{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r01-natural-number-0001","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=0、k=0。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[0]","explanation_zh":"含起點後繼窗=[0]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111001,"parameters":{"n":0,"k":0,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[0,0]"}},"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"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r01-natural-number-0003","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=1、k=2。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[1, 2, 3]","explanation_zh":"含起點後繼窗=[1, 2, 3]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111003,"parameters":{"n":1,"k":2,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[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-r01-natural-number-0004","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=2、k=3。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[2, 3, 4, 5]","explanation_zh":"含起點後繼窗=[2, 3, 4, 5]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111004,"parameters":{"n":2,"k":3,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[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-r01-natural-number-0005","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=3、k=4。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[3, 4, 5, 6, 7]","explanation_zh":"含起點後繼窗=[3, 4, 5, 6, 7]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111005,"parameters":{"n":3,"k":4,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[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-r01-natural-number-0006","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=4、k=5。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[4, 5, 6, 7, 8, 9]","explanation_zh":"含起點後繼窗=[4, 5, 6, 7, 8, 9]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111006,"parameters":{"n":4,"k":5,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[4,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r01-natural-number-0007","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=5、k=6。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[5, 6, 7, 8, 9, 10, 11]","explanation_zh":"含起點後繼窗=[5, 6, 7, 8, 9, 10, 11]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111007,"parameters":{"n":5,"k":6,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[5,6]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r01-natural-number-0008","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=6、k=7。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[6, 7, 8, 9, 10, 11, 12, 13]","explanation_zh":"含起點後繼窗=[6, 7, 8, 9, 10, 11, 12, 13]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111008,"parameters":{"n":6,"k":7,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[6,7]"}},"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-r01-natural-number-0009","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=7、k=8。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[7, 8, 9, 10, 11, 12, 13, 14, 15]","explanation_zh":"含起點後繼窗=[7, 8, 9, 10, 11, 12, 13, 14, 15]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111009,"parameters":{"n":7,"k":8,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[7,8]"}},"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-r01-natural-number-0010","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=8、k=0。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[8]","explanation_zh":"含起點後繼窗=[8]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111010,"parameters":{"n":8,"k":0,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[8,0]"}},"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-r01-natural-number-0011","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=9、k=1。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[9, 10]","explanation_zh":"含起點後繼窗=[9, 10]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111011,"parameters":{"n":9,"k":1,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[9,1]"}},"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-r01-natural-number-0012","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=10、k=2。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[10, 11, 12]","explanation_zh":"含起點後繼窗=[10, 11, 12]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111012,"parameters":{"n":10,"k":2,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[10,2]"}},"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-r01-natural-number-0013","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=11、k=3。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[11, 12, 13, 14]","explanation_zh":"含起點後繼窗=[11, 12, 13, 14]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111013,"parameters":{"n":11,"k":3,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[11,3]"}},"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-r01-natural-number-0014","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=12、k=4。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[12, 13, 14, 15, 16]","explanation_zh":"含起點後繼窗=[12, 13, 14, 15, 16]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111014,"parameters":{"n":12,"k":4,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[12,4]"}},"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-r01-natural-number-0015","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=13、k=5。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[13, 14, 15, 16, 17, 18]","explanation_zh":"含起點後繼窗=[13, 14, 15, 16, 17, 18]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111015,"parameters":{"n":13,"k":5,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[13,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r01-natural-number-0016","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=14、k=6。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[14, 15, 16, 17, 18, 19, 20]","explanation_zh":"含起點後繼窗=[14, 15, 16, 17, 18, 19, 20]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111016,"parameters":{"n":14,"k":6,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[14,6]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r01-natural-number-0017","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=15、k=7。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[15, 16, 17, 18, 19, 20, 21, 22]","explanation_zh":"含起點後繼窗=[15, 16, 17, 18, 19, 20, 21, 22]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111017,"parameters":{"n":15,"k":7,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[15,7]"}},"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-r01-natural-number-0018","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=16、k=8。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[16, 17, 18, 19, 20, 21, 22, 23, 24]","explanation_zh":"含起點後繼窗=[16, 17, 18, 19, 20, 21, 22, 23, 24]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111018,"parameters":{"n":16,"k":8,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[16,8]"}},"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-r01-natural-number-0019","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=17、k=0。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[17]","explanation_zh":"含起點後繼窗=[17]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111019,"parameters":{"n":17,"k":0,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[17,0]"}},"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-r01-natural-number-0020","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=18、k=1。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[18, 19]","explanation_zh":"含起點後繼窗=[18, 19]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111020,"parameters":{"n":18,"k":1,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[18,1]"}},"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-r01-natural-number-0021","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=19、k=2。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[19, 20, 21]","explanation_zh":"含起點後繼窗=[19, 20, 21]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111021,"parameters":{"n":19,"k":2,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[19,2]"}},"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-r01-natural-number-0022","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=20、k=3。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[20, 21, 22, 23]","explanation_zh":"含起點後繼窗=[20, 21, 22, 23]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111022,"parameters":{"n":20,"k":3,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[20,3]"}},"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-r01-natural-number-0023","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=21、k=4。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[21, 22, 23, 24, 25]","explanation_zh":"含起點後繼窗=[21, 22, 23, 24, 25]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111023,"parameters":{"n":21,"k":4,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[21,4]"}},"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-r01-natural-number-0024","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=22、k=5。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[22, 23, 24, 25, 26, 27]","explanation_zh":"含起點後繼窗=[22, 23, 24, 25, 26, 27]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111024,"parameters":{"n":22,"k":5,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[22,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r01-natural-number-0025","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然數的後繼窗","stem_zh":"N包含0；n=23、k=6。列含起點n並做k次後繼的完整窗，按次序寫整數列表。","answer":{"kind":"expression","value":"[23, 24, 25, 26, 27, 28, 29]","explanation_zh":"含起點後繼窗=[23, 24, 25, 26, 27, 28, 29]。從n開始逐次加一，故有k+1項；k=0仍含起點，零在N內。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然數的後繼窗"]},"variant":{"family_id":"qf-density11-natural-successor-window","seed":202610111025,"parameters":{"n":23,"k":6,"operation":"density11_natural_successor_window","domain":"N_including_zero","semantic_template":"qf-density11-natural-successor-window/[23,6]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r01-natural-number-0026","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[0, 0]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(無, 無)","explanation_zh":"部分自然前驅=(無, 無)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111026,"parameters":{"values":[0,0],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[0,0]"}},"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-r01-natural-number-0027","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[0, 1]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(無, 0)","explanation_zh":"部分自然前驅=(無, 0)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111027,"parameters":{"values":[0,1],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[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-r01-natural-number-0028","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[2, 2]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(1, 1)","explanation_zh":"部分自然前驅=(1, 1)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111028,"parameters":{"values":[2,2],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[2,2]"}},"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-r01-natural-number-0029","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[3, 3]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(2, 2)","explanation_zh":"部分自然前驅=(2, 2)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111029,"parameters":{"values":[3,3],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[3,3]"}},"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-r01-natural-number-0030","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[4, 4]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(3, 3)","explanation_zh":"部分自然前驅=(3, 3)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111030,"parameters":{"values":[4,4],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[4,4]"}},"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-r01-natural-number-0031","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[5, 0]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(4, 無)","explanation_zh":"部分自然前驅=(4, 無)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111031,"parameters":{"values":[5,0],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[5,0]"}},"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-r01-natural-number-0032","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[6, 1]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(5, 0)","explanation_zh":"部分自然前驅=(5, 0)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111032,"parameters":{"values":[6,1],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[6,1]"}},"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-r01-natural-number-0033","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[7, 2]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(6, 1)","explanation_zh":"部分自然前驅=(6, 1)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111033,"parameters":{"values":[7,2],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[7,2]"}},"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-r01-natural-number-0034","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[8, 3]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(7, 2)","explanation_zh":"部分自然前驅=(7, 2)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111034,"parameters":{"values":[8,3],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[8,3]"}},"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-r01-natural-number-0035","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[9, 4]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(8, 3)","explanation_zh":"部分自然前驅=(8, 3)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111035,"parameters":{"values":[9,4],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[9,4]"}},"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-r01-natural-number-0036","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[10, 0]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(9, 無)","explanation_zh":"部分自然前驅=(9, 無)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111036,"parameters":{"values":[10,0],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[10,0]"}},"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-r01-natural-number-0037","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[11, 1]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(10, 0)","explanation_zh":"部分自然前驅=(10, 0)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111037,"parameters":{"values":[11,1],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[11,1]"}},"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-r01-natural-number-0038","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[12, 2]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(11, 1)","explanation_zh":"部分自然前驅=(11, 1)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111038,"parameters":{"values":[12,2],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[12,2]"}},"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-r01-natural-number-0039","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[13, 3]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(12, 2)","explanation_zh":"部分自然前驅=(12, 2)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111039,"parameters":{"values":[13,3],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[13,3]"}},"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-r01-natural-number-0040","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[14, 4]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(13, 3)","explanation_zh":"部分自然前驅=(13, 3)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111040,"parameters":{"values":[14,4],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[14,4]"}},"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-r01-natural-number-0041","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[15, 0]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(14, 無)","explanation_zh":"部分自然前驅=(14, 無)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111041,"parameters":{"values":[15,0],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[15,0]"}},"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-r01-natural-number-0042","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[16, 1]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(15, 0)","explanation_zh":"部分自然前驅=(15, 0)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111042,"parameters":{"values":[16,1],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[16,1]"}},"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-r01-natural-number-0043","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[17, 2]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(16, 1)","explanation_zh":"部分自然前驅=(16, 1)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111043,"parameters":{"values":[17,2],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[17,2]"}},"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-r01-natural-number-0044","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[18, 3]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(17, 2)","explanation_zh":"部分自然前驅=(17, 2)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111044,"parameters":{"values":[18,3],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[18,3]"}},"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-r01-natural-number-0045","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[19, 4]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(18, 3)","explanation_zh":"部分自然前驅=(18, 3)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111045,"parameters":{"values":[19,4],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[19,4]"}},"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-r01-natural-number-0046","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[20, 0]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(19, 無)","explanation_zh":"部分自然前驅=(19, 無)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111046,"parameters":{"values":[20,0],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[20,0]"}},"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-r01-natural-number-0047","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[21, 1]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(20, 0)","explanation_zh":"部分自然前驅=(20, 0)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111047,"parameters":{"values":[21,1],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[21,1]"}},"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-r01-natural-number-0048","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[22, 2]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(21, 1)","explanation_zh":"部分自然前驅=(21, 1)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111048,"parameters":{"values":[22,2],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[22,2]"}},"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-r01-natural-number-0049","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[23, 3]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(22, 2)","explanation_zh":"部分自然前驅=(22, 2)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111049,"parameters":{"values":[23,3],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[23,3]"}},"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-r01-natural-number-0050","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"部分自然前驅的有序對","stem_zh":"N包含0；有序值=[24, 4]。各求N中的前驅，若沒有則寫無；保留兩欄次序。","answer":{"kind":"expression","value":"(23, 3)","explanation_zh":"部分自然前驅=(23, 3)。正值減一，零沒有自然前驅；這是非截斷pred，不能把0的前驅記成0。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["部分自然前驅的有序對"]},"variant":{"family_id":"qf-density11-natural-predecessor-pair","seed":202610111050,"parameters":{"values":[24,4],"operation":"density11_natural_predecessor_pair","domain":"N_including_zero","semantic_template":"qf-density11-natural-predecessor-pair/[24,4]"}},"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-r01-natural-number-0051","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=5、hi=4。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"0","explanation_zh":"N閉區間大小=0。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111051,"parameters":{"lo":5,"hi":4,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"empty_interval\"]"}},"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-r01-natural-number-0052","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=0、hi=0。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"1","explanation_zh":"N閉區間大小=1。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111052,"parameters":{"lo":0,"hi":0,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"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-r01-natural-number-0053","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=2、hi=3。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"2","explanation_zh":"N閉區間大小=2。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111053,"parameters":{"lo":2,"hi":3,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"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-r01-natural-number-0054","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=0、hi=2。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"3","explanation_zh":"N閉區間大小=3。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111054,"parameters":{"lo":0,"hi":2,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"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-r01-natural-number-0055","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=1、hi=4。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"4","explanation_zh":"N閉區間大小=4。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111055,"parameters":{"lo":1,"hi":4,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"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-r01-natural-number-0056","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=2、hi=6。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"5","explanation_zh":"N閉區間大小=5。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111056,"parameters":{"lo":2,"hi":6,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r01-natural-number-0057","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=0、hi=5。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"6","explanation_zh":"N閉區間大小=6。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111057,"parameters":{"lo":0,"hi":5,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"6\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r01-natural-number-0058","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=1、hi=7。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"7","explanation_zh":"N閉區間大小=7。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111058,"parameters":{"lo":1,"hi":7,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"7\"]"}},"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-r01-natural-number-0059","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=2、hi=9。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"8","explanation_zh":"N閉區間大小=8。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111059,"parameters":{"lo":2,"hi":9,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"8\"]"}},"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-r01-natural-number-0060","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=0、hi=8。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"9","explanation_zh":"N閉區間大小=9。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111060,"parameters":{"lo":0,"hi":8,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"9\"]"}},"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-r01-natural-number-0061","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=1、hi=10。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"10","explanation_zh":"N閉區間大小=10。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111061,"parameters":{"lo":1,"hi":10,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"10\"]"}},"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-r01-natural-number-0062","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=2、hi=12。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"11","explanation_zh":"N閉區間大小=11。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111062,"parameters":{"lo":2,"hi":12,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"11\"]"}},"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-r01-natural-number-0063","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=0、hi=11。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"12","explanation_zh":"N閉區間大小=12。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111063,"parameters":{"lo":0,"hi":11,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"12\"]"}},"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-r01-natural-number-0064","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=1、hi=13。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"13","explanation_zh":"N閉區間大小=13。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111064,"parameters":{"lo":1,"hi":13,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"13\"]"}},"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-r01-natural-number-0065","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=2、hi=15。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"14","explanation_zh":"N閉區間大小=14。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111065,"parameters":{"lo":2,"hi":15,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"14\"]"}},"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-r01-natural-number-0066","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=0、hi=14。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"15","explanation_zh":"N閉區間大小=15。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111066,"parameters":{"lo":0,"hi":14,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"15\"]"}},"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-r01-natural-number-0067","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=1、hi=16。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"16","explanation_zh":"N閉區間大小=16。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111067,"parameters":{"lo":1,"hi":16,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"16\"]"}},"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-r01-natural-number-0068","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=2、hi=18。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"17","explanation_zh":"N閉區間大小=17。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111068,"parameters":{"lo":2,"hi":18,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"17\"]"}},"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-r01-natural-number-0069","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=0、hi=17。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"18","explanation_zh":"N閉區間大小=18。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111069,"parameters":{"lo":0,"hi":17,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"18\"]"}},"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-r01-natural-number-0070","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=1、hi=19。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"19","explanation_zh":"N閉區間大小=19。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111070,"parameters":{"lo":1,"hi":19,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"19\"]"}},"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-r01-natural-number-0071","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=2、hi=21。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"20","explanation_zh":"N閉區間大小=20。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111071,"parameters":{"lo":2,"hi":21,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"20\"]"}},"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-r01-natural-number-0072","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=0、hi=20。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"21","explanation_zh":"N閉區間大小=21。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111072,"parameters":{"lo":0,"hi":20,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"21\"]"}},"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-r01-natural-number-0073","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=1、hi=22。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"22","explanation_zh":"N閉區間大小=22。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111073,"parameters":{"lo":1,"hi":22,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"22\"]"}},"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-r01-natural-number-0074","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=2、hi=24。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"23","explanation_zh":"N閉區間大小=23。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111074,"parameters":{"lo":2,"hi":24,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"23\"]"}},"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-r01-natural-number-0075","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"含零自然閉區間的大小","stem_zh":"N包含0；lo=0、hi=23。求 {n∈N | lo≤n≤hi} 的大小，端點均含。","answer":{"kind":"expression","value":"24","explanation_zh":"N閉區間大小=24。有效閉區間逐個計數含兩端；上端小於下端則無元素，大小零，不交換端點。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["含零自然閉區間的大小"]},"variant":{"family_id":"qf-density11-natural-interval-size","seed":202610111075,"parameters":{"lo":0,"hi":23,"operation":"density11_natural_interval_size","domain":"N_including_zero","semantic_template":"qf-density11-natural-interval-size/[\"cardinality\",\"24\"]"}},"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-r01-natural-number-0076","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[0/1, -1/1, 1/2]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111076,"parameters":{"fractions":[[0,1],[-1,1],[1,2]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"0/1\",\"-1/1\",\"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-r01-natural-number-0077","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[1/1, 0/1, -1/2]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 是, 否]","explanation_zh":"N含0資格=[是, 是, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111077,"parameters":{"fractions":[[2,2],[0,3],[-1,2]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"1/1\",\"0/1\",\"-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-r01-natural-number-0078","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[2/1, -2/1, 3/5]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111078,"parameters":{"fractions":[[2,1],[-2,1],[3,5]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"2/1\",\"-2/1\",\"3/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r01-natural-number-0079","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[3/1, -3/1, 4/7]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111079,"parameters":{"fractions":[[3,1],[-3,1],[4,7]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"3/1\",\"-3/1\",\"4/7\"]"}},"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-r01-natural-number-0080","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[4/1, -4/1, 5/9]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111080,"parameters":{"fractions":[[4,1],[-4,1],[5,9]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"4/1\",\"-4/1\",\"5/9\"]"}},"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-r01-natural-number-0081","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[5/1, -5/1, 6/11]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111081,"parameters":{"fractions":[[5,1],[-5,1],[6,11]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"5/1\",\"-5/1\",\"6/11\"]"}},"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-r01-natural-number-0082","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[6/1, -6/1, 7/13]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111082,"parameters":{"fractions":[[6,1],[-6,1],[7,13]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"6/1\",\"-6/1\",\"7/13\"]"}},"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-r01-natural-number-0083","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[7/1, -7/1, 8/15]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111083,"parameters":{"fractions":[[7,1],[-7,1],[8,15]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"7/1\",\"-7/1\",\"8/15\"]"}},"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-r01-natural-number-0084","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[8/1, -8/1, 9/17]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111084,"parameters":{"fractions":[[8,1],[-8,1],[9,17]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"8/1\",\"-8/1\",\"9/17\"]"}},"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-r01-natural-number-0085","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[9/1, -9/1, 10/19]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111085,"parameters":{"fractions":[[9,1],[-9,1],[10,19]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"9/1\",\"-9/1\",\"10/19\"]"}},"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-r01-natural-number-0086","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[10/1, -10/1, 11/21]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111086,"parameters":{"fractions":[[10,1],[-10,1],[11,21]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"10/1\",\"-10/1\",\"11/21\"]"}},"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-r01-natural-number-0087","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[11/1, -11/1, 12/23]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111087,"parameters":{"fractions":[[11,1],[-11,1],[12,23]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"11/1\",\"-11/1\",\"12/23\"]"}},"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-r01-natural-number-0088","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[12/1, -12/1, 13/25]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111088,"parameters":{"fractions":[[12,1],[-12,1],[13,25]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"12/1\",\"-12/1\",\"13/25\"]"}},"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-r01-natural-number-0089","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[13/1, -13/1, 14/27]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111089,"parameters":{"fractions":[[13,1],[-13,1],[14,27]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"13/1\",\"-13/1\",\"14/27\"]"}},"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-r01-natural-number-0090","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[14/1, -14/1, 15/29]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111090,"parameters":{"fractions":[[14,1],[-14,1],[15,29]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"14/1\",\"-14/1\",\"15/29\"]"}},"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-r01-natural-number-0091","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[15/1, -15/1, 16/31]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111091,"parameters":{"fractions":[[15,1],[-15,1],[16,31]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"15/1\",\"-15/1\",\"16/31\"]"}},"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-r01-natural-number-0092","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[16/1, -16/1, 17/33]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111092,"parameters":{"fractions":[[16,1],[-16,1],[17,33]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"16/1\",\"-16/1\",\"17/33\"]"}},"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-r01-natural-number-0093","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[17/1, -17/1, 18/35]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111093,"parameters":{"fractions":[[17,1],[-17,1],[18,35]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"17/1\",\"-17/1\",\"18/35\"]"}},"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-r01-natural-number-0094","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[18/1, -18/1, 19/37]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111094,"parameters":{"fractions":[[18,1],[-18,1],[19,37]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"18/1\",\"-18/1\",\"19/37\"]"}},"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-r01-natural-number-0095","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[19/1, -19/1, 20/39]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111095,"parameters":{"fractions":[[19,1],[-19,1],[20,39]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"19/1\",\"-19/1\",\"20/39\"]"}},"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-r01-natural-number-0096","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[20/1, -20/1, 21/41]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111096,"parameters":{"fractions":[[20,1],[-20,1],[21,41]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"20/1\",\"-20/1\",\"21/41\"]"}},"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-r01-natural-number-0097","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[21/1, -21/1, 22/43]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111097,"parameters":{"fractions":[[21,1],[-21,1],[22,43]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"21/1\",\"-21/1\",\"22/43\"]"}},"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-r01-natural-number-0098","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[22/1, -22/1, 23/45]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111098,"parameters":{"fractions":[[22,1],[-22,1],[23,45]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"22/1\",\"-22/1\",\"23/45\"]"}},"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-r01-natural-number-0099","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[23/1, -23/1, 24/47]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111099,"parameters":{"fractions":[[23,1],[-23,1],[24,47]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"23/1\",\"-23/1\",\"24/47\"]"}},"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-r01-natural-number-0100","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","language":"zh-Hant","source_mko_ids":["mko-natural-number"],"title_zh":"三個有理候選的含零N資格","stem_zh":"N包含0；有序有理數候選=[24/1, -24/1, 25/49]。逐項答值是否屬N，以是／否列表表示。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"N含0資格=[是, 否, 否]。先精確約分，再要求非負且分母為一；零通過，負整數與非整數分數不通過。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三個有理候選的含零N資格"]},"variant":{"family_id":"qf-density11-natural-rational-membership-mask","seed":202610111100,"parameters":{"fractions":[[24,1],[-24,1],[25,49]],"operation":"density11_natural_rational_membership_mask","domain":"Q_candidates_for_N0","semantic_template":"qf-density11-natural-rational-membership-mask/[\"24/1\",\"-24/1\",\"25/49\"]"}},"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"}}