{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r02-counterexample-0001","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U=∅，整數條件 P(x)：x≥0，待檢驗全稱 ∀x∈U P(x)。候選 c=0 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=否；整數條件值=真。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042001,"parameters":{"U":[],"predicate":{"kind":"ge","value":0},"c":0,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[],\"c\":0,\"operation\":\"witness_validity\",\"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-r02-counterexample-0002","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={-2, -1, 0}，整數條件 P(x)：x≥0，待檢驗全稱 ∀x∈U P(x)。候選 c=-1 是否為有效反例？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈U=是；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042002,"parameters":{"U":[-2,-1,0],"predicate":{"kind":"ge","value":0},"c":-1,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[-2,-1,0],\"c\":-1,\"operation\":\"witness_validity\",\"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-r02-counterexample-0003","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={-2, -1, 0}，整數條件 P(x)：x≥0，待檢驗全稱 ∀x∈U P(x)。候選 c=0 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=是；整數條件值=真。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042003,"parameters":{"U":[-2,-1,0],"predicate":{"kind":"ge","value":0},"c":0,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[-2,-1,0],\"c\":0,\"operation\":\"witness_validity\",\"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-r02-counterexample-0004","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={-2, -1, 0}，整數條件 P(x)：x≥0，待檢驗全稱 ∀x∈U P(x)。候選 c=-3 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=否；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042004,"parameters":{"U":[-2,-1,0],"predicate":{"kind":"ge","value":0},"c":-3,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[-2,-1,0],\"c\":-3,\"operation\":\"witness_validity\",\"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-r02-counterexample-0005","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={-8, -7, -6}，整數條件 P(x)：x≥-7，待檢驗全稱 ∀x∈U P(x)。候選 c=-8 是否為有效反例？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈U=是；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042005,"parameters":{"U":[-8,-7,-6],"predicate":{"kind":"ge","value":-7},"c":-8,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[-8,-7,-6],\"c\":-8,\"operation\":\"witness_validity\",\"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-r02-counterexample-0006","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={-7, -6, -5}，整數條件 P(x)：x≤-6，待檢驗全稱 ∀x∈U P(x)。候選 c=-6 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=是；整數條件值=真。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042006,"parameters":{"U":[-7,-6,-5],"predicate":{"kind":"le","value":-6},"c":-6,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[-7,-6,-5],\"c\":-6,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"le\",\"value\":-6}}"}},"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-r02-counterexample-0007","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={-6, -5, -4}，整數條件 P(x)：x=-5，待檢驗全稱 ∀x∈U P(x)。候選 c=-2 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=否；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042007,"parameters":{"U":[-6,-5,-4],"predicate":{"kind":"eq","value":-5},"c":-2,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[-6,-5,-4],\"c\":-2,\"operation\":\"witness_validity\",\"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-r02-counterexample-0008","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={-5, -4, -3}，整數條件 P(x)：x 是非零整數 3 的整數倍，待檢驗全稱 ∀x∈U P(x)。候選 c=-5 是否為有效反例？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈U=是；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042008,"parameters":{"U":[-5,-4,-3],"predicate":{"kind":"multiple","value":3},"c":-5,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[-5,-4,-3],\"c\":-5,\"operation\":\"witness_validity\",\"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-r02-counterexample-0009","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={-4, -3, -2}，整數條件 P(x)：x≥-3，待檢驗全稱 ∀x∈U P(x)。候選 c=-3 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=是；整數條件值=真。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042009,"parameters":{"U":[-4,-3,-2],"predicate":{"kind":"ge","value":-3},"c":-3,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[-4,-3,-2],\"c\":-3,\"operation\":\"witness_validity\",\"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-r02-counterexample-0010","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={-3, -2, -1}，整數條件 P(x)：x≤-2，待檢驗全稱 ∀x∈U P(x)。候選 c=1 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=否；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042010,"parameters":{"U":[-3,-2,-1],"predicate":{"kind":"le","value":-2},"c":1,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[-3,-2,-1],\"c\":1,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"le\",\"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-r02-counterexample-0011","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={-2, -1, 0}，整數條件 P(x)：x=-1，待檢驗全稱 ∀x∈U P(x)。候選 c=-2 是否為有效反例？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈U=是；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042011,"parameters":{"U":[-2,-1,0],"predicate":{"kind":"eq","value":-1},"c":-2,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[-2,-1,0],\"c\":-2,\"operation\":\"witness_validity\",\"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-r02-counterexample-0012","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={-1, 0, 1}，整數條件 P(x)：x 是非零整數 4 的整數倍，待檢驗全稱 ∀x∈U P(x)。候選 c=0 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=是；整數條件值=真。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042012,"parameters":{"U":[-1,0,1],"predicate":{"kind":"multiple","value":4},"c":0,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[-1,0,1],\"c\":0,\"operation\":\"witness_validity\",\"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-r02-counterexample-0013","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={0, 1, 2}，整數條件 P(x)：x≥1，待檢驗全稱 ∀x∈U P(x)。候選 c=4 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=否；整數條件值=真。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042013,"parameters":{"U":[0,1,2],"predicate":{"kind":"ge","value":1},"c":4,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[0,1,2],\"c\":4,\"operation\":\"witness_validity\",\"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-r02-counterexample-0014","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={1, 2, 3}，整數條件 P(x)：x≤2，待檢驗全稱 ∀x∈U P(x)。候選 c=1 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=是；整數條件值=真。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042014,"parameters":{"U":[1,2,3],"predicate":{"kind":"le","value":2},"c":1,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[1,2,3],\"c\":1,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"le\",\"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-r02-counterexample-0015","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={2, 3, 4}，整數條件 P(x)：x=3，待檢驗全稱 ∀x∈U P(x)。候選 c=3 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=是；整數條件值=真。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042015,"parameters":{"U":[2,3,4],"predicate":{"kind":"eq","value":3},"c":3,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[2,3,4],\"c\":3,\"operation\":\"witness_validity\",\"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-r02-counterexample-0016","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={3, 4, 5}，整數條件 P(x)：x 是非零整數 2 的整數倍，待檢驗全稱 ∀x∈U P(x)。候選 c=7 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=否；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042016,"parameters":{"U":[3,4,5],"predicate":{"kind":"multiple","value":2},"c":7,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[3,4,5],\"c\":7,\"operation\":\"witness_validity\",\"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-r02-counterexample-0017","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={4, 5, 6}，整數條件 P(x)：x≥5，待檢驗全稱 ∀x∈U P(x)。候選 c=4 是否為有效反例？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈U=是；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042017,"parameters":{"U":[4,5,6],"predicate":{"kind":"ge","value":5},"c":4,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[4,5,6],\"c\":4,\"operation\":\"witness_validity\",\"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-r02-counterexample-0018","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={5, 6, 7}，整數條件 P(x)：x≤6，待檢驗全稱 ∀x∈U P(x)。候選 c=6 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=是；整數條件值=真。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042018,"parameters":{"U":[5,6,7],"predicate":{"kind":"le","value":6},"c":6,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[5,6,7],\"c\":6,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"le\",\"value\":6}}"}},"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-r02-counterexample-0019","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={6, 7, 8}，整數條件 P(x)：x=7，待檢驗全稱 ∀x∈U P(x)。候選 c=10 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=否；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042019,"parameters":{"U":[6,7,8],"predicate":{"kind":"eq","value":7},"c":10,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[6,7,8],\"c\":10,\"operation\":\"witness_validity\",\"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-r02-counterexample-0020","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={7, 8, 9}，整數條件 P(x)：x 是非零整數 3 的整數倍，待檢驗全稱 ∀x∈U P(x)。候選 c=7 是否為有效反例？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈U=是；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042020,"parameters":{"U":[7,8,9],"predicate":{"kind":"multiple","value":3},"c":7,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[7,8,9],\"c\":7,\"operation\":\"witness_validity\",\"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-r02-counterexample-0021","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={8, 9, 10}，整數條件 P(x)：x≥9，待檢驗全稱 ∀x∈U P(x)。候選 c=9 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=是；整數條件值=真。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042021,"parameters":{"U":[8,9,10],"predicate":{"kind":"ge","value":9},"c":9,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[8,9,10],\"c\":9,\"operation\":\"witness_validity\",\"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-r02-counterexample-0022","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={9, 10, 11}，整數條件 P(x)：x≤10，待檢驗全稱 ∀x∈U P(x)。候選 c=13 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=否；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042022,"parameters":{"U":[9,10,11],"predicate":{"kind":"le","value":10},"c":13,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[9,10,11],\"c\":13,\"operation\":\"witness_validity\",\"predicate\":{\"kind\":\"le\",\"value\":10}}"}},"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-r02-counterexample-0023","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={10, 11, 12}，整數條件 P(x)：x=11，待檢驗全稱 ∀x∈U P(x)。候選 c=10 是否為有效反例？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈U=是；整數條件值=假。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042023,"parameters":{"U":[10,11,12],"predicate":{"kind":"eq","value":11},"c":10,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[10,11,12],\"c\":10,\"operation\":\"witness_validity\",\"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-r02-counterexample-0024","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={11, 12, 13}，整數條件 P(x)：x 是非零整數 4 的整數倍，待檢驗全稱 ∀x∈U P(x)。候選 c=12 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=是；整數條件值=真。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042024,"parameters":{"U":[11,12,13],"predicate":{"kind":"multiple","value":4},"c":12,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[11,12,13],\"c\":12,\"operation\":\"witness_validity\",\"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-r02-counterexample-0025","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"候選反例的論域與失敗資格","stem_zh":"明示有限論域 U={12, 13, 14}，整數條件 P(x)：x≥13，待檢驗全稱 ∀x∈U P(x)。候選 c=16 是否為有效反例？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈U=否；整數條件值=真。先檢查論域資格；域外整數公式值即使失敗也不反駁受限全稱。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選反例的論域與失敗資格"]},"variant":{"family_id":"qf-counterexample-witness-validity","seed":202610042025,"parameters":{"U":[12,13,14],"predicate":{"kind":"ge","value":13},"c":16,"operation":"witness_validity","semantic_template":"qf-counterexample-witness-validity/{\"U\":[12,13,14],\"c\":16,\"operation\":\"witness_validity\",\"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-r02-counterexample-0026","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U=∅，整數條件 P(x)：x≥0，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"反例集=∅。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042026,"parameters":{"U":[],"predicate":{"kind":"ge","value":0},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[],\"operation\":\"first_witness\",\"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-r02-counterexample-0027","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={-2, -1, 0}，整數條件 P(x)：x≥-2，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"不存在","explanation_zh":"反例集=∅。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042027,"parameters":{"U":[-2,-1,0],"predicate":{"kind":"ge","value":-2},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[-2,-1,0],\"operation\":\"first_witness\",\"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-r02-counterexample-0028","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={1, 2, 3}，整數條件 P(x)：x≥4，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"1","explanation_zh":"反例集={1, 2, 3}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042028,"parameters":{"U":[1,2,3],"predicate":{"kind":"ge","value":4},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[1,2,3],\"operation\":\"first_witness\",\"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-r02-counterexample-0029","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"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，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"-1","explanation_zh":"反例集={-1, 1}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042029,"parameters":{"U":[-1,0,1],"predicate":{"kind":"multiple","value":2},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[-1,0,1],\"operation\":\"first_witness\",\"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-r02-counterexample-0030","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={-8, -7, -5, -3}，整數條件 P(x)：x≥-7，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"-8","explanation_zh":"反例集={-8}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042030,"parameters":{"U":[-8,-7,-5,-3],"predicate":{"kind":"ge","value":-7},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[-8,-7,-5,-3],\"operation\":\"first_witness\",\"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-r02-counterexample-0031","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={-7, -6, -4, -2}，整數條件 P(x)：x≤-4，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"-2","explanation_zh":"反例集={-2}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042031,"parameters":{"U":[-7,-6,-4,-2],"predicate":{"kind":"le","value":-4},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[-7,-6,-4,-2],\"operation\":\"first_witness\",\"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-r02-counterexample-0032","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={-6, -5, -3, -1}，整數條件 P(x)：x=-5，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"-6","explanation_zh":"反例集={-6, -3, -1}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042032,"parameters":{"U":[-6,-5,-3,-1],"predicate":{"kind":"eq","value":-5},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[-6,-5,-3,-1],\"operation\":\"first_witness\",\"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-r02-counterexample-0033","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"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，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"-5","explanation_zh":"反例集={-5, -4, -2}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042033,"parameters":{"U":[-5,-4,-2,0],"predicate":{"kind":"multiple","value":3},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[-5,-4,-2,0],\"operation\":\"first_witness\",\"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-r02-counterexample-0034","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={-4, -3, -1, 1}，整數條件 P(x)：x≥-3，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"-4","explanation_zh":"反例集={-4}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042034,"parameters":{"U":[-4,-3,-1,1],"predicate":{"kind":"ge","value":-3},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[-4,-3,-1,1],\"operation\":\"first_witness\",\"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-r02-counterexample-0035","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={-3, -2, 0, 2}，整數條件 P(x)：x≤0，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"2","explanation_zh":"反例集={2}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042035,"parameters":{"U":[-3,-2,0,2],"predicate":{"kind":"le","value":0},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[-3,-2,0,2],\"operation\":\"first_witness\",\"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-r02-counterexample-0036","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={-2, -1, 1, 3}，整數條件 P(x)：x=-1，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"-2","explanation_zh":"反例集={-2, 1, 3}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042036,"parameters":{"U":[-2,-1,1,3],"predicate":{"kind":"eq","value":-1},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[-2,-1,1,3],\"operation\":\"first_witness\",\"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-r02-counterexample-0037","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"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，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"-1","explanation_zh":"反例集={-1, 2}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042037,"parameters":{"U":[-1,0,2,4],"predicate":{"kind":"multiple","value":4},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[-1,0,2,4],\"operation\":\"first_witness\",\"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-r02-counterexample-0038","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={0, 1, 3, 5}，整數條件 P(x)：x≥1，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"0","explanation_zh":"反例集={0}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042038,"parameters":{"U":[0,1,3,5],"predicate":{"kind":"ge","value":1},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[0,1,3,5],\"operation\":\"first_witness\",\"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-r02-counterexample-0039","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={1, 2, 4, 6}，整數條件 P(x)：x≤4，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"6","explanation_zh":"反例集={6}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042039,"parameters":{"U":[1,2,4,6],"predicate":{"kind":"le","value":4},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[1,2,4,6],\"operation\":\"first_witness\",\"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-r02-counterexample-0040","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={2, 3, 5, 7}，整數條件 P(x)：x=3，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"2","explanation_zh":"反例集={2, 5, 7}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042040,"parameters":{"U":[2,3,5,7],"predicate":{"kind":"eq","value":3},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[2,3,5,7],\"operation\":\"first_witness\",\"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-r02-counterexample-0041","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"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，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"3","explanation_zh":"反例集={3}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042041,"parameters":{"U":[3,4,6,8],"predicate":{"kind":"multiple","value":2},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[3,4,6,8],\"operation\":\"first_witness\",\"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-r02-counterexample-0042","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={4, 5, 7, 9}，整數條件 P(x)：x≥5，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"4","explanation_zh":"反例集={4}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042042,"parameters":{"U":[4,5,7,9],"predicate":{"kind":"ge","value":5},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[4,5,7,9],\"operation\":\"first_witness\",\"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-r02-counterexample-0043","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={5, 6, 8, 10}，整數條件 P(x)：x≤8，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"10","explanation_zh":"反例集={10}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042043,"parameters":{"U":[5,6,8,10],"predicate":{"kind":"le","value":8},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[5,6,8,10],\"operation\":\"first_witness\",\"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-r02-counterexample-0044","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={6, 7, 9, 11}，整數條件 P(x)：x=7，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"6","explanation_zh":"反例集={6, 9, 11}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042044,"parameters":{"U":[6,7,9,11],"predicate":{"kind":"eq","value":7},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[6,7,9,11],\"operation\":\"first_witness\",\"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-r02-counterexample-0045","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"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，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"7","explanation_zh":"反例集={7, 8, 10}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042045,"parameters":{"U":[7,8,10,12],"predicate":{"kind":"multiple","value":3},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[7,8,10,12],\"operation\":\"first_witness\",\"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-r02-counterexample-0046","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={8, 9, 11, 13}，整數條件 P(x)：x≥9，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"8","explanation_zh":"反例集={8}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042046,"parameters":{"U":[8,9,11,13],"predicate":{"kind":"ge","value":9},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[8,9,11,13],\"operation\":\"first_witness\",\"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-r02-counterexample-0047","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={9, 10, 12, 14}，整數條件 P(x)：x≤12，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"14","explanation_zh":"反例集={14}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042047,"parameters":{"U":[9,10,12,14],"predicate":{"kind":"le","value":12},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[9,10,12,14],\"operation\":\"first_witness\",\"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-r02-counterexample-0048","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={10, 11, 13, 15}，整數條件 P(x)：x=11，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"10","explanation_zh":"反例集={10, 13, 15}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042048,"parameters":{"U":[10,11,13,15],"predicate":{"kind":"eq","value":11},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[10,11,13,15],\"operation\":\"first_witness\",\"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-r02-counterexample-0049","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"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，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"11","explanation_zh":"反例集={11, 14}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042049,"parameters":{"U":[11,12,14,16],"predicate":{"kind":"multiple","value":4},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[11,12,14,16],\"operation\":\"first_witness\",\"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-r02-counterexample-0050","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"有限論域的數值最小反例","stem_zh":"明示有限論域 U={12, 13, 15, 17}，整數條件 P(x)：x≥13，待檢驗全稱 ∀x∈U P(x)。數值最小的合法反例為何？沒有反例時答「不存在」。","answer":{"kind":"short_text","value":"12","explanation_zh":"反例集={12}。按數值遞增選最小成員；空失敗集合明示不存在。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限論域的數值最小反例"]},"variant":{"family_id":"qf-counterexample-first-witness","seed":202610042050,"parameters":{"U":[12,13,15,17],"predicate":{"kind":"ge","value":13},"operation":"first_witness","semantic_template":"qf-counterexample-first-witness/{\"U\":[12,13,15,17],\"operation\":\"first_witness\",\"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-r02-counterexample-0051","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U=∅，整數條件 P(x)：x=0，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-2, 0, 4} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"合法候選反例=∅。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042051,"parameters":{"U":[],"predicate":{"kind":"eq","value":0},"C":[-2,0,4],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-2,0,4],\"U\":[],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"eq\",\"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-r02-counterexample-0052","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={-2, -1, 0}，整數條件 P(x)：x≥-2，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-3, -2, 0, 5} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"合法候選反例=∅。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042052,"parameters":{"U":[-2,-1,0],"predicate":{"kind":"ge","value":-2},"C":[-3,-2,0,5],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-3,-2,0,5],\"U\":[-2,-1,0],\"operation\":\"candidate_set\",\"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-r02-counterexample-0053","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={-10, -9, -7}，整數條件 P(x)：x=-9，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-12, -10, -9, -7, -3} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{-10, -7}","explanation_zh":"合法候選反例={-10, -7}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042053,"parameters":{"U":[-10,-9,-7],"predicate":{"kind":"eq","value":-9},"C":[-12,-10,-9,-7,-3],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-12,-10,-9,-7,-3],\"U\":[-10,-9,-7],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"eq\",\"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-r02-counterexample-0054","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={-9, -8, -6}，整數條件 P(x)：x 是非零整數 2 的整數倍，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-11, -9, -8, -6, -2} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{-9}","explanation_zh":"合法候選反例={-9}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042054,"parameters":{"U":[-9,-8,-6],"predicate":{"kind":"multiple","value":2},"C":[-11,-9,-8,-6,-2],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-11,-9,-8,-6,-2],\"U\":[-9,-8,-6],\"operation\":\"candidate_set\",\"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-r02-counterexample-0055","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={-8, -7, -5}，整數條件 P(x)：x≥-7，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-10, -8, -7, -5, -1} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{-8}","explanation_zh":"合法候選反例={-8}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042055,"parameters":{"U":[-8,-7,-5],"predicate":{"kind":"ge","value":-7},"C":[-10,-8,-7,-5,-1],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-10,-8,-7,-5,-1],\"U\":[-8,-7,-5],\"operation\":\"candidate_set\",\"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-r02-counterexample-0056","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={-7, -6, -4}，整數條件 P(x)：x≤-6，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-9, -7, -6, -4, 0} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{-4}","explanation_zh":"合法候選反例={-4}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042056,"parameters":{"U":[-7,-6,-4],"predicate":{"kind":"le","value":-6},"C":[-9,-7,-6,-4,0],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-9,-7,-6,-4,0],\"U\":[-7,-6,-4],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"le\",\"value\":-6}}"}},"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-r02-counterexample-0057","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={-6, -5, -3}，整數條件 P(x)：x=-5，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-8, -6, -5, -3, 1} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{-6, -3}","explanation_zh":"合法候選反例={-6, -3}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042057,"parameters":{"U":[-6,-5,-3],"predicate":{"kind":"eq","value":-5},"C":[-8,-6,-5,-3,1],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-8,-6,-5,-3,1],\"U\":[-6,-5,-3],\"operation\":\"candidate_set\",\"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-r02-counterexample-0058","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={-5, -4, -2}，整數條件 P(x)：x 是非零整數 3 的整數倍，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-7, -5, -4, -2, 2} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{-5, -4, -2}","explanation_zh":"合法候選反例={-5, -4, -2}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042058,"parameters":{"U":[-5,-4,-2],"predicate":{"kind":"multiple","value":3},"C":[-7,-5,-4,-2,2],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-7,-5,-4,-2,2],\"U\":[-5,-4,-2],\"operation\":\"candidate_set\",\"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-r02-counterexample-0059","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={-4, -3, -1}，整數條件 P(x)：x≥-3，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-6, -4, -3, -1, 3} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{-4}","explanation_zh":"合法候選反例={-4}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042059,"parameters":{"U":[-4,-3,-1],"predicate":{"kind":"ge","value":-3},"C":[-6,-4,-3,-1,3],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-6,-4,-3,-1,3],\"U\":[-4,-3,-1],\"operation\":\"candidate_set\",\"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-r02-counterexample-0060","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={-3, -2, 0}，整數條件 P(x)：x≤-2，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-5, -3, -2, 0, 4} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{0}","explanation_zh":"合法候選反例={0}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042060,"parameters":{"U":[-3,-2,0],"predicate":{"kind":"le","value":-2},"C":[-5,-3,-2,0,4],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-5,-3,-2,0,4],\"U\":[-3,-2,0],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"le\",\"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-r02-counterexample-0061","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={-2, -1, 1}，整數條件 P(x)：x=-1，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-4, -2, -1, 1, 5} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{-2, 1}","explanation_zh":"合法候選反例={-2, 1}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042061,"parameters":{"U":[-2,-1,1],"predicate":{"kind":"eq","value":-1},"C":[-4,-2,-1,1,5],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-4,-2,-1,1,5],\"U\":[-2,-1,1],\"operation\":\"candidate_set\",\"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-r02-counterexample-0062","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={-1, 0, 2}，整數條件 P(x)：x 是非零整數 4 的整數倍，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-3, -1, 0, 2, 6} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{-1, 2}","explanation_zh":"合法候選反例={-1, 2}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042062,"parameters":{"U":[-1,0,2],"predicate":{"kind":"multiple","value":4},"C":[-3,-1,0,2,6],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-3,-1,0,2,6],\"U\":[-1,0,2],\"operation\":\"candidate_set\",\"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-r02-counterexample-0063","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={0, 1, 3}，整數條件 P(x)：x≥1，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-2, 0, 1, 3, 7} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{0}","explanation_zh":"合法候選反例={0}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042063,"parameters":{"U":[0,1,3],"predicate":{"kind":"ge","value":1},"C":[-2,0,1,3,7],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-2,0,1,3,7],\"U\":[0,1,3],\"operation\":\"candidate_set\",\"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-r02-counterexample-0064","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={1, 2, 4}，整數條件 P(x)：x≤2，待檢驗全稱 ∀x∈U P(x)。候選集合 C={-1, 1, 2, 4, 8} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{4}","explanation_zh":"合法候選反例={4}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042064,"parameters":{"U":[1,2,4],"predicate":{"kind":"le","value":2},"C":[-1,1,2,4,8],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[-1,1,2,4,8],\"U\":[1,2,4],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"le\",\"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-r02-counterexample-0065","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={2, 3, 5}，整數條件 P(x)：x=3，待檢驗全稱 ∀x∈U P(x)。候選集合 C={0, 2, 3, 5, 9} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{2, 5}","explanation_zh":"合法候選反例={2, 5}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042065,"parameters":{"U":[2,3,5],"predicate":{"kind":"eq","value":3},"C":[0,2,3,5,9],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[0,2,3,5,9],\"U\":[2,3,5],\"operation\":\"candidate_set\",\"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-r02-counterexample-0066","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={3, 4, 6}，整數條件 P(x)：x 是非零整數 2 的整數倍，待檢驗全稱 ∀x∈U P(x)。候選集合 C={1, 3, 4, 6, 10} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{3}","explanation_zh":"合法候選反例={3}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042066,"parameters":{"U":[3,4,6],"predicate":{"kind":"multiple","value":2},"C":[1,3,4,6,10],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[1,3,4,6,10],\"U\":[3,4,6],\"operation\":\"candidate_set\",\"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-r02-counterexample-0067","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={4, 5, 7}，整數條件 P(x)：x≥5，待檢驗全稱 ∀x∈U P(x)。候選集合 C={2, 4, 5, 7, 11} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{4}","explanation_zh":"合法候選反例={4}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042067,"parameters":{"U":[4,5,7],"predicate":{"kind":"ge","value":5},"C":[2,4,5,7,11],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[2,4,5,7,11],\"U\":[4,5,7],\"operation\":\"candidate_set\",\"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-r02-counterexample-0068","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={5, 6, 8}，整數條件 P(x)：x≤6，待檢驗全稱 ∀x∈U P(x)。候選集合 C={3, 5, 6, 8, 12} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{8}","explanation_zh":"合法候選反例={8}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042068,"parameters":{"U":[5,6,8],"predicate":{"kind":"le","value":6},"C":[3,5,6,8,12],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[3,5,6,8,12],\"U\":[5,6,8],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"le\",\"value\":6}}"}},"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-r02-counterexample-0069","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={6, 7, 9}，整數條件 P(x)：x=7，待檢驗全稱 ∀x∈U P(x)。候選集合 C={4, 6, 7, 9, 13} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{6, 9}","explanation_zh":"合法候選反例={6, 9}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042069,"parameters":{"U":[6,7,9],"predicate":{"kind":"eq","value":7},"C":[4,6,7,9,13],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[4,6,7,9,13],\"U\":[6,7,9],\"operation\":\"candidate_set\",\"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-r02-counterexample-0070","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={7, 8, 10}，整數條件 P(x)：x 是非零整數 3 的整數倍，待檢驗全稱 ∀x∈U P(x)。候選集合 C={5, 7, 8, 10, 14} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{7, 8, 10}","explanation_zh":"合法候選反例={7, 8, 10}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042070,"parameters":{"U":[7,8,10],"predicate":{"kind":"multiple","value":3},"C":[5,7,8,10,14],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[5,7,8,10,14],\"U\":[7,8,10],\"operation\":\"candidate_set\",\"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-r02-counterexample-0071","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={8, 9, 11}，整數條件 P(x)：x≥9，待檢驗全稱 ∀x∈U P(x)。候選集合 C={6, 8, 9, 11, 15} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{8}","explanation_zh":"合法候選反例={8}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042071,"parameters":{"U":[8,9,11],"predicate":{"kind":"ge","value":9},"C":[6,8,9,11,15],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[6,8,9,11,15],\"U\":[8,9,11],\"operation\":\"candidate_set\",\"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-r02-counterexample-0072","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={9, 10, 12}，整數條件 P(x)：x≤10，待檢驗全稱 ∀x∈U P(x)。候選集合 C={7, 9, 10, 12, 16} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{12}","explanation_zh":"合法候選反例={12}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042072,"parameters":{"U":[9,10,12],"predicate":{"kind":"le","value":10},"C":[7,9,10,12,16],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[7,9,10,12,16],\"U\":[9,10,12],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"le\",\"value\":10}}"}},"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-r02-counterexample-0073","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={10, 11, 13}，整數條件 P(x)：x=11，待檢驗全稱 ∀x∈U P(x)。候選集合 C={8, 10, 11, 13, 17} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{10, 13}","explanation_zh":"合法候選反例={10, 13}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042073,"parameters":{"U":[10,11,13],"predicate":{"kind":"eq","value":11},"C":[8,10,11,13,17],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[8,10,11,13,17],\"U\":[10,11,13],\"operation\":\"candidate_set\",\"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-r02-counterexample-0074","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={11, 12, 14}，整數條件 P(x)：x 是非零整數 4 的整數倍，待檢驗全稱 ∀x∈U P(x)。候選集合 C={9, 11, 12, 14, 18} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{11, 14}","explanation_zh":"合法候選反例={11, 14}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042074,"parameters":{"U":[11,12,14],"predicate":{"kind":"multiple","value":4},"C":[9,11,12,14,18],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[9,11,12,14,18],\"U\":[11,12,14],\"operation\":\"candidate_set\",\"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-r02-counterexample-0075","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"篩出候選集合中的合法反例","stem_zh":"明示有限論域 U={12, 13, 15}，整數條件 P(x)：x≥13，待檢驗全稱 ∀x∈U P(x)。候選集合 C={10, 12, 13, 15, 19} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。","answer":{"kind":"expression","value":"{12}","explanation_zh":"合法候選反例={12}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["篩出候選集合中的合法反例"]},"variant":{"family_id":"qf-counterexample-candidate-set","seed":202610042075,"parameters":{"U":[12,13,15],"predicate":{"kind":"ge","value":13},"C":[10,12,13,15,19],"operation":"candidate_set","semantic_template":"qf-counterexample-candidate-set/{\"C\":[10,12,13,15,19],\"U\":[12,13,15],\"operation\":\"candidate_set\",\"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-r02-counterexample-0076","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={-12, -11, -10}，整數條件 P(x)：x≥-11，待檢驗全稱 ∀x∈U P(x)。c=-12 原本是合法反例；改成明示子論域 V={-12} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈V=是；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042076,"parameters":{"U":[-12,-11,-10],"V":[-12],"predicate":{"kind":"ge","value":-11},"c":-12,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-12,-11,-10],\"V\":[-12],\"c\":-12,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"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-r02-counterexample-0077","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={-11, -10, -9}，整數條件 P(x)：x≥-9，待檢驗全稱 ∀x∈U P(x)。c=-11 原本是合法反例；改成明示子論域 V={-10, -9} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042077,"parameters":{"U":[-11,-10,-9],"V":[-10,-9],"predicate":{"kind":"ge","value":-9},"c":-11,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-11,-10,-9],\"V\":[-10,-9],\"c\":-11,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0078","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={-10, -9, -8}，整數條件 P(x)：x≥-9，待檢驗全稱 ∀x∈U P(x)。c=-10 原本是合法反例；改成明示子論域 V=∅ 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042078,"parameters":{"U":[-10,-9,-8],"V":[],"predicate":{"kind":"ge","value":-9},"c":-10,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-10,-9,-8],\"V\":[],\"c\":-10,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0079","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={-9, -8, -7}，整數條件 P(x)：x≥-7，待檢驗全稱 ∀x∈U P(x)。c=-9 原本是合法反例；改成明示子論域 V={-9} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈V=是；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042079,"parameters":{"U":[-9,-8,-7],"V":[-9],"predicate":{"kind":"ge","value":-7},"c":-9,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-9,-8,-7],\"V\":[-9],\"c\":-9,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0080","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={-8, -7, -6}，整數條件 P(x)：x≥-7，待檢驗全稱 ∀x∈U P(x)。c=-8 原本是合法反例；改成明示子論域 V={-7, -6} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042080,"parameters":{"U":[-8,-7,-6],"V":[-7,-6],"predicate":{"kind":"ge","value":-7},"c":-8,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-8,-7,-6],\"V\":[-7,-6],\"c\":-8,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0081","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={-7, -6, -5}，整數條件 P(x)：x≥-5，待檢驗全稱 ∀x∈U P(x)。c=-7 原本是合法反例；改成明示子論域 V=∅ 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042081,"parameters":{"U":[-7,-6,-5],"V":[],"predicate":{"kind":"ge","value":-5},"c":-7,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-7,-6,-5],\"V\":[],\"c\":-7,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0082","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={-6, -5, -4}，整數條件 P(x)：x≥-5，待檢驗全稱 ∀x∈U P(x)。c=-6 原本是合法反例；改成明示子論域 V={-6} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈V=是；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042082,"parameters":{"U":[-6,-5,-4],"V":[-6],"predicate":{"kind":"ge","value":-5},"c":-6,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-6,-5,-4],\"V\":[-6],\"c\":-6,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0083","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={-5, -4, -3}，整數條件 P(x)：x≥-3，待檢驗全稱 ∀x∈U P(x)。c=-5 原本是合法反例；改成明示子論域 V={-4, -3} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042083,"parameters":{"U":[-5,-4,-3],"V":[-4,-3],"predicate":{"kind":"ge","value":-3},"c":-5,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-5,-4,-3],\"V\":[-4,-3],\"c\":-5,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0084","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={-4, -3, -2}，整數條件 P(x)：x≥-3，待檢驗全稱 ∀x∈U P(x)。c=-4 原本是合法反例；改成明示子論域 V=∅ 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042084,"parameters":{"U":[-4,-3,-2],"V":[],"predicate":{"kind":"ge","value":-3},"c":-4,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-4,-3,-2],\"V\":[],\"c\":-4,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0085","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={-3, -2, -1}，整數條件 P(x)：x≥-1，待檢驗全稱 ∀x∈U P(x)。c=-3 原本是合法反例；改成明示子論域 V={-3} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈V=是；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042085,"parameters":{"U":[-3,-2,-1],"V":[-3],"predicate":{"kind":"ge","value":-1},"c":-3,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-3,-2,-1],\"V\":[-3],\"c\":-3,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0086","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={-2, -1, 0}，整數條件 P(x)：x≥-1，待檢驗全稱 ∀x∈U P(x)。c=-2 原本是合法反例；改成明示子論域 V={-1, 0} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042086,"parameters":{"U":[-2,-1,0],"V":[-1,0],"predicate":{"kind":"ge","value":-1},"c":-2,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-2,-1,0],\"V\":[-1,0],\"c\":-2,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0087","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={-1, 0, 1}，整數條件 P(x)：x≥1，待檢驗全稱 ∀x∈U P(x)。c=-1 原本是合法反例；改成明示子論域 V=∅ 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042087,"parameters":{"U":[-1,0,1],"V":[],"predicate":{"kind":"ge","value":1},"c":-1,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[-1,0,1],\"V\":[],\"c\":-1,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0088","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={0, 1, 2}，整數條件 P(x)：x≥1，待檢驗全稱 ∀x∈U P(x)。c=0 原本是合法反例；改成明示子論域 V={0} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈V=是；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042088,"parameters":{"U":[0,1,2],"V":[0],"predicate":{"kind":"ge","value":1},"c":0,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[0,1,2],\"V\":[0],\"c\":0,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0089","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={1, 2, 3}，整數條件 P(x)：x≥3，待檢驗全稱 ∀x∈U P(x)。c=1 原本是合法反例；改成明示子論域 V={2, 3} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042089,"parameters":{"U":[1,2,3],"V":[2,3],"predicate":{"kind":"ge","value":3},"c":1,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[1,2,3],\"V\":[2,3],\"c\":1,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0090","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={2, 3, 4}，整數條件 P(x)：x≥3，待檢驗全稱 ∀x∈U P(x)。c=2 原本是合法反例；改成明示子論域 V=∅ 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042090,"parameters":{"U":[2,3,4],"V":[],"predicate":{"kind":"ge","value":3},"c":2,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[2,3,4],\"V\":[],\"c\":2,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0091","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={3, 4, 5}，整數條件 P(x)：x≥5，待檢驗全稱 ∀x∈U P(x)。c=3 原本是合法反例；改成明示子論域 V={3} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈V=是；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042091,"parameters":{"U":[3,4,5],"V":[3],"predicate":{"kind":"ge","value":5},"c":3,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[3,4,5],\"V\":[3],\"c\":3,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0092","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={4, 5, 6}，整數條件 P(x)：x≥5，待檢驗全稱 ∀x∈U P(x)。c=4 原本是合法反例；改成明示子論域 V={5, 6} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042092,"parameters":{"U":[4,5,6],"V":[5,6],"predicate":{"kind":"ge","value":5},"c":4,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[4,5,6],\"V\":[5,6],\"c\":4,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0093","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={5, 6, 7}，整數條件 P(x)：x≥7，待檢驗全稱 ∀x∈U P(x)。c=5 原本是合法反例；改成明示子論域 V=∅ 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042093,"parameters":{"U":[5,6,7],"V":[],"predicate":{"kind":"ge","value":7},"c":5,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[5,6,7],\"V\":[],\"c\":5,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0094","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={6, 7, 8}，整數條件 P(x)：x≥7，待檢驗全稱 ∀x∈U P(x)。c=6 原本是合法反例；改成明示子論域 V={6} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈V=是；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042094,"parameters":{"U":[6,7,8],"V":[6],"predicate":{"kind":"ge","value":7},"c":6,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[6,7,8],\"V\":[6],\"c\":6,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0095","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={7, 8, 9}，整數條件 P(x)：x≥9，待檢驗全稱 ∀x∈U P(x)。c=7 原本是合法反例；改成明示子論域 V={8, 9} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042095,"parameters":{"U":[7,8,9],"V":[8,9],"predicate":{"kind":"ge","value":9},"c":7,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[7,8,9],\"V\":[8,9],\"c\":7,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0096","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={8, 9, 10}，整數條件 P(x)：x≥9，待檢驗全稱 ∀x∈U P(x)。c=8 原本是合法反例；改成明示子論域 V=∅ 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042096,"parameters":{"U":[8,9,10],"V":[],"predicate":{"kind":"ge","value":9},"c":8,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[8,9,10],\"V\":[],\"c\":8,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0097","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={9, 10, 11}，整數條件 P(x)：x≥11，待檢驗全稱 ∀x∈U P(x)。c=9 原本是合法反例；改成明示子論域 V={9} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈V=是；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042097,"parameters":{"U":[9,10,11],"V":[9],"predicate":{"kind":"ge","value":11},"c":9,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[9,10,11],\"V\":[9],\"c\":9,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"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-r02-counterexample-0098","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={10, 11, 12}，整數條件 P(x)：x≥11，待檢驗全稱 ∀x∈U P(x)。c=10 原本是合法反例；改成明示子論域 V={11, 12} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042098,"parameters":{"U":[10,11,12],"V":[11,12],"predicate":{"kind":"ge","value":11},"c":10,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[10,11,12],\"V\":[11,12],\"c\":10,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"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-r02-counterexample-0099","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={11, 12, 13}，整數條件 P(x)：x≥13，待檢驗全稱 ∀x∈U P(x)。c=11 原本是合法反例；改成明示子論域 V=∅ 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"否","explanation_zh":"c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042099,"parameters":{"U":[11,12,13],"V":[],"predicate":{"kind":"ge","value":13},"c":11,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[11,12,13],\"V\":[],\"c\":11,\"operation\":\"restricted_witness\",\"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-r02-counterexample-0100","batch_id":"batch-20261003-advance-20261004-r02-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後原反例的資格","stem_zh":"明示有限論域 U={12, 13, 14}，整數條件 P(x)：x≥13，待檢驗全稱 ∀x∈U P(x)。c=12 原本是合法反例；改成明示子論域 V={12} 後，c 還是合法反例嗎？","answer":{"kind":"short_text","value":"是","explanation_zh":"c∈V=是；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後原反例的資格"]},"variant":{"family_id":"qf-counterexample-domain-restriction","seed":202610042100,"parameters":{"U":[12,13,14],"V":[12],"predicate":{"kind":"ge","value":13},"c":12,"operation":"restricted_witness","semantic_template":"qf-counterexample-domain-restriction/{\"U\":[12,13,14],\"V\":[12],\"c\":12,\"operation\":\"restricted_witness\",\"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"}}