{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0001","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U=∅，整數條件 P(x)：x≥0。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=0；失敗=0；論域=0。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041001,"parameters":{"U":[],"predicate":{"kind":"ge","value":0},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0002","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={-2, -1, 0}，整數條件 P(x)：x≥-2。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=0；論域=3。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041002,"parameters":{"U":[-2,-1,0],"predicate":{"kind":"ge","value":-2},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[-2,-1,0],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":-2}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0003","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={1, 2, 3}，整數條件 P(x)：x≥4。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=0；失敗=3；論域=3。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041003,"parameters":{"U":[1,2,3],"predicate":{"kind":"ge","value":4},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[1,2,3],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0004","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={-1, 0, 1}，整數條件 P(x)：x 是非零整數 2 的整數倍。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=2；論域=3。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041004,"parameters":{"U":[-1,0,1],"predicate":{"kind":"multiple","value":2},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[-1,0,1],\"operation\":\"forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0005","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={-8, -7, -5, -3}，整數條件 P(x)：x≥-7。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041005,"parameters":{"U":[-8,-7,-5,-3],"predicate":{"kind":"ge","value":-7},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[-8,-7,-5,-3],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0006","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={-7, -6, -4, -2}，整數條件 P(x)：x≤-4。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041006,"parameters":{"U":[-7,-6,-4,-2],"predicate":{"kind":"le","value":-4},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[-7,-6,-4,-2],\"operation\":\"forall\",\"predicate\":{\"kind\":\"le\",\"value\":-4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0007","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={-6, -5, -3, -1}，整數條件 P(x)：x=-5。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041007,"parameters":{"U":[-6,-5,-3,-1],"predicate":{"kind":"eq","value":-5},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[-6,-5,-3,-1],\"operation\":\"forall\",\"predicate\":{\"kind\":\"eq\",\"value\":-5}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0008","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={-5, -4, -2, 0}，整數條件 P(x)：x 是非零整數 3 的整數倍。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041008,"parameters":{"U":[-5,-4,-2,0],"predicate":{"kind":"multiple","value":3},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[-5,-4,-2,0],\"operation\":\"forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0009","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={-4, -3, -1, 1}，整數條件 P(x)：x≥-3。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041009,"parameters":{"U":[-4,-3,-1,1],"predicate":{"kind":"ge","value":-3},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[-4,-3,-1,1],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0010","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={-3, -2, 0, 2}，整數條件 P(x)：x≤0。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041010,"parameters":{"U":[-3,-2,0,2],"predicate":{"kind":"le","value":0},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[-3,-2,0,2],\"operation\":\"forall\",\"predicate\":{\"kind\":\"le\",\"value\":0}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0011","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={-2, -1, 1, 3}，整數條件 P(x)：x=-1。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041011,"parameters":{"U":[-2,-1,1,3],"predicate":{"kind":"eq","value":-1},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[-2,-1,1,3],\"operation\":\"forall\",\"predicate\":{\"kind\":\"eq\",\"value\":-1}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0012","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={-1, 0, 2, 4}，整數條件 P(x)：x 是非零整數 4 的整數倍。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=2；失敗=2；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041012,"parameters":{"U":[-1,0,2,4],"predicate":{"kind":"multiple","value":4},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[-1,0,2,4],\"operation\":\"forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0013","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={0, 1, 3, 5}，整數條件 P(x)：x≥1。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041013,"parameters":{"U":[0,1,3,5],"predicate":{"kind":"ge","value":1},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[0,1,3,5],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0014","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={1, 2, 4, 6}，整數條件 P(x)：x≤4。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041014,"parameters":{"U":[1,2,4,6],"predicate":{"kind":"le","value":4},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[1,2,4,6],\"operation\":\"forall\",\"predicate\":{\"kind\":\"le\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0015","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={2, 3, 5, 7}，整數條件 P(x)：x=3。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041015,"parameters":{"U":[2,3,5,7],"predicate":{"kind":"eq","value":3},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[2,3,5,7],\"operation\":\"forall\",\"predicate\":{\"kind\":\"eq\",\"value\":3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0016","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={3, 4, 6, 8}，整數條件 P(x)：x 是非零整數 2 的整數倍。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041016,"parameters":{"U":[3,4,6,8],"predicate":{"kind":"multiple","value":2},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[3,4,6,8],\"operation\":\"forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0017","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={4, 5, 7, 9}，整數條件 P(x)：x≥5。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041017,"parameters":{"U":[4,5,7,9],"predicate":{"kind":"ge","value":5},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[4,5,7,9],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0018","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={5, 6, 8, 10}，整數條件 P(x)：x≤8。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041018,"parameters":{"U":[5,6,8,10],"predicate":{"kind":"le","value":8},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[5,6,8,10],\"operation\":\"forall\",\"predicate\":{\"kind\":\"le\",\"value\":8}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0019","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={6, 7, 9, 11}，整數條件 P(x)：x=7。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041019,"parameters":{"U":[6,7,9,11],"predicate":{"kind":"eq","value":7},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[6,7,9,11],\"operation\":\"forall\",\"predicate\":{\"kind\":\"eq\",\"value\":7}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0020","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={7, 8, 10, 12}，整數條件 P(x)：x 是非零整數 3 的整數倍。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041020,"parameters":{"U":[7,8,10,12],"predicate":{"kind":"multiple","value":3},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[7,8,10,12],\"operation\":\"forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0021","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={8, 9, 11, 13}，整數條件 P(x)：x≥9。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041021,"parameters":{"U":[8,9,11,13],"predicate":{"kind":"ge","value":9},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[8,9,11,13],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0022","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={9, 10, 12, 14}，整數條件 P(x)：x≤12。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041022,"parameters":{"U":[9,10,12,14],"predicate":{"kind":"le","value":12},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[9,10,12,14],\"operation\":\"forall\",\"predicate\":{\"kind\":\"le\",\"value\":12}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0023","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={10, 11, 13, 15}，整數條件 P(x)：x=11。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041023,"parameters":{"U":[10,11,13,15],"predicate":{"kind":"eq","value":11},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[10,11,13,15],\"operation\":\"forall\",\"predicate\":{\"kind\":\"eq\",\"value\":11}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0024","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={11, 12, 14, 16}，整數條件 P(x)：x 是非零整數 4 的整數倍。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=2；失敗=2；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041024,"parameters":{"U":[11,12,14,16],"predicate":{"kind":"multiple","value":4},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[11,12,14,16],\"operation\":\"forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0025","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的全稱真值","stem_zh":"採古典真值，明示有限論域 U={12, 13, 15, 17}，整數條件 P(x)：x≥13。∀x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的全稱真值"]},"variant":{"family_id":"qf-quantifier-finite-universal","seed":202610041025,"parameters":{"U":[12,13,15,17],"predicate":{"kind":"ge","value":13},"operation":"forall","semantic_template":"qf-quantifier-finite-universal/{\"U\":[12,13,15,17],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0026","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U=∅，整數條件 P(x)：x≥0。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=0；失敗=0；論域=0。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041026,"parameters":{"U":[],"predicate":{"kind":"ge","value":0},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0027","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={-2, -1, 0}，整數條件 P(x)：x≥-2。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=0；論域=3。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041027,"parameters":{"U":[-2,-1,0],"predicate":{"kind":"ge","value":-2},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[-2,-1,0],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":-2}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0028","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={1, 2, 3}，整數條件 P(x)：x≥4。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=0；失敗=3；論域=3。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041028,"parameters":{"U":[1,2,3],"predicate":{"kind":"ge","value":4},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[1,2,3],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0029","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={-1, 0, 1}，整數條件 P(x)：x 是非零整數 2 的整數倍。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=2；論域=3。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041029,"parameters":{"U":[-1,0,1],"predicate":{"kind":"multiple","value":2},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[-1,0,1],\"operation\":\"exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0030","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={-8, -7, -5, -3}，整數條件 P(x)：x≥-7。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041030,"parameters":{"U":[-8,-7,-5,-3],"predicate":{"kind":"ge","value":-7},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[-8,-7,-5,-3],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0031","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={-7, -6, -4, -2}，整數條件 P(x)：x≤-4。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041031,"parameters":{"U":[-7,-6,-4,-2],"predicate":{"kind":"le","value":-4},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[-7,-6,-4,-2],\"operation\":\"exists\",\"predicate\":{\"kind\":\"le\",\"value\":-4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0032","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={-6, -5, -3, -1}，整數條件 P(x)：x=-5。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041032,"parameters":{"U":[-6,-5,-3,-1],"predicate":{"kind":"eq","value":-5},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[-6,-5,-3,-1],\"operation\":\"exists\",\"predicate\":{\"kind\":\"eq\",\"value\":-5}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0033","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={-5, -4, -2, 0}，整數條件 P(x)：x 是非零整數 3 的整數倍。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041033,"parameters":{"U":[-5,-4,-2,0],"predicate":{"kind":"multiple","value":3},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[-5,-4,-2,0],\"operation\":\"exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0034","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={-4, -3, -1, 1}，整數條件 P(x)：x≥-3。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041034,"parameters":{"U":[-4,-3,-1,1],"predicate":{"kind":"ge","value":-3},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[-4,-3,-1,1],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0035","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={-3, -2, 0, 2}，整數條件 P(x)：x≤0。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041035,"parameters":{"U":[-3,-2,0,2],"predicate":{"kind":"le","value":0},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[-3,-2,0,2],\"operation\":\"exists\",\"predicate\":{\"kind\":\"le\",\"value\":0}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0036","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={-2, -1, 1, 3}，整數條件 P(x)：x=-1。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041036,"parameters":{"U":[-2,-1,1,3],"predicate":{"kind":"eq","value":-1},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[-2,-1,1,3],\"operation\":\"exists\",\"predicate\":{\"kind\":\"eq\",\"value\":-1}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0037","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={-1, 0, 2, 4}，整數條件 P(x)：x 是非零整數 4 的整數倍。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=2；失敗=2；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041037,"parameters":{"U":[-1,0,2,4],"predicate":{"kind":"multiple","value":4},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[-1,0,2,4],\"operation\":\"exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0038","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={0, 1, 3, 5}，整數條件 P(x)：x≥1。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041038,"parameters":{"U":[0,1,3,5],"predicate":{"kind":"ge","value":1},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[0,1,3,5],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0039","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={1, 2, 4, 6}，整數條件 P(x)：x≤4。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041039,"parameters":{"U":[1,2,4,6],"predicate":{"kind":"le","value":4},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[1,2,4,6],\"operation\":\"exists\",\"predicate\":{\"kind\":\"le\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0040","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={2, 3, 5, 7}，整數條件 P(x)：x=3。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041040,"parameters":{"U":[2,3,5,7],"predicate":{"kind":"eq","value":3},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[2,3,5,7],\"operation\":\"exists\",\"predicate\":{\"kind\":\"eq\",\"value\":3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0041","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={3, 4, 6, 8}，整數條件 P(x)：x 是非零整數 2 的整數倍。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041041,"parameters":{"U":[3,4,6,8],"predicate":{"kind":"multiple","value":2},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[3,4,6,8],\"operation\":\"exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0042","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={4, 5, 7, 9}，整數條件 P(x)：x≥5。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041042,"parameters":{"U":[4,5,7,9],"predicate":{"kind":"ge","value":5},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[4,5,7,9],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0043","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={5, 6, 8, 10}，整數條件 P(x)：x≤8。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041043,"parameters":{"U":[5,6,8,10],"predicate":{"kind":"le","value":8},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[5,6,8,10],\"operation\":\"exists\",\"predicate\":{\"kind\":\"le\",\"value\":8}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0044","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={6, 7, 9, 11}，整數條件 P(x)：x=7。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041044,"parameters":{"U":[6,7,9,11],"predicate":{"kind":"eq","value":7},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[6,7,9,11],\"operation\":\"exists\",\"predicate\":{\"kind\":\"eq\",\"value\":7}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0045","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={7, 8, 10, 12}，整數條件 P(x)：x 是非零整數 3 的整數倍。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041045,"parameters":{"U":[7,8,10,12],"predicate":{"kind":"multiple","value":3},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[7,8,10,12],\"operation\":\"exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0046","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={8, 9, 11, 13}，整數條件 P(x)：x≥9。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041046,"parameters":{"U":[8,9,11,13],"predicate":{"kind":"ge","value":9},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[8,9,11,13],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0047","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={9, 10, 12, 14}，整數條件 P(x)：x≤12。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041047,"parameters":{"U":[9,10,12,14],"predicate":{"kind":"le","value":12},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[9,10,12,14],\"operation\":\"exists\",\"predicate\":{\"kind\":\"le\",\"value\":12}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0048","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={10, 11, 13, 15}，整數條件 P(x)：x=11。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041048,"parameters":{"U":[10,11,13,15],"predicate":{"kind":"eq","value":11},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[10,11,13,15],\"operation\":\"exists\",\"predicate\":{\"kind\":\"eq\",\"value\":11}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0049","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={11, 12, 14, 16}，整數條件 P(x)：x 是非零整數 4 的整數倍。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=2；失敗=2；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041049,"parameters":{"U":[11,12,14,16],"predicate":{"kind":"multiple","value":4},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[11,12,14,16],\"operation\":\"exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0050","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限論域的存在真值","stem_zh":"採古典真值，明示有限論域 U={12, 13, 15, 17}，整數條件 P(x)：x≥13。∃x∈U P(x) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的存在真值"]},"variant":{"family_id":"qf-quantifier-finite-existential","seed":202610041050,"parameters":{"U":[12,13,15,17],"predicate":{"kind":"ge","value":13},"operation":"exists","semantic_template":"qf-quantifier-finite-existential/{\"U\":[12,13,15,17],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0051","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U=∅，整數條件 P(x)：x≥0。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=0；失敗=0；論域=0。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041051,"parameters":{"U":[],"predicate":{"kind":"ge","value":0},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0052","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={-2, -1, 0}，整數條件 P(x)：x≥-2。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=0；論域=3。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041052,"parameters":{"U":[-2,-1,0],"predicate":{"kind":"ge","value":-2},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[-2,-1,0],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":-2}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0053","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={1, 2, 3}，整數條件 P(x)：x≥4。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=0；失敗=3；論域=3。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041053,"parameters":{"U":[1,2,3],"predicate":{"kind":"ge","value":4},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[1,2,3],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0054","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={-1, 0, 1}，整數條件 P(x)：x 是非零整數 2 的整數倍。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=2；論域=3。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041054,"parameters":{"U":[-1,0,1],"predicate":{"kind":"multiple","value":2},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[-1,0,1],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0055","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={-8, -7, -5, -3}，整數條件 P(x)：x≥-7。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041055,"parameters":{"U":[-8,-7,-5,-3],"predicate":{"kind":"ge","value":-7},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[-8,-7,-5,-3],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0056","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={-7, -6, -4, -2}，整數條件 P(x)：x≤-4。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041056,"parameters":{"U":[-7,-6,-4,-2],"predicate":{"kind":"le","value":-4},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[-7,-6,-4,-2],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"le\",\"value\":-4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0057","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={-6, -5, -3, -1}，整數條件 P(x)：x=-5。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041057,"parameters":{"U":[-6,-5,-3,-1],"predicate":{"kind":"eq","value":-5},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[-6,-5,-3,-1],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"eq\",\"value\":-5}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0058","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={-5, -4, -2, 0}，整數條件 P(x)：x 是非零整數 3 的整數倍。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041058,"parameters":{"U":[-5,-4,-2,0],"predicate":{"kind":"multiple","value":3},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[-5,-4,-2,0],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0059","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={-4, -3, -1, 1}，整數條件 P(x)：x≥-3。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041059,"parameters":{"U":[-4,-3,-1,1],"predicate":{"kind":"ge","value":-3},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[-4,-3,-1,1],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0060","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={-3, -2, 0, 2}，整數條件 P(x)：x≤0。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041060,"parameters":{"U":[-3,-2,0,2],"predicate":{"kind":"le","value":0},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[-3,-2,0,2],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"le\",\"value\":0}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0061","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={-2, -1, 1, 3}，整數條件 P(x)：x=-1。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041061,"parameters":{"U":[-2,-1,1,3],"predicate":{"kind":"eq","value":-1},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[-2,-1,1,3],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"eq\",\"value\":-1}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0062","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={-1, 0, 2, 4}，整數條件 P(x)：x 是非零整數 4 的整數倍。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=2；失敗=2；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041062,"parameters":{"U":[-1,0,2,4],"predicate":{"kind":"multiple","value":4},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[-1,0,2,4],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0063","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={0, 1, 3, 5}，整數條件 P(x)：x≥1。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041063,"parameters":{"U":[0,1,3,5],"predicate":{"kind":"ge","value":1},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[0,1,3,5],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0064","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={1, 2, 4, 6}，整數條件 P(x)：x≤4。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041064,"parameters":{"U":[1,2,4,6],"predicate":{"kind":"le","value":4},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[1,2,4,6],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"le\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0065","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={2, 3, 5, 7}，整數條件 P(x)：x=3。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041065,"parameters":{"U":[2,3,5,7],"predicate":{"kind":"eq","value":3},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[2,3,5,7],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"eq\",\"value\":3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0066","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={3, 4, 6, 8}，整數條件 P(x)：x 是非零整數 2 的整數倍。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041066,"parameters":{"U":[3,4,6,8],"predicate":{"kind":"multiple","value":2},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[3,4,6,8],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0067","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={4, 5, 7, 9}，整數條件 P(x)：x≥5。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041067,"parameters":{"U":[4,5,7,9],"predicate":{"kind":"ge","value":5},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[4,5,7,9],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0068","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={5, 6, 8, 10}，整數條件 P(x)：x≤8。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041068,"parameters":{"U":[5,6,8,10],"predicate":{"kind":"le","value":8},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[5,6,8,10],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"le\",\"value\":8}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0069","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={6, 7, 9, 11}，整數條件 P(x)：x=7。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041069,"parameters":{"U":[6,7,9,11],"predicate":{"kind":"eq","value":7},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[6,7,9,11],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"eq\",\"value\":7}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0070","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={7, 8, 10, 12}，整數條件 P(x)：x 是非零整數 3 的整數倍。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041070,"parameters":{"U":[7,8,10,12],"predicate":{"kind":"multiple","value":3},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[7,8,10,12],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0071","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={8, 9, 11, 13}，整數條件 P(x)：x≥9。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041071,"parameters":{"U":[8,9,11,13],"predicate":{"kind":"ge","value":9},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[8,9,11,13],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0072","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={9, 10, 12, 14}，整數條件 P(x)：x≤12。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041072,"parameters":{"U":[9,10,12,14],"predicate":{"kind":"le","value":12},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[9,10,12,14],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"le\",\"value\":12}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0073","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={10, 11, 13, 15}，整數條件 P(x)：x=11。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041073,"parameters":{"U":[10,11,13,15],"predicate":{"kind":"eq","value":11},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[10,11,13,15],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"eq\",\"value\":11}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0074","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={11, 12, 14, 16}，整數條件 P(x)：x 是非零整數 4 的整數倍。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=2；失敗=2；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041074,"parameters":{"U":[11,12,14,16],"predicate":{"kind":"multiple","value":4},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[11,12,14,16],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0075","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限全稱的否定真值","stem_zh":"採古典真值，明示有限論域 U={12, 13, 15, 17}，整數條件 P(x)：x≥13。¬(∀x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限全稱的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-universal","seed":202610041075,"parameters":{"U":[12,13,15,17],"predicate":{"kind":"ge","value":13},"operation":"not_forall","semantic_template":"qf-quantifier-negated-universal/{\"U\":[12,13,15,17],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0076","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U=∅，整數條件 P(x)：x≥0。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=0；失敗=0；論域=0。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041076,"parameters":{"U":[],"predicate":{"kind":"ge","value":0},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0077","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={-2, -1, 0}，整數條件 P(x)：x≥-2。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=0；論域=3。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041077,"parameters":{"U":[-2,-1,0],"predicate":{"kind":"ge","value":-2},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[-2,-1,0],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":-2}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0078","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={1, 2, 3}，整數條件 P(x)：x≥4。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"是","explanation_zh":"滿足=0；失敗=3；論域=3。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041078,"parameters":{"U":[1,2,3],"predicate":{"kind":"ge","value":4},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[1,2,3],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0079","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={-1, 0, 1}，整數條件 P(x)：x 是非零整數 2 的整數倍。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=2；論域=3。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041079,"parameters":{"U":[-1,0,1],"predicate":{"kind":"multiple","value":2},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[-1,0,1],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0080","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={-8, -7, -5, -3}，整數條件 P(x)：x≥-7。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041080,"parameters":{"U":[-8,-7,-5,-3],"predicate":{"kind":"ge","value":-7},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[-8,-7,-5,-3],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0081","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={-7, -6, -4, -2}，整數條件 P(x)：x≤-4。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041081,"parameters":{"U":[-7,-6,-4,-2],"predicate":{"kind":"le","value":-4},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[-7,-6,-4,-2],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"le\",\"value\":-4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0082","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={-6, -5, -3, -1}，整數條件 P(x)：x=-5。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041082,"parameters":{"U":[-6,-5,-3,-1],"predicate":{"kind":"eq","value":-5},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[-6,-5,-3,-1],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"eq\",\"value\":-5}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0083","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={-5, -4, -2, 0}，整數條件 P(x)：x 是非零整數 3 的整數倍。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041083,"parameters":{"U":[-5,-4,-2,0],"predicate":{"kind":"multiple","value":3},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[-5,-4,-2,0],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0084","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={-4, -3, -1, 1}，整數條件 P(x)：x≥-3。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041084,"parameters":{"U":[-4,-3,-1,1],"predicate":{"kind":"ge","value":-3},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[-4,-3,-1,1],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0085","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={-3, -2, 0, 2}，整數條件 P(x)：x≤0。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041085,"parameters":{"U":[-3,-2,0,2],"predicate":{"kind":"le","value":0},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[-3,-2,0,2],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"le\",\"value\":0}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0086","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={-2, -1, 1, 3}，整數條件 P(x)：x=-1。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041086,"parameters":{"U":[-2,-1,1,3],"predicate":{"kind":"eq","value":-1},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[-2,-1,1,3],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"eq\",\"value\":-1}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0087","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={-1, 0, 2, 4}，整數條件 P(x)：x 是非零整數 4 的整數倍。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=2；失敗=2；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041087,"parameters":{"U":[-1,0,2,4],"predicate":{"kind":"multiple","value":4},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[-1,0,2,4],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0088","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={0, 1, 3, 5}，整數條件 P(x)：x≥1。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041088,"parameters":{"U":[0,1,3,5],"predicate":{"kind":"ge","value":1},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[0,1,3,5],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0089","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={1, 2, 4, 6}，整數條件 P(x)：x≤4。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041089,"parameters":{"U":[1,2,4,6],"predicate":{"kind":"le","value":4},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[1,2,4,6],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"le\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0090","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={2, 3, 5, 7}，整數條件 P(x)：x=3。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041090,"parameters":{"U":[2,3,5,7],"predicate":{"kind":"eq","value":3},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[2,3,5,7],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"eq\",\"value\":3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0091","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={3, 4, 6, 8}，整數條件 P(x)：x 是非零整數 2 的整數倍。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041091,"parameters":{"U":[3,4,6,8],"predicate":{"kind":"multiple","value":2},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[3,4,6,8],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0092","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={4, 5, 7, 9}，整數條件 P(x)：x≥5。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041092,"parameters":{"U":[4,5,7,9],"predicate":{"kind":"ge","value":5},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[4,5,7,9],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0093","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={5, 6, 8, 10}，整數條件 P(x)：x≤8。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041093,"parameters":{"U":[5,6,8,10],"predicate":{"kind":"le","value":8},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[5,6,8,10],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"le\",\"value\":8}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0094","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={6, 7, 9, 11}，整數條件 P(x)：x=7。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041094,"parameters":{"U":[6,7,9,11],"predicate":{"kind":"eq","value":7},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[6,7,9,11],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"eq\",\"value\":7}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0095","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={7, 8, 10, 12}，整數條件 P(x)：x 是非零整數 3 的整數倍。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041095,"parameters":{"U":[7,8,10,12],"predicate":{"kind":"multiple","value":3},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[7,8,10,12],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0096","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={8, 9, 11, 13}，整數條件 P(x)：x≥9。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041096,"parameters":{"U":[8,9,11,13],"predicate":{"kind":"ge","value":9},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[8,9,11,13],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0097","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={9, 10, 12, 14}，整數條件 P(x)：x≤12。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041097,"parameters":{"U":[9,10,12,14],"predicate":{"kind":"le","value":12},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[9,10,12,14],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"le\",\"value\":12}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0098","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={10, 11, 13, 15}，整數條件 P(x)：x=11。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=1；失敗=3；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041098,"parameters":{"U":[10,11,13,15],"predicate":{"kind":"eq","value":11},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[10,11,13,15],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"eq\",\"value\":11}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0099","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={11, 12, 14, 16}，整數條件 P(x)：x 是非零整數 4 的整數倍。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=2；失敗=2；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041099,"parameters":{"U":[11,12,14,16],"predicate":{"kind":"multiple","value":4},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[11,12,14,16],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r01-quantifier-0100","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限存在的否定真值","stem_zh":"採古典真值，明示有限論域 U={12, 13, 15, 17}，整數條件 P(x)：x≥13。¬(∃x∈U P(x)) 是否為真？","answer":{"kind":"short_text","value":"否","explanation_zh":"滿足=3；失敗=1；論域=4。完整檢查明示論域；空論域全稱為真、存在為假，否定後交換真假。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限存在的否定真值"]},"variant":{"family_id":"qf-quantifier-negated-existential","seed":202610041100,"parameters":{"U":[12,13,15,17],"predicate":{"kind":"ge","value":13},"operation":"not_exists","semantic_template":"qf-quantifier-negated-existential/{\"U\":[12,13,15,17],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}