{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0001","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=0，允許索引=[]；只在這個有限域給定a_n=1+(2)n。原序候選索引=[-1, 0, 1]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(0, [否, 否, 否], [不合索引, 不合索引, 不合索引])","explanation_zh":"有限數列索引資格與值=(0, [否, 否, 否], [不合索引, 不合索引, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155001,"parameters":{"s":1,"d":2,"L":0,"queries":[-1,0,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[0,[[\"untyped\"],[\"untyped\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0002","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=1，允許索引=[0]；只在這個有限域給定a_n=0+(0)n。原序候選索引=[0, 1, -1]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(1, [是, 否, 否], [0, 不合索引, 不合索引])","explanation_zh":"有限數列索引資格與值=(1, [是, 否, 否], [0, 不合索引, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155002,"parameters":{"s":0,"d":0,"L":1,"queries":[0,1,-1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[1,[[\"typed\",\"0\"],[\"untyped\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0003","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=-10+(1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [-10, -7, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [-10, -7, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155003,"parameters":{"s":-10,"d":1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-10\"],[\"typed\",\"-7\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0004","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=-9+(-1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [-9, -12, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [-9, -12, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155004,"parameters":{"s":-9,"d":-1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-9\"],[\"typed\",\"-12\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0005","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=-8+(0)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [-8, -8, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [-8, -8, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155005,"parameters":{"s":-8,"d":0,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-8\"],[\"typed\",\"-8\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0006","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=-7+(1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [-7, -4, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [-7, -4, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155006,"parameters":{"s":-7,"d":1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-7\"],[\"typed\",\"-4\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0007","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=-6+(-1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [-6, -9, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [-6, -9, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155007,"parameters":{"s":-6,"d":-1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-6\"],[\"typed\",\"-9\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0008","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=-5+(0)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [-5, -5, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [-5, -5, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155008,"parameters":{"s":-5,"d":0,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-5\"],[\"typed\",\"-5\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0009","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=-4+(1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [-4, -1, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [-4, -1, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155009,"parameters":{"s":-4,"d":1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-4\"],[\"typed\",\"-1\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0010","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=-3+(-1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [-3, -6, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [-3, -6, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155010,"parameters":{"s":-3,"d":-1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-3\"],[\"typed\",\"-6\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0011","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=-2+(0)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [-2, -2, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [-2, -2, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155011,"parameters":{"s":-2,"d":0,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-2\"],[\"typed\",\"-2\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0012","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=-1+(1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [-1, 2, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [-1, 2, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155012,"parameters":{"s":-1,"d":1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"-1\"],[\"typed\",\"2\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0013","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=0+(-1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [0, -3, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [0, -3, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155013,"parameters":{"s":0,"d":-1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"0\"],[\"typed\",\"-3\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0014","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=1+(0)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [1, 1, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [1, 1, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155014,"parameters":{"s":1,"d":0,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"1\"],[\"typed\",\"1\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0015","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=2+(1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [2, 5, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [2, 5, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155015,"parameters":{"s":2,"d":1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"2\"],[\"typed\",\"5\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0016","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=3+(-1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [3, 0, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [3, 0, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155016,"parameters":{"s":3,"d":-1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"3\"],[\"typed\",\"0\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0017","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=4+(0)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [4, 4, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [4, 4, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155017,"parameters":{"s":4,"d":0,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"4\"],[\"typed\",\"4\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0018","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=5+(1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [5, 8, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [5, 8, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155018,"parameters":{"s":5,"d":1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"5\"],[\"typed\",\"8\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0019","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=6+(-1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [6, 3, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [6, 3, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155019,"parameters":{"s":6,"d":-1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"6\"],[\"typed\",\"3\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0020","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=7+(0)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [7, 7, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [7, 7, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155020,"parameters":{"s":7,"d":0,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"7\"],[\"typed\",\"7\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0021","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=8+(1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [8, 11, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [8, 11, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155021,"parameters":{"s":8,"d":1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"8\"],[\"typed\",\"11\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0022","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=9+(-1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [9, 6, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [9, 6, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155022,"parameters":{"s":9,"d":-1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"9\"],[\"typed\",\"6\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0023","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=10+(0)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [10, 10, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [10, 10, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155023,"parameters":{"s":10,"d":0,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"10\"],[\"typed\",\"10\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0024","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=11+(1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [11, 14, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [11, 14, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155024,"parameters":{"s":11,"d":1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"11\"],[\"typed\",\"14\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0025","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"有限數列的索引邊界","stem_zh":"N從0起，明示有限索引。L=4，允許索引=[0, 1, 2, 3]；只在這個有限域給定a_n=12+(-1)n。原序候選索引=[0, 3, 4]。求 (L, 索引資格, 合法項值或不合索引)。","answer":{"kind":"expression","value":"(4, [是, 是, 否], [12, 9, 不合索引])","explanation_zh":"有限數列索引資格與值=(4, [是, 是, 否], [12, 9, 不合索引])。先核0≤索引<L，只有合格索引才代入項公式，域外不冒稱已有項值。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限數列的索引邊界"]},"variant":{"family_id":"qf-density15-sequence-window-boundary-queries","seed":202610155025,"parameters":{"s":12,"d":-1,"L":4,"queries":[0,3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_window_boundary_queries","semantic_template":"qf-density15-sequence-window-boundary-queries/[4,[[\"typed\",\"12\"],[\"typed\",\"9\"],[\"untyped\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0026","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=0，a_(n+1)=a_n+(1)，窗口start=0、length=0。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([], [], [])","explanation_zh":"明示遞迴窗口與差列=([], [], [])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155026,"parameters":{"s":0,"d":1,"start":0,"length":0,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[\"empty_window\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0027","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=0，a_(n+1)=a_n+(7)，窗口start=0、length=1。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([0], [0], [])","explanation_zh":"明示遞迴窗口與差列=([0], [0], [])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155027,"parameters":{"s":0,"d":7,"start":0,"length":1,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[0],[\"0\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0028","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=-10，a_(n+1)=a_n+(1)，窗口start=0、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([0, 1, 2], [-10, -9, -8], [1, 1])","explanation_zh":"明示遞迴窗口與差列=([0, 1, 2], [-10, -9, -8], [1, 1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155028,"parameters":{"s":-10,"d":1,"start":0,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[0,1,2],[\"-10\",\"-9\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0029","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=-9，a_(n+1)=a_n+(-1)，窗口start=1、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([1, 2, 3], [-10, -11, -12], [-1, -1])","explanation_zh":"明示遞迴窗口與差列=([1, 2, 3], [-10, -11, -12], [-1, -1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155029,"parameters":{"s":-9,"d":-1,"start":1,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[1,2,3],[\"-10\",\"-11\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0030","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=-8，a_(n+1)=a_n+(0)，窗口start=2、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([2, 3, 4], [-8, -8, -8], [0, 0])","explanation_zh":"明示遞迴窗口與差列=([2, 3, 4], [-8, -8, -8], [0, 0])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155030,"parameters":{"s":-8,"d":0,"start":2,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[2,3,4],[\"-8\",\"-8\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0031","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=-7，a_(n+1)=a_n+(1)，窗口start=3、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([3, 4, 5], [-4, -3, -2], [1, 1])","explanation_zh":"明示遞迴窗口與差列=([3, 4, 5], [-4, -3, -2], [1, 1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155031,"parameters":{"s":-7,"d":1,"start":3,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[3,4,5],[\"-4\",\"-3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0032","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=-6，a_(n+1)=a_n+(-1)，窗口start=4、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([4, 5, 6], [-10, -11, -12], [-1, -1])","explanation_zh":"明示遞迴窗口與差列=([4, 5, 6], [-10, -11, -12], [-1, -1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155032,"parameters":{"s":-6,"d":-1,"start":4,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[4,5,6],[\"-10\",\"-11\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0033","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=-5，a_(n+1)=a_n+(0)，窗口start=5、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([5, 6, 7], [-5, -5, -5], [0, 0])","explanation_zh":"明示遞迴窗口與差列=([5, 6, 7], [-5, -5, -5], [0, 0])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155033,"parameters":{"s":-5,"d":0,"start":5,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[5,6,7],[\"-5\",\"-5\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0034","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=-4，a_(n+1)=a_n+(1)，窗口start=6、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([6, 7, 8], [2, 3, 4], [1, 1])","explanation_zh":"明示遞迴窗口與差列=([6, 7, 8], [2, 3, 4], [1, 1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155034,"parameters":{"s":-4,"d":1,"start":6,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[6,7,8],[\"2\",\"3\",\"4\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0035","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=-3，a_(n+1)=a_n+(-1)，窗口start=7、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([7, 8, 9], [-10, -11, -12], [-1, -1])","explanation_zh":"明示遞迴窗口與差列=([7, 8, 9], [-10, -11, -12], [-1, -1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155035,"parameters":{"s":-3,"d":-1,"start":7,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[7,8,9],[\"-10\",\"-11\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0036","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=-2，a_(n+1)=a_n+(0)，窗口start=8、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([8, 9, 10], [-2, -2, -2], [0, 0])","explanation_zh":"明示遞迴窗口與差列=([8, 9, 10], [-2, -2, -2], [0, 0])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155036,"parameters":{"s":-2,"d":0,"start":8,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[8,9,10],[\"-2\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0037","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=-1，a_(n+1)=a_n+(1)，窗口start=9、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([9, 10, 11], [8, 9, 10], [1, 1])","explanation_zh":"明示遞迴窗口與差列=([9, 10, 11], [8, 9, 10], [1, 1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155037,"parameters":{"s":-1,"d":1,"start":9,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[9,10,11],[\"8\",\"9\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0038","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=0，a_(n+1)=a_n+(-1)，窗口start=10、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([10, 11, 12], [-10, -11, -12], [-1, -1])","explanation_zh":"明示遞迴窗口與差列=([10, 11, 12], [-10, -11, -12], [-1, -1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155038,"parameters":{"s":0,"d":-1,"start":10,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[10,11,12],[\"-10\",\"-11\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0039","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=1，a_(n+1)=a_n+(0)，窗口start=11、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([11, 12, 13], [1, 1, 1], [0, 0])","explanation_zh":"明示遞迴窗口與差列=([11, 12, 13], [1, 1, 1], [0, 0])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155039,"parameters":{"s":1,"d":0,"start":11,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[11,12,13],[\"1\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0040","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=2，a_(n+1)=a_n+(1)，窗口start=12、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([12, 13, 14], [14, 15, 16], [1, 1])","explanation_zh":"明示遞迴窗口與差列=([12, 13, 14], [14, 15, 16], [1, 1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155040,"parameters":{"s":2,"d":1,"start":12,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[12,13,14],[\"14\",\"15\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0041","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=3，a_(n+1)=a_n+(-1)，窗口start=13、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([13, 14, 15], [-10, -11, -12], [-1, -1])","explanation_zh":"明示遞迴窗口與差列=([13, 14, 15], [-10, -11, -12], [-1, -1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155041,"parameters":{"s":3,"d":-1,"start":13,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[13,14,15],[\"-10\",\"-11\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0042","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=4，a_(n+1)=a_n+(0)，窗口start=14、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([14, 15, 16], [4, 4, 4], [0, 0])","explanation_zh":"明示遞迴窗口與差列=([14, 15, 16], [4, 4, 4], [0, 0])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155042,"parameters":{"s":4,"d":0,"start":14,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[14,15,16],[\"4\",\"4\",\"4\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0043","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=5，a_(n+1)=a_n+(1)，窗口start=15、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([15, 16, 17], [20, 21, 22], [1, 1])","explanation_zh":"明示遞迴窗口與差列=([15, 16, 17], [20, 21, 22], [1, 1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155043,"parameters":{"s":5,"d":1,"start":15,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[15,16,17],[\"20\",\"21\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0044","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=6，a_(n+1)=a_n+(-1)，窗口start=16、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([16, 17, 18], [-10, -11, -12], [-1, -1])","explanation_zh":"明示遞迴窗口與差列=([16, 17, 18], [-10, -11, -12], [-1, -1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155044,"parameters":{"s":6,"d":-1,"start":16,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[16,17,18],[\"-10\",\"-11\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0045","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=7，a_(n+1)=a_n+(0)，窗口start=17、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([17, 18, 19], [7, 7, 7], [0, 0])","explanation_zh":"明示遞迴窗口與差列=([17, 18, 19], [7, 7, 7], [0, 0])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155045,"parameters":{"s":7,"d":0,"start":17,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[17,18,19],[\"7\",\"7\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0046","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=8，a_(n+1)=a_n+(1)，窗口start=18、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([18, 19, 20], [26, 27, 28], [1, 1])","explanation_zh":"明示遞迴窗口與差列=([18, 19, 20], [26, 27, 28], [1, 1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155046,"parameters":{"s":8,"d":1,"start":18,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[18,19,20],[\"26\",\"27\",\"28\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0047","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=9，a_(n+1)=a_n+(-1)，窗口start=19、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([19, 20, 21], [-10, -11, -12], [-1, -1])","explanation_zh":"明示遞迴窗口與差列=([19, 20, 21], [-10, -11, -12], [-1, -1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155047,"parameters":{"s":9,"d":-1,"start":19,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[19,20,21],[\"-10\",\"-11\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0048","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=10，a_(n+1)=a_n+(0)，窗口start=20、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([20, 21, 22], [10, 10, 10], [0, 0])","explanation_zh":"明示遞迴窗口與差列=([20, 21, 22], [10, 10, 10], [0, 0])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155048,"parameters":{"s":10,"d":0,"start":20,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[20,21,22],[\"10\",\"10\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0049","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=11，a_(n+1)=a_n+(1)，窗口start=21、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([21, 22, 23], [32, 33, 34], [1, 1])","explanation_zh":"明示遞迴窗口與差列=([21, 22, 23], [32, 33, 34], [1, 1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155049,"parameters":{"s":11,"d":1,"start":21,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[21,22,23],[\"32\",\"33\",\"34\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0050","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"明示遞迴的有限窗口","stem_zh":"N從0起，明示有限索引。a_0=12，a_(n+1)=a_n+(-1)，窗口start=22、length=3。求 (窗口索引, 項值, 相鄰差)，length=0答三個空列。","answer":{"kind":"expression","value":"([22, 23, 24], [-10, -11, -12], [-1, -1])","explanation_zh":"明示遞迴窗口與差列=([22, 23, 24], [-10, -11, -12], [-1, -1])。從已給初始項逐步加步量，抵達窗口才記錄項；保持索引次序。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示遞迴的有限窗口"]},"variant":{"family_id":"qf-density15-sequence-affine-recurrence-window","seed":202610155050,"parameters":{"s":12,"d":-1,"start":22,"length":3,"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_affine_recurrence_window","semantic_template":"qf-density15-sequence-affine-recurrence-window/[[22,23,24],[\"-10\",\"-11\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0051","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[0, 0, 0, 0]，兩個候選nexts=[0, 0]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([0, 0, 0, 0, 0], [0, 0, 0, 0, 0], 是, 是)","explanation_zh":"兩個有限延伸與同前綴=([0, 0, 0, 0, 0], [0, 0, 0, 0, 0], 是, 是)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155051,"parameters":{"prefix":[0,0,0,0],"nexts":[0,0],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[0,0,0,0],[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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0052","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[1, 1, 2, 2]，兩個候選nexts=[3, 4]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([1, 1, 2, 2, 3], [1, 1, 2, 2, 4], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([1, 1, 2, 2, 3], [1, 1, 2, 2, 4], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155052,"parameters":{"prefix":[1,1,2,2],"nexts":[3,4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[1,1,2,2],[3,4]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0053","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[-10, 0, -10, 1]，兩個候選nexts=[2, -1]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([-10, 0, -10, 1, 2], [-10, 0, -10, 1, -1], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([-10, 0, -10, 1, 2], [-10, 0, -10, 1, -1], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155053,"parameters":{"prefix":[-10,0,-10,1],"nexts":[2,-1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[-10,0,-10,1],[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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0054","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[-9, 0, -9, 1]，兩個候選nexts=[3, -2]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([-9, 0, -9, 1, 3], [-9, 0, -9, 1, -2], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([-9, 0, -9, 1, 3], [-9, 0, -9, 1, -2], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155054,"parameters":{"prefix":[-9,0,-9,1],"nexts":[3,-2],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[-9,0,-9,1],[3,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0055","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[-8, 0, -8, 1]，兩個候選nexts=[4, -3]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([-8, 0, -8, 1, 4], [-8, 0, -8, 1, -3], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([-8, 0, -8, 1, 4], [-8, 0, -8, 1, -3], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155055,"parameters":{"prefix":[-8,0,-8,1],"nexts":[4,-3],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[-8,0,-8,1],[4,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0056","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[-7, 0, -7, 1]，兩個候選nexts=[5, -4]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([-7, 0, -7, 1, 5], [-7, 0, -7, 1, -4], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([-7, 0, -7, 1, 5], [-7, 0, -7, 1, -4], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155056,"parameters":{"prefix":[-7,0,-7,1],"nexts":[5,-4],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[-7,0,-7,1],[5,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0057","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[-6, 0, -6, 1]，兩個候選nexts=[6, -5]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([-6, 0, -6, 1, 6], [-6, 0, -6, 1, -5], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([-6, 0, -6, 1, 6], [-6, 0, -6, 1, -5], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155057,"parameters":{"prefix":[-6,0,-6,1],"nexts":[6,-5],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[-6,0,-6,1],[6,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0058","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[-5, 0, -5, 1]，兩個候選nexts=[7, -6]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([-5, 0, -5, 1, 7], [-5, 0, -5, 1, -6], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([-5, 0, -5, 1, 7], [-5, 0, -5, 1, -6], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155058,"parameters":{"prefix":[-5,0,-5,1],"nexts":[7,-6],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[-5,0,-5,1],[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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0059","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[-4, 0, -4, 1]，兩個候選nexts=[8, -7]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([-4, 0, -4, 1, 8], [-4, 0, -4, 1, -7], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([-4, 0, -4, 1, 8], [-4, 0, -4, 1, -7], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155059,"parameters":{"prefix":[-4,0,-4,1],"nexts":[8,-7],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[-4,0,-4,1],[8,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0060","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[-3, 0, -3, 1]，兩個候選nexts=[9, -8]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([-3, 0, -3, 1, 9], [-3, 0, -3, 1, -8], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([-3, 0, -3, 1, 9], [-3, 0, -3, 1, -8], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155060,"parameters":{"prefix":[-3,0,-3,1],"nexts":[9,-8],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[-3,0,-3,1],[9,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0061","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[-2, 0, -2, 1]，兩個候選nexts=[10, -9]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([-2, 0, -2, 1, 10], [-2, 0, -2, 1, -9], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([-2, 0, -2, 1, 10], [-2, 0, -2, 1, -9], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155061,"parameters":{"prefix":[-2,0,-2,1],"nexts":[10,-9],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[-2,0,-2,1],[10,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0062","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[-1, 0, -1, 1]，兩個候選nexts=[11, -10]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([-1, 0, -1, 1, 11], [-1, 0, -1, 1, -10], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([-1, 0, -1, 1, 11], [-1, 0, -1, 1, -10], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155062,"parameters":{"prefix":[-1,0,-1,1],"nexts":[11,-10],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[-1,0,-1,1],[11,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0063","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[0, 0, 0, 1]，兩個候選nexts=[12, -11]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([0, 0, 0, 1, 12], [0, 0, 0, 1, -11], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([0, 0, 0, 1, 12], [0, 0, 0, 1, -11], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155063,"parameters":{"prefix":[0,0,0,1],"nexts":[12,-11],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[0,0,0,1],[12,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0064","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[1, 0, 1, 1]，兩個候選nexts=[13, -12]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([1, 0, 1, 1, 13], [1, 0, 1, 1, -12], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([1, 0, 1, 1, 13], [1, 0, 1, 1, -12], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155064,"parameters":{"prefix":[1,0,1,1],"nexts":[13,-12],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[1,0,1,1],[13,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0065","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[2, 0, 2, 1]，兩個候選nexts=[14, -13]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([2, 0, 2, 1, 14], [2, 0, 2, 1, -13], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([2, 0, 2, 1, 14], [2, 0, 2, 1, -13], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155065,"parameters":{"prefix":[2,0,2,1],"nexts":[14,-13],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[2,0,2,1],[14,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0066","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[3, 0, 3, 1]，兩個候選nexts=[15, -14]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([3, 0, 3, 1, 15], [3, 0, 3, 1, -14], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([3, 0, 3, 1, 15], [3, 0, 3, 1, -14], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155066,"parameters":{"prefix":[3,0,3,1],"nexts":[15,-14],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[3,0,3,1],[15,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0067","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[4, 0, 4, 1]，兩個候選nexts=[16, -15]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([4, 0, 4, 1, 16], [4, 0, 4, 1, -15], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([4, 0, 4, 1, 16], [4, 0, 4, 1, -15], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155067,"parameters":{"prefix":[4,0,4,1],"nexts":[16,-15],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[4,0,4,1],[16,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0068","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[5, 0, 5, 1]，兩個候選nexts=[17, -16]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([5, 0, 5, 1, 17], [5, 0, 5, 1, -16], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([5, 0, 5, 1, 17], [5, 0, 5, 1, -16], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155068,"parameters":{"prefix":[5,0,5,1],"nexts":[17,-16],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[5,0,5,1],[17,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0069","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[6, 0, 6, 1]，兩個候選nexts=[18, -17]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([6, 0, 6, 1, 18], [6, 0, 6, 1, -17], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([6, 0, 6, 1, 18], [6, 0, 6, 1, -17], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155069,"parameters":{"prefix":[6,0,6,1],"nexts":[18,-17],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[6,0,6,1],[18,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0070","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[7, 0, 7, 1]，兩個候選nexts=[19, -18]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([7, 0, 7, 1, 19], [7, 0, 7, 1, -18], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([7, 0, 7, 1, 19], [7, 0, 7, 1, -18], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155070,"parameters":{"prefix":[7,0,7,1],"nexts":[19,-18],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[7,0,7,1],[19,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0071","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[8, 0, 8, 1]，兩個候選nexts=[20, -19]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([8, 0, 8, 1, 20], [8, 0, 8, 1, -19], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([8, 0, 8, 1, 20], [8, 0, 8, 1, -19], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155071,"parameters":{"prefix":[8,0,8,1],"nexts":[20,-19],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[8,0,8,1],[20,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0072","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[9, 0, 9, 1]，兩個候選nexts=[21, -20]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([9, 0, 9, 1, 21], [9, 0, 9, 1, -20], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([9, 0, 9, 1, 21], [9, 0, 9, 1, -20], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155072,"parameters":{"prefix":[9,0,9,1],"nexts":[21,-20],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[9,0,9,1],[21,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0073","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[10, 0, 10, 1]，兩個候選nexts=[22, -21]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([10, 0, 10, 1, 22], [10, 0, 10, 1, -21], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([10, 0, 10, 1, 22], [10, 0, 10, 1, -21], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155073,"parameters":{"prefix":[10,0,10,1],"nexts":[22,-21],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[10,0,10,1],[22,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0074","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[11, 0, 11, 1]，兩個候選nexts=[23, -22]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([11, 0, 11, 1, 23], [11, 0, 11, 1, -22], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([11, 0, 11, 1, 23], [11, 0, 11, 1, -22], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155074,"parameters":{"prefix":[11,0,11,1],"nexts":[23,-22],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[11,0,11,1],[23,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0075","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"同前綴的兩個候選延伸","stem_zh":"N從0起，明示有限索引。已給原序四項prefix=[12, 0, 12, 1]，兩個候選nexts=[24, -23]。只附加各自第五項，求 (第一五項列, 第二五項列, 前四項相同資格, 第五項相同資格)。未指定通項。","answer":{"kind":"expression","value":"([12, 0, 12, 1, 24], [12, 0, 12, 1, -23], 是, 否)","explanation_zh":"兩個有限延伸與同前綴=([12, 0, 12, 1, 24], [12, 0, 12, 1, -23], 是, 否)。各附加一個候選後核對，不從前四項補出唯一規則。相同候選不當不同延伸。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同前綴的兩個候選延伸"]},"variant":{"family_id":"qf-density15-sequence-two-finite-extensions","seed":202610155075,"parameters":{"prefix":[12,0,12,1],"nexts":[24,-23],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_two_finite_extensions","semantic_template":"qf-density15-sequence-two-finite-extensions/[[12,0,12,1],[24,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0076","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[0, 0, 0, 0, 0]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([0, 0, 0, 0, 0], [0, 0, 0, 0], 是)","explanation_zh":"五項前綴和與有序差列=([0, 0, 0, 0, 0], [0, 0, 0, 0], 是)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155076,"parameters":{"values":[0,0,0,0,0],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[0,0,0,0,0]"}},"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0077","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[2, 2, 2, 2, 2]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([2, 4, 6, 8, 10], [0, 0, 0, 0], 是)","explanation_zh":"五項前綴和與有序差列=([2, 4, 6, 8, 10], [0, 0, 0, 0], 是)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155077,"parameters":{"values":[2,2,2,2,2],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[2,2,2,2,2]"}},"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0078","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[-2, -1, 0, 1, 2]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([-2, -3, -3, -2, 0], [1, 1, 1, 1], 否)","explanation_zh":"五項前綴和與有序差列=([-2, -3, -3, -2, 0], [1, 1, 1, 1], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155078,"parameters":{"values":[-2,-1,0,1,2],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-2,-1,0,1,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0079","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[-9, 0, -9, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([-9, -9, -18, -19, -18], [9, -9, 8, 2], 否)","explanation_zh":"五項前綴和與有序差列=([-9, -9, -18, -19, -18], [9, -9, 8, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155079,"parameters":{"values":[-9,0,-9,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-9,0,-9,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0080","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[-8, 0, -8, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([-8, -8, -16, -17, -16], [8, -8, 7, 2], 否)","explanation_zh":"五項前綴和與有序差列=([-8, -8, -16, -17, -16], [8, -8, 7, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155080,"parameters":{"values":[-8,0,-8,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-8,0,-8,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0081","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[-7, 0, -7, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([-7, -7, -14, -15, -14], [7, -7, 6, 2], 否)","explanation_zh":"五項前綴和與有序差列=([-7, -7, -14, -15, -14], [7, -7, 6, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155081,"parameters":{"values":[-7,0,-7,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-7,0,-7,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0082","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[-6, 0, -6, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([-6, -6, -12, -13, -12], [6, -6, 5, 2], 否)","explanation_zh":"五項前綴和與有序差列=([-6, -6, -12, -13, -12], [6, -6, 5, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155082,"parameters":{"values":[-6,0,-6,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-6,0,-6,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0083","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[-5, 0, -5, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([-5, -5, -10, -11, -10], [5, -5, 4, 2], 否)","explanation_zh":"五項前綴和與有序差列=([-5, -5, -10, -11, -10], [5, -5, 4, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155083,"parameters":{"values":[-5,0,-5,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-5,0,-5,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0084","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[-4, 0, -4, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([-4, -4, -8, -9, -8], [4, -4, 3, 2], 否)","explanation_zh":"五項前綴和與有序差列=([-4, -4, -8, -9, -8], [4, -4, 3, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155084,"parameters":{"values":[-4,0,-4,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-4,0,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0085","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[-3, 0, -3, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([-3, -3, -6, -7, -6], [3, -3, 2, 2], 否)","explanation_zh":"五項前綴和與有序差列=([-3, -3, -6, -7, -6], [3, -3, 2, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155085,"parameters":{"values":[-3,0,-3,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-3,0,-3,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0086","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[-2, 0, -2, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([-2, -2, -4, -5, -4], [2, -2, 1, 2], 否)","explanation_zh":"五項前綴和與有序差列=([-2, -2, -4, -5, -4], [2, -2, 1, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155086,"parameters":{"values":[-2,0,-2,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-2,0,-2,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0087","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[-1, 0, -1, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([-1, -1, -2, -3, -2], [1, -1, 0, 2], 否)","explanation_zh":"五項前綴和與有序差列=([-1, -1, -2, -3, -2], [1, -1, 0, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155087,"parameters":{"values":[-1,0,-1,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[-1,0,-1,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0088","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[0, 0, 0, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([0, 0, 0, -1, 0], [0, 0, -1, 2], 否)","explanation_zh":"五項前綴和與有序差列=([0, 0, 0, -1, 0], [0, 0, -1, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155088,"parameters":{"values":[0,0,0,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[0,0,0,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0089","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[1, 0, 1, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([1, 1, 2, 1, 2], [-1, 1, -2, 2], 否)","explanation_zh":"五項前綴和與有序差列=([1, 1, 2, 1, 2], [-1, 1, -2, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155089,"parameters":{"values":[1,0,1,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[1,0,1,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0090","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[2, 0, 2, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([2, 2, 4, 3, 4], [-2, 2, -3, 2], 否)","explanation_zh":"五項前綴和與有序差列=([2, 2, 4, 3, 4], [-2, 2, -3, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155090,"parameters":{"values":[2,0,2,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[2,0,2,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0091","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[3, 0, 3, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([3, 3, 6, 5, 6], [-3, 3, -4, 2], 否)","explanation_zh":"五項前綴和與有序差列=([3, 3, 6, 5, 6], [-3, 3, -4, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155091,"parameters":{"values":[3,0,3,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[3,0,3,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0092","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[4, 0, 4, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([4, 4, 8, 7, 8], [-4, 4, -5, 2], 否)","explanation_zh":"五項前綴和與有序差列=([4, 4, 8, 7, 8], [-4, 4, -5, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155092,"parameters":{"values":[4,0,4,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[4,0,4,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0093","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[5, 0, 5, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([5, 5, 10, 9, 10], [-5, 5, -6, 2], 否)","explanation_zh":"五項前綴和與有序差列=([5, 5, 10, 9, 10], [-5, 5, -6, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155093,"parameters":{"values":[5,0,5,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[5,0,5,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0094","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[6, 0, 6, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([6, 6, 12, 11, 12], [-6, 6, -7, 2], 否)","explanation_zh":"五項前綴和與有序差列=([6, 6, 12, 11, 12], [-6, 6, -7, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155094,"parameters":{"values":[6,0,6,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[6,0,6,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0095","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[7, 0, 7, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([7, 7, 14, 13, 14], [-7, 7, -8, 2], 否)","explanation_zh":"五項前綴和與有序差列=([7, 7, 14, 13, 14], [-7, 7, -8, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155095,"parameters":{"values":[7,0,7,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[7,0,7,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0096","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[8, 0, 8, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([8, 8, 16, 15, 16], [-8, 8, -9, 2], 否)","explanation_zh":"五項前綴和與有序差列=([8, 8, 16, 15, 16], [-8, 8, -9, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155096,"parameters":{"values":[8,0,8,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[8,0,8,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0097","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[9, 0, 9, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([9, 9, 18, 17, 18], [-9, 9, -10, 2], 否)","explanation_zh":"五項前綴和與有序差列=([9, 9, 18, 17, 18], [-9, 9, -10, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155097,"parameters":{"values":[9,0,9,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[9,0,9,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0098","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[10, 0, 10, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([10, 10, 20, 19, 20], [-10, 10, -11, 2], 否)","explanation_zh":"五項前綴和與有序差列=([10, 10, 20, 19, 20], [-10, 10, -11, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155098,"parameters":{"values":[10,0,10,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[10,0,10,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0099","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[11, 0, 11, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([11, 11, 22, 21, 22], [-11, 11, -12, 2], 否)","explanation_zh":"五項前綴和與有序差列=([11, 11, 22, 21, 22], [-11, 11, -12, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155099,"parameters":{"values":[11,0,11,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[11,0,11,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r05-sequence-0100","batch_id":"batch-20261011-advance-20261015-r05-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-sequence"],"title_zh":"五項累加與有序差","stem_zh":"N從0起，明示有限索引。原序五項=[12, 0, 12, -1, 1]。求 (五個截至各位置的前綴和, 四個後項減前項, 五值皆相同資格)。","answer":{"kind":"expression","value":"([12, 12, 24, 23, 24], [-12, 12, -13, 2], 否)","explanation_zh":"五項前綴和與有序差列=([12, 12, 24, 23, 24], [-12, 12, -13, 2], 否)。逐項累加及逐鄰相減，不去重、不排序。有限前綴和不是無窮級數或極限。數列次序與重複保留，有限前綴不推定唯一無限規則。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["五項累加與有序差"]},"variant":{"family_id":"qf-density15-sequence-five-term-prefix-deltas","seed":202610155100,"parameters":{"values":[12,0,12,-1,1],"domain":"finite_zero_based_sequence_protocol","operation":"density15_sequence_five_term_prefix_deltas","semantic_template":"qf-density15-sequence-five-term-prefix-deltas/[12,0,12,-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}