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{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0021","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的像生成集","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[0, 1, 2, 3, 4, 5, 6, 7]，S=[2, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。列像主濾子的生成集f(S)，遞增且去重。","answer":{"kind":"expression","value":"[2, 4]","explanation_zh":"主濾子像生成集=[2, 4]。只把S的每個元素送入合法總映射，再去重；空生成集的像仍空，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的像生成集"]},"variant":{"family_id":"qf-density11-tends-principal-image","seed":202610117021,"parameters":{"n":8,"m":8,"images":[0,1,2,3,4,5,6,7],"S":[2,4],"filter_model":"principal_generators","operation":"density11_tends_principal_image","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-image/[2,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0022","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的像生成集","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[0, 1, 2, 3, 4, 5, 6, 7]，S=[0, 2, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。列像主濾子的生成集f(S)，遞增且去重。","answer":{"kind":"expression","value":"[0, 2, 4]","explanation_zh":"主濾子像生成集=[0, 2, 4]。只把S的每個元素送入合法總映射，再去重；空生成集的像仍空，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的像生成集"]},"variant":{"family_id":"qf-density11-tends-principal-image","seed":202610117022,"parameters":{"n":8,"m":8,"images":[0,1,2,3,4,5,6,7],"S":[0,2,4],"filter_model":"principal_generators","operation":"density11_tends_principal_image","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-image/[0,2,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0023","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的像生成集","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[0, 1, 2, 3, 4, 5, 6, 7]，S=[1, 2, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。列像主濾子的生成集f(S)，遞增且去重。","answer":{"kind":"expression","value":"[1, 2, 4]","explanation_zh":"主濾子像生成集=[1, 2, 4]。只把S的每個元素送入合法總映射，再去重；空生成集的像仍空，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的像生成集"]},"variant":{"family_id":"qf-density11-tends-principal-image","seed":202610117023,"parameters":{"n":8,"m":8,"images":[0,1,2,3,4,5,6,7],"S":[1,2,4],"filter_model":"principal_generators","operation":"density11_tends_principal_image","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-image/[1,2,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0024","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的像生成集","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[0, 1, 2, 3, 4, 5, 6, 7]，S=[0, 1, 2, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。列像主濾子的生成集f(S)，遞增且去重。","answer":{"kind":"expression","value":"[0, 1, 2, 4]","explanation_zh":"主濾子像生成集=[0, 1, 2, 4]。只把S的每個元素送入合法總映射，再去重；空生成集的像仍空，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的像生成集"]},"variant":{"family_id":"qf-density11-tends-principal-image","seed":202610117024,"parameters":{"n":8,"m":8,"images":[0,1,2,3,4,5,6,7],"S":[0,1,2,4],"filter_model":"principal_generators","operation":"density11_tends_principal_image","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-image/[0,1,2,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0025","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的像生成集","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[0, 1, 2, 3, 4, 5, 6, 7]，S=[3, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。列像主濾子的生成集f(S)，遞增且去重。","answer":{"kind":"expression","value":"[3, 4]","explanation_zh":"主濾子像生成集=[3, 4]。只把S的每個元素送入合法總映射，再去重；空生成集的像仍空，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的像生成集"]},"variant":{"family_id":"qf-density11-tends-principal-image","seed":202610117025,"parameters":{"n":8,"m":8,"images":[0,1,2,3,4,5,6,7],"S":[3,4],"filter_model":"principal_generators","operation":"density11_tends_principal_image","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-image/[3,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0026","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(3)→Fin(1) 的表=[0, 0, 0]，S=[0, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[]","explanation_zh":"主濾子趨近阻擋點=[]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117026,"parameters":{"n":3,"m":1,"images":[0,0,0],"S":[0,2],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0027","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[1, 0, 0, 0, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[0]","explanation_zh":"主濾子趨近阻擋點=[0]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117027,"parameters":{"n":8,"m":2,"images":[1,0,0,0,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0028","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[0, 1, 0, 0, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[1]","explanation_zh":"主濾子趨近阻擋點=[1]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117028,"parameters":{"n":8,"m":2,"images":[0,1,0,0,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0029","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[1, 1, 0, 0, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[0, 1]","explanation_zh":"主濾子趨近阻擋點=[0, 1]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117029,"parameters":{"n":8,"m":2,"images":[1,1,0,0,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[0,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0030","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[0, 0, 1, 0, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[2]","explanation_zh":"主濾子趨近阻擋點=[2]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117030,"parameters":{"n":8,"m":2,"images":[0,0,1,0,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0031","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[1, 0, 1, 0, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[0, 2]","explanation_zh":"主濾子趨近阻擋點=[0, 2]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117031,"parameters":{"n":8,"m":2,"images":[1,0,1,0,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[0,2]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0032","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[0, 1, 1, 0, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[1, 2]","explanation_zh":"主濾子趨近阻擋點=[1, 2]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117032,"parameters":{"n":8,"m":2,"images":[0,1,1,0,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[1,2]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0035","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[1, 0, 0, 1, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[0, 3]","explanation_zh":"主濾子趨近阻擋點=[0, 3]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117035,"parameters":{"n":8,"m":2,"images":[1,0,0,1,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[0,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0036","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[0, 1, 0, 1, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[1, 3]","explanation_zh":"主濾子趨近阻擋點=[1, 3]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117036,"parameters":{"n":8,"m":2,"images":[0,1,0,1,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[1,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0037","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[1, 1, 0, 1, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[0, 1, 3]","explanation_zh":"主濾子趨近阻擋點=[0, 1, 3]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117037,"parameters":{"n":8,"m":2,"images":[1,1,0,1,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[0,1,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0038","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[0, 0, 1, 1, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[2, 3]","explanation_zh":"主濾子趨近阻擋點=[2, 3]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117038,"parameters":{"n":8,"m":2,"images":[0,0,1,1,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[2,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0039","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[1, 0, 1, 1, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[0, 2, 3]","explanation_zh":"主濾子趨近阻擋點=[0, 2, 3]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117039,"parameters":{"n":8,"m":2,"images":[1,0,1,1,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[0,2,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0040","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[0, 1, 1, 1, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[1, 2, 3]","explanation_zh":"主濾子趨近阻擋點=[1, 2, 3]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117040,"parameters":{"n":8,"m":2,"images":[0,1,1,1,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[1,2,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0041","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[1, 1, 1, 1, 0, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[0, 1, 2, 3]","explanation_zh":"主濾子趨近阻擋點=[0, 1, 2, 3]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117041,"parameters":{"n":8,"m":2,"images":[1,1,1,1,0,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[0,1,2,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0042","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[0, 0, 0, 0, 1, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[4]","explanation_zh":"主濾子趨近阻擋點=[4]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117042,"parameters":{"n":8,"m":2,"images":[0,0,0,0,1,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0043","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[1, 0, 0, 0, 1, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[0, 4]","explanation_zh":"主濾子趨近阻擋點=[0, 4]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117043,"parameters":{"n":8,"m":2,"images":[1,0,0,0,1,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[0,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0044","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[0, 1, 0, 0, 1, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[1, 4]","explanation_zh":"主濾子趨近阻擋點=[1, 4]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117044,"parameters":{"n":8,"m":2,"images":[0,1,0,0,1,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[1,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0045","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[1, 1, 0, 0, 1, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[0, 1, 4]","explanation_zh":"主濾子趨近阻擋點=[0, 1, 4]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117045,"parameters":{"n":8,"m":2,"images":[1,1,0,0,1,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[0,1,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0046","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[0, 0, 1, 0, 1, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[2, 4]","explanation_zh":"主濾子趨近阻擋點=[2, 4]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117046,"parameters":{"n":8,"m":2,"images":[0,0,1,0,1,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[2,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0047","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[1, 0, 1, 0, 1, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[0, 2, 4]","explanation_zh":"主濾子趨近阻擋點=[0, 2, 4]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117047,"parameters":{"n":8,"m":2,"images":[1,0,1,0,1,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[0,2,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0048","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[0, 1, 1, 0, 1, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[1, 2, 4]","explanation_zh":"主濾子趨近阻擋點=[1, 2, 4]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117048,"parameters":{"n":8,"m":2,"images":[0,1,1,0,1,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[1,2,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0049","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[1, 1, 1, 0, 1, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[0, 1, 2, 4]","explanation_zh":"主濾子趨近阻擋點=[0, 1, 2, 4]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117049,"parameters":{"n":8,"m":2,"images":[1,1,1,0,1,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[0,1,2,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0050","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子趨近的阻擋點","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(2) 的表=[0, 0, 0, 1, 1, 0, 0, 0]，S=[0, 1, 2, 3, 4, 5, 6, 7]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成。列所有x∈S且f(x)∉T的來源索引，遞增；空集表示此有限模型無阻擋。","answer":{"kind":"expression","value":"[3, 4]","explanation_zh":"主濾子趨近阻擋點=[3, 4]。方向是f(S)包含於T，逐個來源點核對，不反轉包含；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子趨近的阻擋點"]},"variant":{"family_id":"qf-density11-tends-principal-obstructions","seed":202610117050,"parameters":{"n":8,"m":2,"images":[0,0,0,1,1,0,0,0],"S":[0,1,2,3,4,5,6,7],"T":[0],"filter_model":"principal_generators","operation":"density11_tends_principal_obstructions","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-obstructions/[3,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0051","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"Fin(0)=∅，[]表示空型別的完整表；有限主濾子；完整總映射 f:Fin(0)→Fin(0) 的表=[]，S=[]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[]；目標principal(T)由包含T的所有目標子集構成；有序候選=[-1, 0, 1]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117051,"parameters":{"n":0,"m":0,"images":[],"S":[],"T":[],"candidates":[-1,0,1],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_type\",\"outside_type\",\"outside_type\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0052","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[3, 4, 5]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117052,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[3,4,5],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_generator\",\"outside_generator\",\"outside_generator\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0053","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[0, 0, 0]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117053,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[0,0,0],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"maps_inside\",\"maps_inside\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0054","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[]；目標principal(T)由包含T的所有目標子集構成；有序候選=[0, 1, 2]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[是, 是, 是]","explanation_zh":"主濾子反例候選資格=[是, 是, 是]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117054,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[],"candidates":[0,1,2],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"obstructs\",\"obstructs\",\"obstructs\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0055","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[3, 4, 102]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117055,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[3,4,102],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_generator\",\"outside_generator\",\"outside_type\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0056","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[0, 0, 102]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117056,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[0,0,102],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"maps_inside\",\"outside_type\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0057","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[1, 2, 102]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[是, 是, 否]","explanation_zh":"主濾子反例候選資格=[是, 是, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117057,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[1,2,102],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"obstructs\",\"obstructs\",\"outside_type\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0058","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[100, 4, 5]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117058,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[100,4,5],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_type\",\"outside_generator\",\"outside_generator\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0059","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[3, 0, 5]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117059,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[3,0,5],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_generator\",\"maps_inside\",\"outside_generator\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0060","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[0, 2, 5]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 是, 否]","explanation_zh":"主濾子反例候選資格=[否, 是, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117060,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[0,2,5],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"obstructs\",\"outside_generator\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0061","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[1, 101, 0]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"主濾子反例候選資格=[是, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117061,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[1,101,0],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"obstructs\",\"outside_type\",\"maps_inside\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0062","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[100, 0, 0]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117062,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[100,0,0],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_type\",\"maps_inside\",\"maps_inside\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0063","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[3, 2, 0]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 是, 否]","explanation_zh":"主濾子反例候選資格=[否, 是, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117063,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[3,2,0],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_generator\",\"obstructs\",\"maps_inside\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0064","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[0, 101, 1]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 是]","explanation_zh":"主濾子反例候選資格=[否, 否, 是]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117064,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[0,101,1],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"outside_type\",\"obstructs\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0065","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[1, 4, 1]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[是, 否, 是]","explanation_zh":"主濾子反例候選資格=[是, 否, 是]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117065,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[1,4,1],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"obstructs\",\"outside_generator\",\"obstructs\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0066","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[100, 2, 1]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 是, 是]","explanation_zh":"主濾子反例候選資格=[否, 是, 是]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117066,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[100,2,1],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_type\",\"obstructs\",\"obstructs\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0067","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[3, 101, 102]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117067,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[3,101,102],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_generator\",\"outside_type\",\"outside_type\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0068","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[0, 4, 102]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117068,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[0,4,102],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"outside_generator\",\"outside_type\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0069","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[1, 0, 102]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"主濾子反例候選資格=[是, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117069,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[1,0,102],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"obstructs\",\"maps_inside\",\"outside_type\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0070","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[100, 101, 5]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117070,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[100,101,5],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_type\",\"outside_type\",\"outside_generator\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0071","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[0, 0, 5]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117071,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[0,0,5],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"maps_inside\",\"outside_generator\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0072","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[1, 2, 5]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[是, 是, 否]","explanation_zh":"主濾子反例候選資格=[是, 是, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117072,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[1,2,5],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"obstructs\",\"obstructs\",\"outside_generator\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0073","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[100, 4, 0]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117073,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[100,4,0],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_type\",\"outside_generator\",\"maps_inside\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0074","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[3, 0, 0]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117074,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[3,0,0],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_generator\",\"maps_inside\",\"maps_inside\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0075","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"有限主濾子的反例候選資格","stem_zh":"有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[0, 2, 0]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。","answer":{"kind":"expression","value":"[否, 是, 否]","explanation_zh":"主濾子反例候選資格=[否, 是, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限主濾子的反例候選資格"]},"variant":{"family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117075,"parameters":{"n":6,"m":2,"images":[0,1,1,0,0,1],"S":[0,1,2],"T":[0],"candidates":[0,2,0],"filter_model":"principal_generators","operation":"density11_tends_principal_candidate_mask","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"obstructs\",\"maps_inside\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0076","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"Fin(0)=∅，[]表示空型別的完整表；有限主濾子；完整總映射 f:Fin(0)→Fin(0) 的表=[]，S=[]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(0)→Fin(0) 的表=[]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([], [])","explanation_zh":"兩路主濾子合成像=([], [])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117076,"parameters":{"n":0,"m":0,"l":0,"images":[],"after":[],"S":[],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0077","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(4)→Fin(3) 的表=[0, 0, 1, 2]，S=[0, 1, 2, 3]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(3)→Fin(8) 的表=[0, 0, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([0], [0])","explanation_zh":"兩路主濾子合成像=([0], [0])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117077,"parameters":{"n":4,"m":3,"l":8,"images":[0,0,1,2],"after":[0,0,0],"S":[0,1,2,3],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0078","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[1]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([1], [1])","explanation_zh":"兩路主濾子合成像=([1], [1])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117078,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[1],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0079","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[0, 1]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([0, 1], [0, 1])","explanation_zh":"兩路主濾子合成像=([0, 1], [0, 1])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117079,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[0,1],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[0,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0080","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([2], [2])","explanation_zh":"兩路主濾子合成像=([2], [2])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117080,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[2],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0081","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[0, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([0, 2], [0, 2])","explanation_zh":"兩路主濾子合成像=([0, 2], [0, 2])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117081,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[0,2],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[0,2]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0082","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([1, 2], [1, 2])","explanation_zh":"兩路主濾子合成像=([1, 2], [1, 2])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117082,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[1,2],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[1,2]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0083","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[0, 1, 2]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([0, 1, 2], [0, 1, 2])","explanation_zh":"兩路主濾子合成像=([0, 1, 2], [0, 1, 2])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117083,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[0,1,2],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[0,1,2]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0084","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[3]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([3], [3])","explanation_zh":"兩路主濾子合成像=([3], [3])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117084,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[3],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0085","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[0, 3]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([0, 3], [0, 3])","explanation_zh":"兩路主濾子合成像=([0, 3], [0, 3])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117085,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[0,3],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[0,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0086","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[1, 3]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([1, 3], [1, 3])","explanation_zh":"兩路主濾子合成像=([1, 3], [1, 3])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117086,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[1,3],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[1,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0087","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[0, 1, 3]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([0, 1, 3], [0, 1, 3])","explanation_zh":"兩路主濾子合成像=([0, 1, 3], [0, 1, 3])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117087,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[0,1,3],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[0,1,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0088","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[2, 3]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([2, 3], [2, 3])","explanation_zh":"兩路主濾子合成像=([2, 3], [2, 3])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117088,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[2,3],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[2,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0089","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[0, 2, 3]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([0, 2, 3], [0, 2, 3])","explanation_zh":"兩路主濾子合成像=([0, 2, 3], [0, 2, 3])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117089,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[0,2,3],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[0,2,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0090","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[1, 2, 3]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([1, 2, 3], [1, 2, 3])","explanation_zh":"兩路主濾子合成像=([1, 2, 3], [1, 2, 3])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117090,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[1,2,3],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[1,2,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0091","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[0, 1, 2, 3]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([0, 1, 2, 3], [0, 1, 2, 3])","explanation_zh":"兩路主濾子合成像=([0, 1, 2, 3], [0, 1, 2, 3])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117091,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[0,1,2,3],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[0,1,2,3]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0092","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([4], [4])","explanation_zh":"兩路主濾子合成像=([4], [4])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117092,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[4],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0093","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[0, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([0, 4], [0, 4])","explanation_zh":"兩路主濾子合成像=([0, 4], [0, 4])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117093,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[0,4],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[0,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0094","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[1, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([1, 4], [1, 4])","explanation_zh":"兩路主濾子合成像=([1, 4], [1, 4])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117094,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[1,4],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[1,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0095","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[0, 1, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([0, 1, 4], [0, 1, 4])","explanation_zh":"兩路主濾子合成像=([0, 1, 4], [0, 1, 4])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117095,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[0,1,4],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[0,1,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0096","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[2, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([2, 4], [2, 4])","explanation_zh":"兩路主濾子合成像=([2, 4], [2, 4])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117096,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[2,4],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[2,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0097","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[0, 2, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([0, 2, 4], [0, 2, 4])","explanation_zh":"兩路主濾子合成像=([0, 2, 4], [0, 2, 4])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117097,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[0,2,4],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[0,2,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0098","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[1, 2, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([1, 2, 4], [1, 2, 4])","explanation_zh":"兩路主濾子合成像=([1, 2, 4], [1, 2, 4])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117098,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[1,2,4],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[1,2,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0099","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[0, 1, 2, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([0, 1, 2, 4], [0, 1, 2, 4])","explanation_zh":"兩路主濾子合成像=([0, 1, 2, 4], [0, 1, 2, 4])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117099,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[0,1,2,4],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[0,1,2,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r07-tends-to-relation-0100","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-tends-to-relation"],"title_zh":"兩條有限主濾子合成像","stem_zh":"有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]，S=[3, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。完整總映射 g:Fin(8)→Fin(8) 的表=[7, 6, 5, 4, 3, 2, 1, 0]。依序列 ((g∘f)(S), g(f(S)))，兩條路各以完整遞增集表示。","answer":{"kind":"expression","value":"([3, 4], [3, 4])","explanation_zh":"兩路主濾子合成像=([3, 4], [3, 4])。第一條逐來源合成，第二條先去重中間像再映射；所有F/G表項仍須總合法，非一般數列或點極限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩條有限主濾子合成像"]},"variant":{"family_id":"qf-density11-tends-principal-composition-images","seed":202610117100,"parameters":{"n":8,"m":8,"l":8,"images":[7,6,5,4,3,2,1,0],"after":[7,6,5,4,3,2,1,0],"S":[3,4],"filter_model":"principal_generators","operation":"density11_tends_principal_composition_images","domain":"finite_principal_filter_model","semantic_template":"qf-density11-tends-principal-composition-images/[3,4]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}