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176060,"parameters":{"n":6,"edges":[[0,2],[0,3],[0,4],[0,5],[1,2]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0061","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 3]], 是)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 3]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176061,"parameters":{"n":6,"edges":[[0,2],[0,3],[0,4],[0,5],[1,3]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0062","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 4]], 是)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 4]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176062,"parameters":{"n":6,"edges":[[0,2],[0,3],[0,4],[0,5],[1,4]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0063","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], 是, 是, 2, 否, 不適用, 不適用, 不求值)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], 是, 是, 2, 否, 不適用, 不適用, 不求值)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176063,"parameters":{"n":6,"edges":[[0,2],[0,3],[0,4],[0,5],[1,5]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0064","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [1, 3]], 是)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [1, 3]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176064,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,4],[0,5],[1,3]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0065","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [1, 3]], 是)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [1, 3]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176065,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,2],[1,3]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,4],[0,5],[1,2],[1,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0066","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [2, 3]], 是)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [2, 3]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176066,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,2],[2,3]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,4],[0,5],[1,2],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0067","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 3]], 是)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 3]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176067,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,3],[2,3]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0068","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 4]], 是)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 4]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176068,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,3],[2,4]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0069","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], 是, 是, 2, 否, 不適用, 不適用, 不求值)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], 是, 是, 2, 否, 不適用, 不適用, 不求值)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176069,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,3],[2,5]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0070","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [2, 3]], 是)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [2, 3]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176070,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,4],[0,5],[2,3]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,2],[0,4],[0,5],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0071","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [1, 3]], 是)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [1, 3]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176071,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,2],[1,3]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,4],[0,5],[1,2],[1,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0072","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [2, 3]], 是)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [2, 3]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176072,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,2],[2,3]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,4],[0,5],[1,2],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0073","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 3], [2, 3]], 是)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 3], [2, 3]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176073,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,3],[2,3]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,4],[0,5],[1,3],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0074","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 4], [2, 3]], 是)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], 是, 是, 1, 是, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 4], [2, 3]], 是)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176074,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,4],[2,3]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,4],[0,5],[1,4],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0075","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"度1葉刪除的原標籤","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]]；各邊兩端皆合法且相異。候選刪除at=5。答 (n, 正規邊集, 原樹資格, at合頂點資格, 原度或不合頂點, 合法樹葉資格, 剩餘原標籤或不適用, 剩餘邊或不適用, 剩餘樹資格或不求值)。必須原圖為樹且度恰1才刪。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], 是, 是, 2, 否, 不適用, 不適用, 不求值)","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], 是, 是, 2, 否, 不適用, 不適用, 不求值)；先核合法度1葉；刪該頂點及相接邊後，以原標籤DFS核剩餘圖。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["度1葉刪除的原標籤"]},"variant":{"family_id":"qf-density17-tree-leaf-deletion","seed":202610176075,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,5],[2,3]],"at":5,"domain":"Fin_tree_qualification","operation":"tree-leaf-deletion","semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,4],[0,5],[1,5],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0076","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(0)=∅，無向邊列項=[]；各邊兩端皆合法且相異。候選新增邊=[0, 1]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(0, [], 否, [否, 否], 是, 否, 否, 不適用, 不適用)","explanation_zh":"離散結構核對=(0, [], 否, [否, 否], 是, 否, 否, 不適用, 不適用)；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176076,"parameters":{"n":0,"edges":[],"edge":[0,1],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[0,1],\"edges\":[],\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0077","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(1)=[0]，無向邊列項=[]；各邊兩端皆合法且相異。候選新增邊=[0, 0]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(1, [], 是, [是, 是], 否, 否, 否, 不適用, 不適用)","explanation_zh":"離散結構核對=(1, [], 是, [是, 是], 否, 否, 否, 不適用, 不適用)；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176077,"parameters":{"n":1,"edges":[],"edge":[0,0],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[0,0],\"edges\":[],\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0078","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(2)=[0, 1]，無向邊列項=[[0, 1]]；各邊兩端皆合法且相異。候選新增邊=[0, 1]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(2, [[0, 1]], 是, [是, 是], 是, 是, 否, 不適用, 不適用)","explanation_zh":"離散結構核對=(2, [[0, 1]], 是, [是, 是], 是, 是, 否, 不適用, 不適用)；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176078,"parameters":{"n":2,"edges":[[0,1]],"edge":[0,1],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[0,1],\"edges\":[[0,1]],\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0079","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(3)=[0, 1, 2]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。候選新增邊=[2, 0]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], 是, [是, 是], 是, 否, 是, [0, 1, 2], [0, 1, 2, 0])","explanation_zh":"離散結構核對=(3, [[0, 1], [1, 2]], 是, [是, 是], 是, 否, 是, [0, 1, 2], [0, 1, 2, 0])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176079,"parameters":{"n":3,"edges":[[0,1],[1,2]],"edge":[2,0],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[0,2],\"edges\":[[0,1],[1,2]],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0080","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(3)=[0, 1, 2]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。候選新增邊=[-1, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], 是, [否, 是], 是, 否, 否, 不適用, 不適用)","explanation_zh":"離散結構核對=(3, [[0, 1], [1, 2]], 是, [否, 是], 是, 否, 否, 不適用, 不適用)；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176080,"parameters":{"n":3,"edges":[[0,1],[1,2]],"edge":[-1,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[-1,2],\"edges\":[[0,1],[1,2]],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0081","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(3)=[0, 1, 2]，無向邊列項=[[0, 1], [1, 2]]；各邊兩端皆合法且相異。候選新增邊=[1, 1]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], 是, [是, 是], 否, 否, 否, 不適用, 不適用)","explanation_zh":"離散結構核對=(3, [[0, 1], [1, 2]], 是, [是, 是], 否, 否, 否, 不適用, 不適用)；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176081,"parameters":{"n":3,"edges":[[0,1],[1,2]],"edge":[1,1],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[1,1],\"edges\":[[0,1],[1,2]],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0082","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(4)=[0, 1, 2, 3]，無向邊列項=[[1, 2], [1, 3], [2, 3]]；各邊兩端皆合法且相異。候選新增邊=[0, 1]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(4, [[1, 2], [1, 3], [2, 3]], 否, [是, 是], 是, 否, 否, 不適用, 不適用)","explanation_zh":"離散結構核對=(4, [[1, 2], [1, 3], [2, 3]], 否, [是, 是], 是, 否, 否, 不適用, 不適用)；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176082,"parameters":{"n":4,"edges":[[1,2],[1,3],[2,3]],"edge":[0,1],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[0,1],\"edges\":[[1,2],[1,3],[2,3]],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0083","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176083,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,3],[0,4],[0,5]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0084","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], 是, [是, 是], 是, 否, 是, [2, 1, 0, 5], [2, 1, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], 是, [是, 是], 是, 否, 是, [2, 1, 0, 5], [2, 1, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176084,"parameters":{"n":6,"edges":[[0,1],[0,3],[0,4],[0,5],[1,2]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,3],[0,4],[0,5],[1,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0085","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176085,"parameters":{"n":6,"edges":[[0,2],[0,3],[0,4],[0,5],[1,2]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0086","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176086,"parameters":{"n":6,"edges":[[0,2],[0,3],[0,4],[0,5],[1,3]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0087","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176087,"parameters":{"n":6,"edges":[[0,2],[0,3],[0,4],[0,5],[1,4]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0088","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176088,"parameters":{"n":6,"edges":[[0,2],[0,3],[0,4],[0,5],[1,5]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0089","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176089,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,4],[0,5],[1,3]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0090","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], 是, [是, 是], 是, 否, 是, [2, 1, 0, 5], [2, 1, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], 是, [是, 是], 是, 否, 是, [2, 1, 0, 5], [2, 1, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176090,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,2],[1,3]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,4],[0,5],[1,2],[1,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0091","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 1, 0, 5], [2, 1, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 1, 0, 5], [2, 1, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176091,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,2],[2,3]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,4],[0,5],[1,2],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0092","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 3, 1, 0, 5], [2, 3, 1, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 3, 1, 0, 5], [2, 3, 1, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176092,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,3],[2,3]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0093","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], 是, [是, 是], 是, 否, 是, [2, 4, 0, 5], [2, 4, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], 是, [是, 是], 是, 否, 是, [2, 4, 0, 5], [2, 4, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176093,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,3],[2,4]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,4]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0094","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], 是, [是, 是], 是, 是, 否, 不適用, 不適用)","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], 是, [是, 是], 是, 是, 否, 不適用, 不適用)；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176094,"parameters":{"n":6,"edges":[[0,1],[0,4],[0,5],[1,3],[2,5]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,5]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0095","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176095,"parameters":{"n":6,"edges":[[0,1],[0,2],[0,4],[0,5],[2,3]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,2],[0,4],[0,5],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0096","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176096,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,2],[1,3]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,4],[0,5],[1,2],[1,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0097","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176097,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,2],[2,3]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,4],[0,5],[1,2],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0098","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176098,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,3],[2,3]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,4],[0,5],[1,3],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0099","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176099,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,4],[2,3]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,4],[0,5],[1,4],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r06-tree-0100","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","language":"zh-Hant","source_mko_ids":["mko-tree"],"title_zh":"不存在邊的門檻與定向閉環","stem_zh":"簡單無向圖，Fin頂點從0起。V=Fin(6)=[0, 1, 2, 3, 4, 5]，無向邊列項=[[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]]；各邊兩端皆合法且相異。候選新增邊=[5, 2]。答 (n, 正規邊集, 原樹資格, 兩端合頂點資格, 端點相異資格, 原邊已存在資格, 合法新非邊資格, 較小的新增邊端點到較大的原道路或不適用, 閉回較小新增邊端點的環或不適用)。不是從環中全域最小頂點起列。","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])","explanation_zh":"離散結構核對=(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], 是, [是, 是], 是, 否, 是, [2, 0, 5], [2, 0, 5, 2])；原樹、合型、相異、原邊不存在全通過才尋原道路，最後加回較小新增邊端點。；有限精確核對不是一般形式證明；空圖非樹單點度0非葉；邊數不單獨判樹；刪除保留原標籤；加邊先核原樹與不存在邊。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["不存在邊的門檻與定向閉環"]},"variant":{"family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176100,"parameters":{"n":6,"edges":[[0,2],[0,4],[0,5],[1,5],[2,3]],"edge":[5,2],"domain":"Fin_tree_qualification","operation":"tree-add-nonedge-cycle","semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,4],[0,5],[1,5],[2,3]],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}