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{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r01-mathematical-induction-0003","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=2。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(2, 3)","explanation_zh":"假設與後繼目標索引=(2, 3)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121003,"parameters":{"n":2,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0004","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=3。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(3, 4)","explanation_zh":"假設與後繼目標索引=(3, 4)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121004,"parameters":{"n":3,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0005","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=4。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(4, 5)","explanation_zh":"假設與後繼目標索引=(4, 5)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121005,"parameters":{"n":4,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0006","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=5。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(5, 6)","explanation_zh":"假設與後繼目標索引=(5, 6)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121006,"parameters":{"n":5,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0007","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=6。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(6, 7)","explanation_zh":"假設與後繼目標索引=(6, 7)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121007,"parameters":{"n":6,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0008","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=7。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(7, 8)","explanation_zh":"假設與後繼目標索引=(7, 8)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121008,"parameters":{"n":7,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0009","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=8。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(8, 9)","explanation_zh":"假設與後繼目標索引=(8, 9)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121009,"parameters":{"n":8,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0010","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=9。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(9, 10)","explanation_zh":"假設與後繼目標索引=(9, 10)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121010,"parameters":{"n":9,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0011","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=10。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(10, 11)","explanation_zh":"假設與後繼目標索引=(10, 11)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121011,"parameters":{"n":10,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0012","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=11。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(11, 12)","explanation_zh":"假設與後繼目標索引=(11, 12)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121012,"parameters":{"n":11,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0013","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=12。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(12, 13)","explanation_zh":"假設與後繼目標索引=(12, 13)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121013,"parameters":{"n":12,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0014","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=13。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(13, 14)","explanation_zh":"假設與後繼目標索引=(13, 14)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121014,"parameters":{"n":13,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0015","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=14。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(14, 15)","explanation_zh":"假設與後繼目標索引=(14, 15)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121015,"parameters":{"n":14,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0016","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=15。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(15, 16)","explanation_zh":"假設與後繼目標索引=(15, 16)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121016,"parameters":{"n":15,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0017","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=16。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(16, 17)","explanation_zh":"假設與後繼目標索引=(16, 17)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121017,"parameters":{"n":16,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0018","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=17。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(17, 18)","explanation_zh":"假設與後繼目標索引=(17, 18)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121018,"parameters":{"n":17,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0019","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=18。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(18, 19)","explanation_zh":"假設與後繼目標索引=(18, 19)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121019,"parameters":{"n":18,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0020","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=19。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(19, 20)","explanation_zh":"假設與後繼目標索引=(19, 20)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121020,"parameters":{"n":19,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0021","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=20。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(20, 21)","explanation_zh":"假設與後繼目標索引=(20, 21)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121021,"parameters":{"n":20,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0022","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=21。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(21, 22)","explanation_zh":"假設與後繼目標索引=(21, 22)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121022,"parameters":{"n":21,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0023","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=22。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(22, 23)","explanation_zh":"假設與後繼目標索引=(22, 23)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121023,"parameters":{"n":22,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0024","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=23。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(23, 24)","explanation_zh":"假設與後繼目標索引=(23, 24)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121024,"parameters":{"n":23,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0025","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"歸納假設與後繼目標索引","stem_zh":"N從0起；有限歸納協議；目前n=24。依序列 (歸納假設P的索引n, 後繼目標的索引n+1)，不宣稱命題已成立。","answer":{"kind":"expression","value":"(24, 25)","explanation_zh":"假設與後繼目標索引=(24, 25)。由非負索引求其後繼；這只標記作用域，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歸納假設與後繼目標索引"]},"variant":{"family_id":"qf-density12-induction-successor-index-pair","seed":202610121025,"parameters":{"n":24,"operation":"density12_induction_successor_index_pair","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-successor-index-pair/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-20261012-r01-mathematical-induction-0026","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=-2、a(k+1)=a(k)+(99)，through=0。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[-2]","explanation_zh":"遞推有限前綴=[-2]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121026,"parameters":{"s":-2,"d":99,"through":0,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[\"base_only\",-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-20261012-r01-mathematical-induction-0027","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=-11、a(k+1)=a(k)+(-1)，through=2。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[-11, -12, -13]","explanation_zh":"遞推有限前綴=[-11, -12, -13]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121027,"parameters":{"s":-11,"d":-1,"through":2,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[-11,-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-20261012-r01-mathematical-induction-0028","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=-10、a(k+1)=a(k)+(0)，through=3。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[-10, -10, -10, -10]","explanation_zh":"遞推有限前綴=[-10, -10, -10, -10]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121028,"parameters":{"s":-10,"d":0,"through":3,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[-10,0,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r01-mathematical-induction-0029","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=-9、a(k+1)=a(k)+(1)，through=4。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[-9, -8, -7, -6, -5]","explanation_zh":"遞推有限前綴=[-9, -8, -7, -6, -5]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121029,"parameters":{"s":-9,"d":1,"through":4,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[-9,1,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r01-mathematical-induction-0030","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=-8、a(k+1)=a(k)+(2)，through=5。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[-8, -6, -4, -2, 0, 2]","explanation_zh":"遞推有限前綴=[-8, -6, -4, -2, 0, 2]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121030,"parameters":{"s":-8,"d":2,"through":5,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[-8,2,5]"}},"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-20261012-r01-mathematical-induction-0031","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=-7、a(k+1)=a(k)+(-2)，through=6。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[-7, -9, -11, -13, -15, -17, -19]","explanation_zh":"遞推有限前綴=[-7, -9, -11, -13, -15, -17, -19]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121031,"parameters":{"s":-7,"d":-2,"through":6,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[-7,-2,6]"}},"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-20261012-r01-mathematical-induction-0032","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=-6、a(k+1)=a(k)+(-1)，through=7。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[-6, -7, -8, -9, -10, -11, -12, -13]","explanation_zh":"遞推有限前綴=[-6, -7, -8, -9, -10, -11, -12, -13]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121032,"parameters":{"s":-6,"d":-1,"through":7,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[-6,-1,7]"}},"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-20261012-r01-mathematical-induction-0033","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=-5、a(k+1)=a(k)+(0)，through=8。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[-5, -5, -5, -5, -5, -5, -5, -5, -5]","explanation_zh":"遞推有限前綴=[-5, -5, -5, -5, -5, -5, -5, -5, -5]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121033,"parameters":{"s":-5,"d":0,"through":8,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[-5,0,8]"}},"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-20261012-r01-mathematical-induction-0034","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=-4、a(k+1)=a(k)+(1)，through=1。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[-4, -3]","explanation_zh":"遞推有限前綴=[-4, -3]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121034,"parameters":{"s":-4,"d":1,"through":1,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[-4,1,1]"}},"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-20261012-r01-mathematical-induction-0035","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=-3、a(k+1)=a(k)+(2)，through=2。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[-3, -1, 1]","explanation_zh":"遞推有限前綴=[-3, -1, 1]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121035,"parameters":{"s":-3,"d":2,"through":2,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[-3,2,2]"}},"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-20261012-r01-mathematical-induction-0036","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=-2、a(k+1)=a(k)+(-2)，through=3。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[-2, -4, -6, -8]","explanation_zh":"遞推有限前綴=[-2, -4, -6, -8]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121036,"parameters":{"s":-2,"d":-2,"through":3,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[-2,-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-20261012-r01-mathematical-induction-0037","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=-1、a(k+1)=a(k)+(-1)，through=4。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[-1, -2, -3, -4, -5]","explanation_zh":"遞推有限前綴=[-1, -2, -3, -4, -5]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121037,"parameters":{"s":-1,"d":-1,"through":4,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[-1,-1,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r01-mathematical-induction-0038","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=0、a(k+1)=a(k)+(0)，through=5。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[0, 0, 0, 0, 0, 0]","explanation_zh":"遞推有限前綴=[0, 0, 0, 0, 0, 0]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121038,"parameters":{"s":0,"d":0,"through":5,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[0,0,5]"}},"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-20261012-r01-mathematical-induction-0039","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=1、a(k+1)=a(k)+(1)，through=6。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[1, 2, 3, 4, 5, 6, 7]","explanation_zh":"遞推有限前綴=[1, 2, 3, 4, 5, 6, 7]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121039,"parameters":{"s":1,"d":1,"through":6,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[1,1,6]"}},"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-20261012-r01-mathematical-induction-0040","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=2、a(k+1)=a(k)+(2)，through=7。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[2, 4, 6, 8, 10, 12, 14, 16]","explanation_zh":"遞推有限前綴=[2, 4, 6, 8, 10, 12, 14, 16]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121040,"parameters":{"s":2,"d":2,"through":7,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[2,2,7]"}},"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-20261012-r01-mathematical-induction-0041","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=3、a(k+1)=a(k)+(-2)，through=8。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[3, 1, -1, -3, -5, -7, -9, -11, -13]","explanation_zh":"遞推有限前綴=[3, 1, -1, -3, -5, -7, -9, -11, -13]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121041,"parameters":{"s":3,"d":-2,"through":8,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[3,-2,8]"}},"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-20261012-r01-mathematical-induction-0042","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=4、a(k+1)=a(k)+(-1)，through=1。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[4, 3]","explanation_zh":"遞推有限前綴=[4, 3]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121042,"parameters":{"s":4,"d":-1,"through":1,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[4,-1,1]"}},"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-20261012-r01-mathematical-induction-0043","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=5、a(k+1)=a(k)+(0)，through=2。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[5, 5, 5]","explanation_zh":"遞推有限前綴=[5, 5, 5]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121043,"parameters":{"s":5,"d":0,"through":2,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[5,0,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r01-mathematical-induction-0044","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=6、a(k+1)=a(k)+(1)，through=3。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[6, 7, 8, 9]","explanation_zh":"遞推有限前綴=[6, 7, 8, 9]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121044,"parameters":{"s":6,"d":1,"through":3,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[6,1,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r01-mathematical-induction-0045","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=7、a(k+1)=a(k)+(2)，through=4。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[7, 9, 11, 13, 15]","explanation_zh":"遞推有限前綴=[7, 9, 11, 13, 15]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121045,"parameters":{"s":7,"d":2,"through":4,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[7,2,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r01-mathematical-induction-0046","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=8、a(k+1)=a(k)+(-2)，through=5。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[8, 6, 4, 2, 0, -2]","explanation_zh":"遞推有限前綴=[8, 6, 4, 2, 0, -2]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121046,"parameters":{"s":8,"d":-2,"through":5,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[8,-2,5]"}},"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-20261012-r01-mathematical-induction-0047","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=9、a(k+1)=a(k)+(-1)，through=6。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[9, 8, 7, 6, 5, 4, 3]","explanation_zh":"遞推有限前綴=[9, 8, 7, 6, 5, 4, 3]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121047,"parameters":{"s":9,"d":-1,"through":6,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[9,-1,6]"}},"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-20261012-r01-mathematical-induction-0048","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=10、a(k+1)=a(k)+(0)，through=7。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[10, 10, 10, 10, 10, 10, 10, 10]","explanation_zh":"遞推有限前綴=[10, 10, 10, 10, 10, 10, 10, 10]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121048,"parameters":{"s":10,"d":0,"through":7,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[10,0,7]"}},"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-20261012-r01-mathematical-induction-0049","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=11、a(k+1)=a(k)+(1)，through=8。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[11, 12, 13, 14, 15, 16, 17, 18, 19]","explanation_zh":"遞推有限前綴=[11, 12, 13, 14, 15, 16, 17, 18, 19]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121049,"parameters":{"s":11,"d":1,"through":8,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[11,1,8]"}},"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-20261012-r01-mathematical-induction-0050","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"整數值遞推的有限前綴","stem_zh":"N從0起；有限歸納協議；a(0)=12、a(k+1)=a(k)+(2)，through=1。列a(0)至a(through)，含兩端。","answer":{"kind":"expression","value":"[12, 14]","explanation_zh":"遞推有限前綴=[12, 14]。從初值逐次加d，零範圍只列初值；索引自然數不要求遞推值非負，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["整數值遞推的有限前綴"]},"variant":{"family_id":"qf-density12-induction-affine-recursive-prefix","seed":202610121050,"parameters":{"s":12,"d":2,"through":1,"operation":"density12_induction_affine_recursive_prefix","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-affine-recursive-prefix/[12,2,1]"}},"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-20261012-r01-mathematical-induction-0051","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=3、步增量d=2；候選c(k)=2k+(3)，n=0。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"候選基礎與步缺陷=(0, 0)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121051,"parameters":{"s":3,"d":2,"a":2,"b":3,"n":0,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"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-20261012-r01-mathematical-induction-0052","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=3、步增量d=2；候選c(k)=2k+(1)，n=5。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(-2, 0)","explanation_zh":"候選基礎與步缺陷=(-2, 0)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121052,"parameters":{"s":3,"d":2,"a":2,"b":1,"n":5,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"-2\",\"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-20261012-r01-mathematical-induction-0053","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=1、步增量d=2；候選c(k)=3k+(1)，n=2。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(0, 1)","explanation_zh":"候選基礎與步缺陷=(0, 1)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121053,"parameters":{"s":1,"d":2,"a":3,"b":1,"n":2,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"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-20261012-r01-mathematical-induction-0054","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=1、步增量d=-1；候選c(k)=-10k+(5)，n=3。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(4, -9)","explanation_zh":"候選基礎與步缺陷=(4, -9)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121054,"parameters":{"s":1,"d":-1,"a":-10,"b":5,"n":3,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"4\",\"-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-20261012-r01-mathematical-induction-0055","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=-2、步增量d=0；候選c(k)=-8k+(3)，n=4。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(5, -8)","explanation_zh":"候選基礎與步缺陷=(5, -8)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121055,"parameters":{"s":-2,"d":0,"a":-8,"b":3,"n":4,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"5\",\"-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-20261012-r01-mathematical-induction-0056","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=-1、步增量d=1；候選c(k)=-6k+(5)，n=5。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(6, -7)","explanation_zh":"候選基礎與步缺陷=(6, -7)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121056,"parameters":{"s":-1,"d":1,"a":-6,"b":5,"n":5,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"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-20261012-r01-mathematical-induction-0057","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=0、步增量d=-1；候選c(k)=-7k+(7)，n=0。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(7, -6)","explanation_zh":"候選基礎與步缺陷=(7, -6)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121057,"parameters":{"s":0,"d":-1,"a":-7,"b":7,"n":0,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"7\",\"-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-20261012-r01-mathematical-induction-0058","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=1、步增量d=0；候選c(k)=-5k+(9)，n=1。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(8, -5)","explanation_zh":"候選基礎與步缺陷=(8, -5)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121058,"parameters":{"s":1,"d":0,"a":-5,"b":9,"n":1,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"8\",\"-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-20261012-r01-mathematical-induction-0059","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=-2、步增量d=1；候選c(k)=-3k+(7)，n=2。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(9, -4)","explanation_zh":"候選基礎與步缺陷=(9, -4)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121059,"parameters":{"s":-2,"d":1,"a":-3,"b":7,"n":2,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"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-20261012-r01-mathematical-induction-0060","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=-1、步增量d=-1；候選c(k)=-4k+(9)，n=3。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(10, -3)","explanation_zh":"候選基礎與步缺陷=(10, -3)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121060,"parameters":{"s":-1,"d":-1,"a":-4,"b":9,"n":3,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"10\",\"-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-20261012-r01-mathematical-induction-0061","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=0、步增量d=0；候選c(k)=-2k+(11)，n=4。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(11, -2)","explanation_zh":"候選基礎與步缺陷=(11, -2)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121061,"parameters":{"s":0,"d":0,"a":-2,"b":11,"n":4,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"11\",\"-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-20261012-r01-mathematical-induction-0062","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=1、步增量d=1；候選c(k)=0k+(13)，n=5。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(12, -1)","explanation_zh":"候選基礎與步缺陷=(12, -1)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121062,"parameters":{"s":1,"d":1,"a":0,"b":13,"n":5,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"12\",\"-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-20261012-r01-mathematical-induction-0063","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=-2、步增量d=-1；候選c(k)=-1k+(11)，n=0。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(13, 0)","explanation_zh":"候選基礎與步缺陷=(13, 0)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121063,"parameters":{"s":-2,"d":-1,"a":-1,"b":11,"n":0,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"13\",\"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-20261012-r01-mathematical-induction-0064","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=-1、步增量d=0；候選c(k)=1k+(13)，n=1。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(14, 1)","explanation_zh":"候選基礎與步缺陷=(14, 1)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121064,"parameters":{"s":-1,"d":0,"a":1,"b":13,"n":1,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"14\",\"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-20261012-r01-mathematical-induction-0065","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=0、步增量d=1；候選c(k)=3k+(15)，n=2。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(15, 2)","explanation_zh":"候選基礎與步缺陷=(15, 2)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121065,"parameters":{"s":0,"d":1,"a":3,"b":15,"n":2,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"15\",\"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-20261012-r01-mathematical-induction-0066","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=1、步增量d=-1；候選c(k)=2k+(17)，n=3。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(16, 3)","explanation_zh":"候選基礎與步缺陷=(16, 3)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121066,"parameters":{"s":1,"d":-1,"a":2,"b":17,"n":3,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"16\",\"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-20261012-r01-mathematical-induction-0067","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=-2、步增量d=0；候選c(k)=4k+(15)，n=4。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(17, 4)","explanation_zh":"候選基礎與步缺陷=(17, 4)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121067,"parameters":{"s":-2,"d":0,"a":4,"b":15,"n":4,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"17\",\"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-20261012-r01-mathematical-induction-0068","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=-1、步增量d=1；候選c(k)=6k+(17)，n=5。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(18, 5)","explanation_zh":"候選基礎與步缺陷=(18, 5)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121068,"parameters":{"s":-1,"d":1,"a":6,"b":17,"n":5,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"18\",\"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-20261012-r01-mathematical-induction-0069","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=0、步增量d=-1；候選c(k)=5k+(19)，n=0。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(19, 6)","explanation_zh":"候選基礎與步缺陷=(19, 6)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121069,"parameters":{"s":0,"d":-1,"a":5,"b":19,"n":0,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"19\",\"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-20261012-r01-mathematical-induction-0070","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=1、步增量d=0；候選c(k)=7k+(21)，n=1。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(20, 7)","explanation_zh":"候選基礎與步缺陷=(20, 7)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121070,"parameters":{"s":1,"d":0,"a":7,"b":21,"n":1,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"20\",\"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-20261012-r01-mathematical-induction-0071","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=-2、步增量d=1；候選c(k)=9k+(19)，n=2。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(21, 8)","explanation_zh":"候選基礎與步缺陷=(21, 8)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121071,"parameters":{"s":-2,"d":1,"a":9,"b":19,"n":2,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"21\",\"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-20261012-r01-mathematical-induction-0072","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=-1、步增量d=-1；候選c(k)=8k+(21)，n=3。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(22, 9)","explanation_zh":"候選基礎與步缺陷=(22, 9)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121072,"parameters":{"s":-1,"d":-1,"a":8,"b":21,"n":3,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"22\",\"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-20261012-r01-mathematical-induction-0073","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=0、步增量d=0；候選c(k)=10k+(23)，n=4。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(23, 10)","explanation_zh":"候選基礎與步缺陷=(23, 10)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121073,"parameters":{"s":0,"d":0,"a":10,"b":23,"n":4,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"23\",\"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-20261012-r01-mathematical-induction-0074","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=1、步增量d=1；候選c(k)=12k+(25)，n=5。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(24, 11)","explanation_zh":"候選基礎與步缺陷=(24, 11)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121074,"parameters":{"s":1,"d":1,"a":12,"b":25,"n":5,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"24\",\"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-20261012-r01-mathematical-induction-0075","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"候選遞推的基礎與步缺陷","stem_zh":"N從0起；有限歸納協議；目標初值s=-2、步增量d=-1；候選c(k)=11k+(23)，n=0。依序求 (c(0)−s, c(n+1)−c(n)−d) 的有號整數。","answer":{"kind":"expression","value":"(25, 12)","explanation_zh":"候選基礎與步缺陷=(25, 12)。分別代入初點與相鄰兩點求殘差，基礎及步缺陷不能混同；有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選遞推的基礎與步缺陷"]},"variant":{"family_id":"qf-density12-induction-candidate-step-defect","seed":202610121075,"parameters":{"s":-2,"d":-1,"a":11,"b":23,"n":0,"operation":"density12_induction_candidate_step_defect","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-candidate-step-defect/[\"25\",\"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-20261012-r01-mathematical-induction-0076","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(0)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(1, 0, 1)","explanation_zh":"有限範圍覆蓋=(1, 0, 1)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121076,"parameters":{"through":0,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0077","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(1000)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(1001, 1000, 1001)","explanation_zh":"有限範圍覆蓋=(1001, 1000, 1001)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121077,"parameters":{"through":1000,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/1000"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-20261012-r01-mathematical-induction-0078","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(1)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(2, 1, 2)","explanation_zh":"有限範圍覆蓋=(2, 1, 2)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121078,"parameters":{"through":1,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0079","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(2)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(3, 2, 3)","explanation_zh":"有限範圍覆蓋=(3, 2, 3)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121079,"parameters":{"through":2,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0080","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(3)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(4, 3, 4)","explanation_zh":"有限範圍覆蓋=(4, 3, 4)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121080,"parameters":{"through":3,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0081","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(4)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(5, 4, 5)","explanation_zh":"有限範圍覆蓋=(5, 4, 5)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121081,"parameters":{"through":4,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0082","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(5)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(6, 5, 6)","explanation_zh":"有限範圍覆蓋=(6, 5, 6)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121082,"parameters":{"through":5,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0083","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(6)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(7, 6, 7)","explanation_zh":"有限範圍覆蓋=(7, 6, 7)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121083,"parameters":{"through":6,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0084","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(7)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(8, 7, 8)","explanation_zh":"有限範圍覆蓋=(8, 7, 8)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121084,"parameters":{"through":7,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0085","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(8)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(9, 8, 9)","explanation_zh":"有限範圍覆蓋=(9, 8, 9)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121085,"parameters":{"through":8,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0086","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(9)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(10, 9, 10)","explanation_zh":"有限範圍覆蓋=(10, 9, 10)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121086,"parameters":{"through":9,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0087","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(10)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(11, 10, 11)","explanation_zh":"有限範圍覆蓋=(11, 10, 11)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121087,"parameters":{"through":10,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0088","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(11)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(12, 11, 12)","explanation_zh":"有限範圍覆蓋=(12, 11, 12)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121088,"parameters":{"through":11,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0089","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(12)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(13, 12, 13)","explanation_zh":"有限範圍覆蓋=(13, 12, 13)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121089,"parameters":{"through":12,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0090","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(13)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(14, 13, 14)","explanation_zh":"有限範圍覆蓋=(14, 13, 14)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121090,"parameters":{"through":13,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0091","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(14)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(15, 14, 15)","explanation_zh":"有限範圍覆蓋=(15, 14, 15)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121091,"parameters":{"through":14,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0092","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(15)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(16, 15, 16)","explanation_zh":"有限範圍覆蓋=(16, 15, 16)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121092,"parameters":{"through":15,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0093","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(16)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(17, 16, 17)","explanation_zh":"有限範圍覆蓋=(17, 16, 17)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121093,"parameters":{"through":16,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0094","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(17)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(18, 17, 18)","explanation_zh":"有限範圍覆蓋=(18, 17, 18)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121094,"parameters":{"through":17,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0095","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(18)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(19, 18, 19)","explanation_zh":"有限範圍覆蓋=(19, 18, 19)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121095,"parameters":{"through":18,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0096","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(19)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(20, 19, 20)","explanation_zh":"有限範圍覆蓋=(20, 19, 20)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121096,"parameters":{"through":19,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0097","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(20)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(21, 20, 21)","explanation_zh":"有限範圍覆蓋=(21, 20, 21)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121097,"parameters":{"through":20,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0098","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(21)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(22, 21, 22)","explanation_zh":"有限範圍覆蓋=(22, 21, 22)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121098,"parameters":{"through":21,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0099","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(22)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(23, 22, 23)","explanation_zh":"有限範圍覆蓋=(23, 22, 23)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121099,"parameters":{"through":22,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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-20261012-r01-mathematical-induction-0100","batch_id":"batch-20261008-advance-20261012-r01-mathematical-induction-foundations","language":"zh-Hant","source_mko_ids":["mko-mathematical-induction"],"title_zh":"有限已檢範圍的覆蓋與下一目標","stem_zh":"N從0起；有限歸納協議；已逐項檢查P(0)到P(23)。依序列 (所含P索引數, 相鄰步數, 下一未覆蓋目標索引)。","answer":{"kind":"expression","value":"(24, 23, 24)","explanation_zh":"有限範圍覆蓋=(24, 23, 24)。含0的索引數比末端多一，相鄰步數等於末端；仍有後續索引，有限檢查非普遍證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限已檢範圍的覆蓋與下一目標"]},"variant":{"family_id":"qf-density12-induction-range-obligations","seed":202610121100,"parameters":{"through":23,"operation":"density12_induction_range_obligations","domain":"finite_induction_protocol","semantic_template":"qf-density12-induction-range-obligations/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"}}