{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0001","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=0。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(1, 1, 1)","explanation_zh":"相鄰階乘與精確比=(1, 1, 1)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145001,"parameters":{"n":0,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0002","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=1。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(1, 2, 2)","explanation_zh":"相鄰階乘與精確比=(1, 2, 2)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145002,"parameters":{"n":1,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0003","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=2。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(2, 6, 3)","explanation_zh":"相鄰階乘與精確比=(2, 6, 3)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145003,"parameters":{"n":2,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0004","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=3。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(6, 24, 4)","explanation_zh":"相鄰階乘與精確比=(6, 24, 4)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145004,"parameters":{"n":3,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0005","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=4。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(24, 120, 5)","explanation_zh":"相鄰階乘與精確比=(24, 120, 5)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145005,"parameters":{"n":4,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0006","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=5。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(120, 720, 6)","explanation_zh":"相鄰階乘與精確比=(120, 720, 6)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145006,"parameters":{"n":5,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0007","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=6。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(720, 5040, 7)","explanation_zh":"相鄰階乘與精確比=(720, 5040, 7)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145007,"parameters":{"n":6,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0008","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=7。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(5040, 40320, 8)","explanation_zh":"相鄰階乘與精確比=(5040, 40320, 8)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145008,"parameters":{"n":7,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0009","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=8。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(40320, 362880, 9)","explanation_zh":"相鄰階乘與精確比=(40320, 362880, 9)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145009,"parameters":{"n":8,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0010","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=9。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(362880, 3628800, 10)","explanation_zh":"相鄰階乘與精確比=(362880, 3628800, 10)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145010,"parameters":{"n":9,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0011","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=10。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(3628800, 39916800, 11)","explanation_zh":"相鄰階乘與精確比=(3628800, 39916800, 11)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145011,"parameters":{"n":10,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0012","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=11。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(39916800, 479001600, 12)","explanation_zh":"相鄰階乘與精確比=(39916800, 479001600, 12)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145012,"parameters":{"n":11,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0013","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=12。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(479001600, 6227020800, 13)","explanation_zh":"相鄰階乘與精確比=(479001600, 6227020800, 13)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145013,"parameters":{"n":12,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0014","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=13。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(6227020800, 87178291200, 14)","explanation_zh":"相鄰階乘與精確比=(6227020800, 87178291200, 14)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145014,"parameters":{"n":13,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0015","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=14。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(87178291200, 1307674368000, 15)","explanation_zh":"相鄰階乘與精確比=(87178291200, 1307674368000, 15)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145015,"parameters":{"n":14,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0016","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=15。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(1307674368000, 20922789888000, 16)","explanation_zh":"相鄰階乘與精確比=(1307674368000, 20922789888000, 16)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145016,"parameters":{"n":15,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0017","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=16。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(20922789888000, 355687428096000, 17)","explanation_zh":"相鄰階乘與精確比=(20922789888000, 355687428096000, 17)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145017,"parameters":{"n":16,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0018","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=17。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(355687428096000, 6402373705728000, 18)","explanation_zh":"相鄰階乘與精確比=(355687428096000, 6402373705728000, 18)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145018,"parameters":{"n":17,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0019","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=18。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(6402373705728000, 121645100408832000, 19)","explanation_zh":"相鄰階乘與精確比=(6402373705728000, 121645100408832000, 19)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145019,"parameters":{"n":18,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0020","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=19。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(121645100408832000, 2432902008176640000, 20)","explanation_zh":"相鄰階乘與精確比=(121645100408832000, 2432902008176640000, 20)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145020,"parameters":{"n":19,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0021","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=20。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(2432902008176640000, 51090942171709440000, 21)","explanation_zh":"相鄰階乘與精確比=(2432902008176640000, 51090942171709440000, 21)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145021,"parameters":{"n":20,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0022","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=21。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(51090942171709440000, 1124000727777607680000, 22)","explanation_zh":"相鄰階乘與精確比=(51090942171709440000, 1124000727777607680000, 22)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145022,"parameters":{"n":21,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0023","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=22。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(1124000727777607680000, 25852016738884976640000, 23)","explanation_zh":"相鄰階乘與精確比=(1124000727777607680000, 25852016738884976640000, 23)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145023,"parameters":{"n":22,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0024","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=23。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(25852016738884976640000, 620448401733239439360000, 24)","explanation_zh":"相鄰階乘與精確比=(25852016738884976640000, 620448401733239439360000, 24)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145024,"parameters":{"n":23,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0025","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"相鄰階乘與精確比","stem_zh":"N從0起；BigInt精確階乘；n=24。依序求 (F(n), F(n+1), F(n+1)/F(n)) 的十進整數，F(0)=1。","answer":{"kind":"expression","value":"(620448401733239439360000, 15511210043330985984000000, 25)","explanation_zh":"相鄰階乘與精確比=(620448401733239439360000, 15511210043330985984000000, 25)。從乘積遞迴計算兩值，再用整數精確除法求比。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["相鄰階乘與精確比"]},"variant":{"family_id":"qf-density14-factorial-adjacent-ratio","seed":202610145025,"parameters":{"n":24,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_adjacent_ratio","semantic_template":"qf-density14-factorial-adjacent-ratio/24"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0026","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=0≤hi=0≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(1, [], 是)","explanation_zh":"階乘窗口乘積與因子列=(1, [], 是)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145026,"parameters":{"lo":0,"hi":0,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0027","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=0≤hi=1≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(1, [1], 否)","explanation_zh":"階乘窗口乘積與因子列=(1, [1], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145027,"parameters":{"lo":0,"hi":1,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[0,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0028","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=0≤hi=20≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(2432902008176640000, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], 否)","explanation_zh":"階乘窗口乘積與因子列=(2432902008176640000, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145028,"parameters":{"lo":0,"hi":20,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[0,20]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0029","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=1≤hi=3≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(6, [2, 3], 否)","explanation_zh":"階乘窗口乘積與因子列=(6, [2, 3], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145029,"parameters":{"lo":1,"hi":3,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[1,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0030","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=2≤hi=5≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(60, [3, 4, 5], 否)","explanation_zh":"階乘窗口乘積與因子列=(60, [3, 4, 5], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145030,"parameters":{"lo":2,"hi":5,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[2,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0031","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=3≤hi=7≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(840, [4, 5, 6, 7], 否)","explanation_zh":"階乘窗口乘積與因子列=(840, [4, 5, 6, 7], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145031,"parameters":{"lo":3,"hi":7,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[3,7]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0032","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=4≤hi=6≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(30, [5, 6], 否)","explanation_zh":"階乘窗口乘積與因子列=(30, [5, 6], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145032,"parameters":{"lo":4,"hi":6,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[4,6]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0033","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=5≤hi=8≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(336, [6, 7, 8], 否)","explanation_zh":"階乘窗口乘積與因子列=(336, [6, 7, 8], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145033,"parameters":{"lo":5,"hi":8,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[5,8]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0034","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=6≤hi=10≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(5040, [7, 8, 9, 10], 否)","explanation_zh":"階乘窗口乘積與因子列=(5040, [7, 8, 9, 10], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145034,"parameters":{"lo":6,"hi":10,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[6,10]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0035","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=7≤hi=9≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(72, [8, 9], 否)","explanation_zh":"階乘窗口乘積與因子列=(72, [8, 9], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145035,"parameters":{"lo":7,"hi":9,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[7,9]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0036","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=8≤hi=11≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(990, [9, 10, 11], 否)","explanation_zh":"階乘窗口乘積與因子列=(990, [9, 10, 11], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145036,"parameters":{"lo":8,"hi":11,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[8,11]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0037","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=9≤hi=13≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(17160, [10, 11, 12, 13], 否)","explanation_zh":"階乘窗口乘積與因子列=(17160, [10, 11, 12, 13], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145037,"parameters":{"lo":9,"hi":13,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[9,13]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0038","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=10≤hi=12≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(132, [11, 12], 否)","explanation_zh":"階乘窗口乘積與因子列=(132, [11, 12], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145038,"parameters":{"lo":10,"hi":12,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[10,12]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0039","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=11≤hi=14≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(2184, [12, 13, 14], 否)","explanation_zh":"階乘窗口乘積與因子列=(2184, [12, 13, 14], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145039,"parameters":{"lo":11,"hi":14,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[11,14]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0040","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=12≤hi=16≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(43680, [13, 14, 15, 16], 否)","explanation_zh":"階乘窗口乘積與因子列=(43680, [13, 14, 15, 16], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145040,"parameters":{"lo":12,"hi":16,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[12,16]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0041","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=13≤hi=15≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(210, [14, 15], 否)","explanation_zh":"階乘窗口乘積與因子列=(210, [14, 15], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145041,"parameters":{"lo":13,"hi":15,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[13,15]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0042","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=14≤hi=17≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(4080, [15, 16, 17], 否)","explanation_zh":"階乘窗口乘積與因子列=(4080, [15, 16, 17], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145042,"parameters":{"lo":14,"hi":17,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[14,17]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0043","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=15≤hi=19≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(93024, [16, 17, 18, 19], 否)","explanation_zh":"階乘窗口乘積與因子列=(93024, [16, 17, 18, 19], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145043,"parameters":{"lo":15,"hi":19,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[15,19]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0044","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=16≤hi=18≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(306, [17, 18], 否)","explanation_zh":"階乘窗口乘積與因子列=(306, [17, 18], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145044,"parameters":{"lo":16,"hi":18,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[16,18]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0045","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=17≤hi=20≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(6840, [18, 19, 20], 否)","explanation_zh":"階乘窗口乘積與因子列=(6840, [18, 19, 20], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145045,"parameters":{"lo":17,"hi":20,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[17,20]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0046","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=18≤hi=22≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(175560, [19, 20, 21, 22], 否)","explanation_zh":"階乘窗口乘積與因子列=(175560, [19, 20, 21, 22], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145046,"parameters":{"lo":18,"hi":22,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[18,22]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0047","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=19≤hi=21≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(420, [20, 21], 否)","explanation_zh":"階乘窗口乘積與因子列=(420, [20, 21], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145047,"parameters":{"lo":19,"hi":21,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[19,21]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0048","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=20≤hi=23≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(10626, [21, 22, 23], 否)","explanation_zh":"階乘窗口乘積與因子列=(10626, [21, 22, 23], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145048,"parameters":{"lo":20,"hi":23,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[20,23]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0049","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=21≤hi=25≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(303600, [22, 23, 24, 25], 否)","explanation_zh":"階乘窗口乘積與因子列=(303600, [22, 23, 24, 25], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145049,"parameters":{"lo":21,"hi":25,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[21,25]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0050","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"階乘窗口的乘積與因子索引","stem_zh":"N從0起；BigInt精確階乘；0≤lo=22≤hi=24≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。","answer":{"kind":"expression","value":"(552, [23, 24], 否)","explanation_zh":"階乘窗口乘積與因子列=(552, [23, 24], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["階乘窗口的乘積與因子索引"]},"variant":{"family_id":"qf-density14-factorial-window-product","seed":202610145050,"parameters":{"lo":22,"hi":24,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_window_product","semantic_template":"qf-density14-factorial-window-product/[22,24]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0051","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=0，候選十進字串u=1、v=1。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(1, 1, 0, 0, 是)","explanation_zh":"候選階乘雙見證與缺陷=(1, 1, 0, 0, 是)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145051,"parameters":{"n":0,"u":"1","v":"1","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0052","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=3，候選十進字串u=12、v=48。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(6, 24, 6, 0, 否)","explanation_zh":"候選階乘雙見證與缺陷=(6, 24, 6, 0, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145052,"parameters":{"n":3,"u":"12","v":"48","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[3,\"12\",\"48\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0053","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=2，候選十進字串u=2、v=7。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(2, 6, 0, 1, 否)","explanation_zh":"候選階乘雙見證與缺陷=(2, 6, 0, 1, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145053,"parameters":{"n":2,"u":"2","v":"7","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[2,\"2\",\"7\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0054","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=3，候選十進字串u=6、v=24。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(6, 24, 0, 0, 是)","explanation_zh":"候選階乘雙見證與缺陷=(6, 24, 0, 0, 是)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145054,"parameters":{"n":3,"u":"6","v":"24","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[3,\"6\",\"24\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0055","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=4，候選十進字串u=25、v=125。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(24, 120, 1, 0, 否)","explanation_zh":"候選階乘雙見證與缺陷=(24, 120, 1, 0, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145055,"parameters":{"n":4,"u":"25","v":"125","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[4,\"25\",\"125\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0056","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=5，候選十進字串u=120、v=721。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(120, 720, 0, 1, 否)","explanation_zh":"候選階乘雙見證與缺陷=(120, 720, 0, 1, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145056,"parameters":{"n":5,"u":"120","v":"721","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[5,\"120\",\"721\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0057","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=6，候選十進字串u=720、v=5040。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(720, 5040, 0, 0, 是)","explanation_zh":"候選階乘雙見證與缺陷=(720, 5040, 0, 0, 是)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145057,"parameters":{"n":6,"u":"720","v":"5040","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[6,\"720\",\"5040\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0058","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=7，候選十進字串u=5041、v=40328。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(5040, 40320, 1, 0, 否)","explanation_zh":"候選階乘雙見證與缺陷=(5040, 40320, 1, 0, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145058,"parameters":{"n":7,"u":"5041","v":"40328","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[7,\"5041\",\"40328\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0059","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=8，候選十進字串u=40320、v=362881。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(40320, 362880, 0, 1, 否)","explanation_zh":"候選階乘雙見證與缺陷=(40320, 362880, 0, 1, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145059,"parameters":{"n":8,"u":"40320","v":"362881","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[8,\"40320\",\"362881\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0060","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=9，候選十進字串u=362880、v=3628800。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(362880, 3628800, 0, 0, 是)","explanation_zh":"候選階乘雙見證與缺陷=(362880, 3628800, 0, 0, 是)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145060,"parameters":{"n":9,"u":"362880","v":"3628800","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[9,\"362880\",\"3628800\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0061","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=10，候選十進字串u=3628801、v=39916811。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(3628800, 39916800, 1, 0, 否)","explanation_zh":"候選階乘雙見證與缺陷=(3628800, 39916800, 1, 0, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145061,"parameters":{"n":10,"u":"3628801","v":"39916811","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[10,\"3628801\",\"39916811\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0062","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=11，候選十進字串u=39916800、v=479001601。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(39916800, 479001600, 0, 1, 否)","explanation_zh":"候選階乘雙見證與缺陷=(39916800, 479001600, 0, 1, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145062,"parameters":{"n":11,"u":"39916800","v":"479001601","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[11,\"39916800\",\"479001601\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0063","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=12，候選十進字串u=479001600、v=6227020800。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(479001600, 6227020800, 0, 0, 是)","explanation_zh":"候選階乘雙見證與缺陷=(479001600, 6227020800, 0, 0, 是)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145063,"parameters":{"n":12,"u":"479001600","v":"6227020800","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[12,\"479001600\",\"6227020800\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0064","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=13，候選十進字串u=6227020801、v=87178291214。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(6227020800, 87178291200, 1, 0, 否)","explanation_zh":"候選階乘雙見證與缺陷=(6227020800, 87178291200, 1, 0, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145064,"parameters":{"n":13,"u":"6227020801","v":"87178291214","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[13,\"6227020801\",\"87178291214\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0065","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=14，候選十進字串u=87178291200、v=1307674368001。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(87178291200, 1307674368000, 0, 1, 否)","explanation_zh":"候選階乘雙見證與缺陷=(87178291200, 1307674368000, 0, 1, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145065,"parameters":{"n":14,"u":"87178291200","v":"1307674368001","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[14,\"87178291200\",\"1307674368001\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0066","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=15，候選十進字串u=1307674368000、v=20922789888000。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(1307674368000, 20922789888000, 0, 0, 是)","explanation_zh":"候選階乘雙見證與缺陷=(1307674368000, 20922789888000, 0, 0, 是)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145066,"parameters":{"n":15,"u":"1307674368000","v":"20922789888000","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[15,\"1307674368000\",\"20922789888000\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0067","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=16，候選十進字串u=20922789888001、v=355687428096017。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(20922789888000, 355687428096000, 1, 0, 否)","explanation_zh":"候選階乘雙見證與缺陷=(20922789888000, 355687428096000, 1, 0, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145067,"parameters":{"n":16,"u":"20922789888001","v":"355687428096017","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[16,\"20922789888001\",\"355687428096017\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0068","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=17，候選十進字串u=355687428096000、v=6402373705728001。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(355687428096000, 6402373705728000, 0, 1, 否)","explanation_zh":"候選階乘雙見證與缺陷=(355687428096000, 6402373705728000, 0, 1, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145068,"parameters":{"n":17,"u":"355687428096000","v":"6402373705728001","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[17,\"355687428096000\",\"6402373705728001\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0069","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=18，候選十進字串u=6402373705728000、v=121645100408832000。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(6402373705728000, 121645100408832000, 0, 0, 是)","explanation_zh":"候選階乘雙見證與缺陷=(6402373705728000, 121645100408832000, 0, 0, 是)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145069,"parameters":{"n":18,"u":"6402373705728000","v":"121645100408832000","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[18,\"6402373705728000\",\"121645100408832000\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0070","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=19，候選十進字串u=121645100408832001、v=2432902008176640020。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(121645100408832000, 2432902008176640000, 1, 0, 否)","explanation_zh":"候選階乘雙見證與缺陷=(121645100408832000, 2432902008176640000, 1, 0, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145070,"parameters":{"n":19,"u":"121645100408832001","v":"2432902008176640020","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[19,\"121645100408832001\",\"2432902008176640020\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0071","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=20，候選十進字串u=2432902008176640000、v=51090942171709440001。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(2432902008176640000, 51090942171709440000, 0, 1, 否)","explanation_zh":"候選階乘雙見證與缺陷=(2432902008176640000, 51090942171709440000, 0, 1, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145071,"parameters":{"n":20,"u":"2432902008176640000","v":"51090942171709440001","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[20,\"2432902008176640000\",\"51090942171709440001\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0072","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=21，候選十進字串u=51090942171709440000、v=1124000727777607680000。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(51090942171709440000, 1124000727777607680000, 0, 0, 是)","explanation_zh":"候選階乘雙見證與缺陷=(51090942171709440000, 1124000727777607680000, 0, 0, 是)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145072,"parameters":{"n":21,"u":"51090942171709440000","v":"1124000727777607680000","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[21,\"51090942171709440000\",\"1124000727777607680000\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0073","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=22，候選十進字串u=1124000727777607680001、v=25852016738884976640023。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(1124000727777607680000, 25852016738884976640000, 1, 0, 否)","explanation_zh":"候選階乘雙見證與缺陷=(1124000727777607680000, 25852016738884976640000, 1, 0, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145073,"parameters":{"n":22,"u":"1124000727777607680001","v":"25852016738884976640023","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[22,\"1124000727777607680001\",\"25852016738884976640023\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0074","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=23，候選十進字串u=25852016738884976640000、v=620448401733239439360001。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(25852016738884976640000, 620448401733239439360000, 0, 1, 否)","explanation_zh":"候選階乘雙見證與缺陷=(25852016738884976640000, 620448401733239439360000, 0, 1, 否)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145074,"parameters":{"n":23,"u":"25852016738884976640000","v":"620448401733239439360001","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[23,\"25852016738884976640000\",\"620448401733239439360001\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0075","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"候選階乘報告的兩個見證","stem_zh":"N從0起；BigInt精確階乘；n=24，候選十進字串u=620448401733239439360000、v=15511210043330985984000000。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。","answer":{"kind":"expression","value":"(620448401733239439360000, 15511210043330985984000000, 0, 0, 是)","explanation_zh":"候選階乘雙見證與缺陷=(620448401733239439360000, 15511210043330985984000000, 0, 0, 是)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選階乘報告的兩個見證"]},"variant":{"family_id":"qf-density14-factorial-reported-successor-audit","seed":202610145075,"parameters":{"n":24,"u":"620448401733239439360000","v":"15511210043330985984000000","domain":"finite_N_factorial_protocol","operation":"density14_factorial_reported_successor_audit","semantic_template":"qf-density14-factorial-reported-successor-audit/[24,\"620448401733239439360000\",\"15511210043330985984000000\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0076","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=1、k=1∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(1, 1, 0)","explanation_zh":"省去單因子乘積與回乘=(1, 1, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145076,"parameters":{"n":1,"k":1,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0077","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=19、k=19∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(121645100408832000, 6402373705728000, 0)","explanation_zh":"省去單因子乘積與回乘=(121645100408832000, 6402373705728000, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145077,"parameters":{"n":19,"k":19,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[19,19]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0078","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=4、k=3∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(24, 8, 0)","explanation_zh":"省去單因子乘積與回乘=(24, 8, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145078,"parameters":{"n":4,"k":3,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0079","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=5、k=4∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(120, 30, 0)","explanation_zh":"省去單因子乘積與回乘=(120, 30, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145079,"parameters":{"n":5,"k":4,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0080","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=6、k=5∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(720, 144, 0)","explanation_zh":"省去單因子乘積與回乘=(720, 144, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145080,"parameters":{"n":6,"k":5,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0081","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=7、k=6∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(5040, 840, 0)","explanation_zh":"省去單因子乘積與回乘=(5040, 840, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145081,"parameters":{"n":7,"k":6,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0082","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=8、k=7∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(40320, 5760, 0)","explanation_zh":"省去單因子乘積與回乘=(40320, 5760, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145082,"parameters":{"n":8,"k":7,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0083","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=9、k=8∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(362880, 45360, 0)","explanation_zh":"省去單因子乘積與回乘=(362880, 45360, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145083,"parameters":{"n":9,"k":8,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0084","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=10、k=9∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(3628800, 403200, 0)","explanation_zh":"省去單因子乘積與回乘=(3628800, 403200, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145084,"parameters":{"n":10,"k":9,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0085","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=11、k=10∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(39916800, 3991680, 0)","explanation_zh":"省去單因子乘積與回乘=(39916800, 3991680, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145085,"parameters":{"n":11,"k":10,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0086","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=12、k=11∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(479001600, 43545600, 0)","explanation_zh":"省去單因子乘積與回乘=(479001600, 43545600, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145086,"parameters":{"n":12,"k":11,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0087","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=13、k=12∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(6227020800, 518918400, 0)","explanation_zh":"省去單因子乘積與回乘=(6227020800, 518918400, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145087,"parameters":{"n":13,"k":12,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0088","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=14、k=13∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(87178291200, 6706022400, 0)","explanation_zh":"省去單因子乘積與回乘=(87178291200, 6706022400, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145088,"parameters":{"n":14,"k":13,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0089","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=15、k=14∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(1307674368000, 93405312000, 0)","explanation_zh":"省去單因子乘積與回乘=(1307674368000, 93405312000, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145089,"parameters":{"n":15,"k":14,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0090","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=16、k=15∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(20922789888000, 1394852659200, 0)","explanation_zh":"省去單因子乘積與回乘=(20922789888000, 1394852659200, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145090,"parameters":{"n":16,"k":15,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0091","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=17、k=16∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(355687428096000, 22230464256000, 0)","explanation_zh":"省去單因子乘積與回乘=(355687428096000, 22230464256000, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145091,"parameters":{"n":17,"k":16,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0092","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=18、k=17∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(6402373705728000, 376610217984000, 0)","explanation_zh":"省去單因子乘積與回乘=(6402373705728000, 376610217984000, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145092,"parameters":{"n":18,"k":17,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0093","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=19、k=18∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(121645100408832000, 6758061133824000, 0)","explanation_zh":"省去單因子乘積與回乘=(121645100408832000, 6758061133824000, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145093,"parameters":{"n":19,"k":18,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0094","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=20、k=19∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(2432902008176640000, 128047474114560000, 0)","explanation_zh":"省去單因子乘積與回乘=(2432902008176640000, 128047474114560000, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145094,"parameters":{"n":20,"k":19,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0095","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=21、k=20∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(51090942171709440000, 2554547108585472000, 0)","explanation_zh":"省去單因子乘積與回乘=(51090942171709440000, 2554547108585472000, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145095,"parameters":{"n":21,"k":20,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0096","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=22、k=21∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(1124000727777607680000, 53523844179886080000, 0)","explanation_zh":"省去單因子乘積與回乘=(1124000727777607680000, 53523844179886080000, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145096,"parameters":{"n":22,"k":21,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0097","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=23、k=22∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(25852016738884976640000, 1175091669949317120000, 0)","explanation_zh":"省去單因子乘積與回乘=(25852016738884976640000, 1175091669949317120000, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145097,"parameters":{"n":23,"k":22,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0098","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=24、k=23∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(620448401733239439360000, 26976017466662584320000, 0)","explanation_zh":"省去單因子乘積與回乘=(620448401733239439360000, 26976017466662584320000, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145098,"parameters":{"n":24,"k":23,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0099","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=25、k=24∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(15511210043330985984000000, 646300418472124416000000, 0)","explanation_zh":"省去單因子乘積與回乘=(15511210043330985984000000, 646300418472124416000000, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145099,"parameters":{"n":25,"k":24,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[25,24]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r05-factorial-0100","batch_id":"batch-20261010-advance-20261014-r05-factorial-foundations","language":"zh-Hant","source_mko_ids":["mko-factorial"],"title_zh":"省去一次因子的乘積","stem_zh":"N從0起；BigInt精確階乘；n=26、k=25∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。","answer":{"kind":"expression","value":"(403291461126605635584000000, 16131658445064225423360000, 0)","explanation_zh":"省去單因子乘積與回乘=(403291461126605635584000000, 16131658445064225423360000, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["省去一次因子的乘積"]},"variant":{"family_id":"qf-density14-factorial-single-factor-omission","seed":202610145100,"parameters":{"n":26,"k":25,"domain":"finite_N_factorial_protocol","operation":"density14_factorial_single_factor_omission","semantic_template":"qf-density14-factorial-single-factor-omission/[26,25]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}