{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0001","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(0)=∅，依索引列P的完整布林述詞表=[]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([], [], 是, 不存在)","explanation_zh":"離散結構核對=([], [], 是, 不存在)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177001,"parameters":{"n":0,"truth":[],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":0,\"truth\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0002","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(3)=[0, 1, 2]，依索引列P的完整布林述詞表=[是, 是, 是]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 是], [1, 是], [2, 是]], [], 是, 不存在)","explanation_zh":"離散結構核對=([[0, 是], [1, 是], [2, 是]], [], 是, 不存在)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177002,"parameters":{"n":3,"truth":[true,true,true],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":3,\"truth\":[true,true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0003","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(3)=[0, 1, 2]，依索引列P的完整布林述詞表=[否, 否, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 否], [1, 否], [2, 否]], [0, 1, 2], 否, 0)","explanation_zh":"離散結構核對=([[0, 否], [1, 否], [2, 否]], [0, 1, 2], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177003,"parameters":{"n":3,"truth":[false,false,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":3,\"truth\":[false,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0004","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 否, 否, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 否], [1, 否], [2, 否], [3, 否], [4, 否]], [0, 1, 2, 3, 4], 否, 0)","explanation_zh":"離散結構核對=([[0, 否], [1, 否], [2, 否], [3, 否], [4, 否]], [0, 1, 2, 3, 4], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177004,"parameters":{"n":5,"truth":[false,false,false,false,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,false,false,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0005","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 否, 否, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 是], [1, 否], [2, 否], [3, 否], [4, 否]], [1, 2, 3, 4], 否, 1)","explanation_zh":"離散結構核對=([[0, 是], [1, 否], [2, 否], [3, 否], [4, 否]], [1, 2, 3, 4], 否, 1)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177005,"parameters":{"n":5,"truth":[true,false,false,false,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,false,false,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0006","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 否, 否, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 否], [1, 是], [2, 否], [3, 否], [4, 否]], [0, 2, 3, 4], 否, 0)","explanation_zh":"離散結構核對=([[0, 否], [1, 是], [2, 否], [3, 否], [4, 否]], [0, 2, 3, 4], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177006,"parameters":{"n":5,"truth":[false,true,false,false,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,true,false,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0007","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 否, 否, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 是], [1, 是], [2, 否], [3, 否], [4, 否]], [2, 3, 4], 否, 2)","explanation_zh":"離散結構核對=([[0, 是], [1, 是], [2, 否], [3, 否], [4, 否]], [2, 3, 4], 否, 2)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177007,"parameters":{"n":5,"truth":[true,true,false,false,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,true,false,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0008","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 是, 否, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 否], [1, 否], [2, 是], [3, 否], [4, 否]], [0, 1, 3, 4], 否, 0)","explanation_zh":"離散結構核對=([[0, 否], [1, 否], [2, 是], [3, 否], [4, 否]], [0, 1, 3, 4], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177008,"parameters":{"n":5,"truth":[false,false,true,false,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,false,true,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0009","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 是, 否, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 是], [1, 否], [2, 是], [3, 否], [4, 否]], [1, 3, 4], 否, 1)","explanation_zh":"離散結構核對=([[0, 是], [1, 否], [2, 是], [3, 否], [4, 否]], [1, 3, 4], 否, 1)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177009,"parameters":{"n":5,"truth":[true,false,true,false,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,false,true,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0010","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 是, 否, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 否], [1, 是], [2, 是], [3, 否], [4, 否]], [0, 3, 4], 否, 0)","explanation_zh":"離散結構核對=([[0, 否], [1, 是], [2, 是], [3, 否], [4, 否]], [0, 3, 4], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177010,"parameters":{"n":5,"truth":[false,true,true,false,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,true,true,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0011","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 是, 否, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 是], [1, 是], [2, 是], [3, 否], [4, 否]], [3, 4], 否, 3)","explanation_zh":"離散結構核對=([[0, 是], [1, 是], [2, 是], [3, 否], [4, 否]], [3, 4], 否, 3)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177011,"parameters":{"n":5,"truth":[true,true,true,false,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,true,true,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0012","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 否, 是, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 否], [1, 否], [2, 否], [3, 是], [4, 否]], [0, 1, 2, 4], 否, 0)","explanation_zh":"離散結構核對=([[0, 否], [1, 否], [2, 否], [3, 是], [4, 否]], [0, 1, 2, 4], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177012,"parameters":{"n":5,"truth":[false,false,false,true,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,false,false,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0013","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 否, 是, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 是], [1, 否], [2, 否], [3, 是], [4, 否]], [1, 2, 4], 否, 1)","explanation_zh":"離散結構核對=([[0, 是], [1, 否], [2, 否], [3, 是], [4, 否]], [1, 2, 4], 否, 1)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177013,"parameters":{"n":5,"truth":[true,false,false,true,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,false,false,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0014","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 否, 是, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 否], [1, 是], [2, 否], [3, 是], [4, 否]], [0, 2, 4], 否, 0)","explanation_zh":"離散結構核對=([[0, 否], [1, 是], [2, 否], [3, 是], [4, 否]], [0, 2, 4], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177014,"parameters":{"n":5,"truth":[false,true,false,true,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,true,false,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0015","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 否, 是, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 是], [1, 是], [2, 否], [3, 是], [4, 否]], [2, 4], 否, 2)","explanation_zh":"離散結構核對=([[0, 是], [1, 是], [2, 否], [3, 是], [4, 否]], [2, 4], 否, 2)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177015,"parameters":{"n":5,"truth":[true,true,false,true,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,true,false,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0016","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 是, 是, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 否], [1, 否], [2, 是], [3, 是], [4, 否]], [0, 1, 4], 否, 0)","explanation_zh":"離散結構核對=([[0, 否], [1, 否], [2, 是], [3, 是], [4, 否]], [0, 1, 4], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177016,"parameters":{"n":5,"truth":[false,false,true,true,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,false,true,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0017","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 是, 是, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 是], [1, 否], [2, 是], [3, 是], [4, 否]], [1, 4], 否, 1)","explanation_zh":"離散結構核對=([[0, 是], [1, 否], [2, 是], [3, 是], [4, 否]], [1, 4], 否, 1)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177017,"parameters":{"n":5,"truth":[true,false,true,true,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,false,true,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0018","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 是, 是, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 否], [1, 是], [2, 是], [3, 是], [4, 否]], [0, 4], 否, 0)","explanation_zh":"離散結構核對=([[0, 否], [1, 是], [2, 是], [3, 是], [4, 否]], [0, 4], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177018,"parameters":{"n":5,"truth":[false,true,true,true,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,true,true,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0019","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 是, 是, 否]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 是], [1, 是], [2, 是], [3, 是], [4, 否]], [4], 否, 4)","explanation_zh":"離散結構核對=([[0, 是], [1, 是], [2, 是], [3, 是], [4, 否]], [4], 否, 4)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177019,"parameters":{"n":5,"truth":[true,true,true,true,false],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,true,true,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0020","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 否, 否, 是]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 否], [1, 否], [2, 否], [3, 否], [4, 是]], [0, 1, 2, 3], 否, 0)","explanation_zh":"離散結構核對=([[0, 否], [1, 否], [2, 否], [3, 否], [4, 是]], [0, 1, 2, 3], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177020,"parameters":{"n":5,"truth":[false,false,false,false,true],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,false,false,false,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0021","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 否, 否, 是]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 是], [1, 否], [2, 否], [3, 否], [4, 是]], [1, 2, 3], 否, 1)","explanation_zh":"離散結構核對=([[0, 是], [1, 否], [2, 否], [3, 否], [4, 是]], [1, 2, 3], 否, 1)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177021,"parameters":{"n":5,"truth":[true,false,false,false,true],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,false,false,false,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0022","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 否, 否, 是]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 否], [1, 是], [2, 否], [3, 否], [4, 是]], [0, 2, 3], 否, 0)","explanation_zh":"離散結構核對=([[0, 否], [1, 是], [2, 否], [3, 否], [4, 是]], [0, 2, 3], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177022,"parameters":{"n":5,"truth":[false,true,false,false,true],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,true,false,false,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0023","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 否, 否, 是]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 是], [1, 是], [2, 否], [3, 否], [4, 是]], [2, 3], 否, 2)","explanation_zh":"離散結構核對=([[0, 是], [1, 是], [2, 否], [3, 否], [4, 是]], [2, 3], 否, 2)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177023,"parameters":{"n":5,"truth":[true,true,false,false,true],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,true,false,false,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0024","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 是, 否, 是]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 否], [1, 否], [2, 是], [3, 否], [4, 是]], [0, 1, 3], 否, 0)","explanation_zh":"離散結構核對=([[0, 否], [1, 否], [2, 是], [3, 否], [4, 是]], [0, 1, 3], 否, 0)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177024,"parameters":{"n":5,"truth":[false,false,true,false,true],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[false,false,true,false,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0025","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"全有限失敗集與最小見證","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 是, 否, 是]，是表示真、否表示假。答 (按索引列[i,P(i)]的完整述詞表, 所有失敗索引, 全稱為真資格, 最小失敗索引或不存在)。布林欄以是／否回答。","answer":{"kind":"expression","value":"([[0, 是], [1, 否], [2, 是], [3, 否], [4, 是]], [1, 3], 否, 1)","explanation_zh":"離散結構核對=([[0, 是], [1, 否], [2, 是], [3, 否], [4, 是]], [1, 3], 否, 1)；枚舉論域每個位置，收齊假值索引再取最小。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全有限失敗集與最小見證"]},"variant":{"family_id":"qf-density17-counter-full-failure-min","seed":202610177025,"parameters":{"n":5,"truth":[true,false,true,false,true],"domain":"Fin_predicate_table","operation":"counter-full-failure-min","semantic_template":"qf-density17-counter-full-failure-min/{\"n\":5,\"truth\":[true,false,true,false,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0026","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(0)=∅，依索引列P的完整布林述詞表=[]，是表示真、否表示假。有序候選=[-1, 0, 1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([], [否, 否, 否], [不合論域, 不合論域, 不合論域], [否, 否, 否])","explanation_zh":"離散結構核對=([], [否, 否, 否], [不合論域, 不合論域, 不合論域], [否, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177026,"parameters":{"n":0,"truth":[],"queries":[-1,0,1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":0,\"queries\":[\"untyped\",\"untyped\",\"untyped\"],\"truth\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0027","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[-1, -1, -1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [否, 否, 否], [不合論域, 不合論域, 不合論域], [否, 否, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [否, 否, 否], [不合論域, 不合論域, 不合論域], [否, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177027,"parameters":{"n":2,"truth":[true,false],"queries":[-1,-1,-1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",\"untyped\",\"untyped\"],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0028","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[0, -1, -1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 否, 否], [是, 不合論域, 不合論域], [否, 否, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 否, 否], [是, 不合論域, 不合論域], [否, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177028,"parameters":{"n":2,"truth":[true,false],"queries":[0,-1,-1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,\"untyped\",\"untyped\"],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0029","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[1, -1, -1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 否, 否], [否, 不合論域, 不合論域], [是, 否, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 否, 否], [否, 不合論域, 不合論域], [是, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177029,"parameters":{"n":2,"truth":[true,false],"queries":[1,-1,-1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,\"untyped\",\"untyped\"],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0030","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[-1, 0, -1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [否, 是, 否], [不合論域, 是, 不合論域], [否, 否, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [否, 是, 否], [不合論域, 是, 不合論域], [否, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177030,"parameters":{"n":2,"truth":[true,false],"queries":[-1,0,-1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",0,\"untyped\"],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0031","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[0, 0, -1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 是, 否], [是, 是, 不合論域], [否, 否, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 是, 否], [是, 是, 不合論域], [否, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177031,"parameters":{"n":2,"truth":[true,false],"queries":[0,0,-1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,0,\"untyped\"],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0032","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[1, 0, -1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 是, 否], [否, 是, 不合論域], [是, 否, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 是, 否], [否, 是, 不合論域], [是, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177032,"parameters":{"n":2,"truth":[true,false],"queries":[1,0,-1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,0,\"untyped\"],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0033","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[-1, 1, -1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [否, 是, 否], [不合論域, 否, 不合論域], [否, 是, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [否, 是, 否], [不合論域, 否, 不合論域], [否, 是, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177033,"parameters":{"n":2,"truth":[true,false],"queries":[-1,1,-1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",1,\"untyped\"],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0034","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[0, 1, -1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 是, 否], [是, 否, 不合論域], [否, 是, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 是, 否], [是, 否, 不合論域], [否, 是, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177034,"parameters":{"n":2,"truth":[true,false],"queries":[0,1,-1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,1,\"untyped\"],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0035","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[1, 1, -1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 是, 否], [否, 否, 不合論域], [是, 是, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 是, 否], [否, 否, 不合論域], [是, 是, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177035,"parameters":{"n":2,"truth":[true,false],"queries":[1,1,-1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,1,\"untyped\"],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0036","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[-1, -1, 0]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [否, 否, 是], [不合論域, 不合論域, 是], [否, 否, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [否, 否, 是], [不合論域, 不合論域, 是], [否, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177036,"parameters":{"n":2,"truth":[true,false],"queries":[-1,-1,0],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",\"untyped\",0],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0037","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[0, -1, 0]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 否, 是], [是, 不合論域, 是], [否, 否, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 否, 是], [是, 不合論域, 是], [否, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177037,"parameters":{"n":2,"truth":[true,false],"queries":[0,-1,0],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,\"untyped\",0],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0038","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[1, -1, 0]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 否, 是], [否, 不合論域, 是], [是, 否, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 否, 是], [否, 不合論域, 是], [是, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177038,"parameters":{"n":2,"truth":[true,false],"queries":[1,-1,0],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,\"untyped\",0],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0039","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[-1, 0, 0]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [否, 是, 是], [不合論域, 是, 是], [否, 否, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [否, 是, 是], [不合論域, 是, 是], [否, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177039,"parameters":{"n":2,"truth":[true,false],"queries":[-1,0,0],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",0,0],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0040","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[0, 0, 0]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 是, 是], [是, 是, 是], [否, 否, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 是, 是], [是, 是, 是], [否, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177040,"parameters":{"n":2,"truth":[true,false],"queries":[0,0,0],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,0,0],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0041","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[1, 0, 0]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 是, 是], [否, 是, 是], [是, 否, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 是, 是], [否, 是, 是], [是, 否, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177041,"parameters":{"n":2,"truth":[true,false],"queries":[1,0,0],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,0,0],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0042","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[-1, 1, 0]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [否, 是, 是], [不合論域, 否, 是], [否, 是, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [否, 是, 是], [不合論域, 否, 是], [否, 是, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177042,"parameters":{"n":2,"truth":[true,false],"queries":[-1,1,0],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",1,0],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0043","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[0, 1, 0]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 是, 是], [是, 否, 是], [否, 是, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 是, 是], [是, 否, 是], [否, 是, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177043,"parameters":{"n":2,"truth":[true,false],"queries":[0,1,0],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,1,0],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0044","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[1, 1, 0]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 是, 是], [否, 否, 是], [是, 是, 否])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 是, 是], [否, 否, 是], [是, 是, 否])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177044,"parameters":{"n":2,"truth":[true,false],"queries":[1,1,0],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,1,0],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0045","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[-1, -1, 1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [否, 否, 是], [不合論域, 不合論域, 否], [否, 否, 是])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [否, 否, 是], [不合論域, 不合論域, 否], [否, 否, 是])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177045,"parameters":{"n":2,"truth":[true,false],"queries":[-1,-1,1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",\"untyped\",1],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0046","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[0, -1, 1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 否, 是], [是, 不合論域, 否], [否, 否, 是])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 否, 是], [是, 不合論域, 否], [否, 否, 是])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177046,"parameters":{"n":2,"truth":[true,false],"queries":[0,-1,1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,\"untyped\",1],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0047","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[1, -1, 1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 否, 是], [否, 不合論域, 否], [是, 否, 是])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 否, 是], [否, 不合論域, 否], [是, 否, 是])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177047,"parameters":{"n":2,"truth":[true,false],"queries":[1,-1,1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,\"untyped\",1],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0048","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[-1, 0, 1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [否, 是, 是], [不合論域, 是, 否], [否, 否, 是])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [否, 是, 是], [不合論域, 是, 否], [否, 否, 是])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177048,"parameters":{"n":2,"truth":[true,false],"queries":[-1,0,1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[\"untyped\",0,1],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0049","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[0, 0, 1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 是, 是], [是, 是, 否], [否, 否, 是])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 是, 是], [是, 是, 否], [否, 否, 是])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177049,"parameters":{"n":2,"truth":[true,false],"queries":[0,0,1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[0,0,1],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0050","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"三候選先論域後失敗","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(2)=[0, 1]，依索引列P的完整布林述詞表=[是, 否]，是表示真、否表示假。有序候選=[1, 0, 1]。答 (按索引列[i,P(i)]的完整述詞表, 三論域資格, 三P值或不合論域, 三合法反例資格)。域外不讀述詞表，也不給P假值。","answer":{"kind":"expression","value":"([[0, 是], [1, 否]], [是, 是, 是], [否, 是, 否], [是, 否, 是])","explanation_zh":"離散結構核對=([[0, 是], [1, 否]], [是, 是, 是], [否, 是, 否], [是, 否, 是])；先判每候選在Fin(n)內；合法且P為假才是反例。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三候選先論域後失敗"]},"variant":{"family_id":"qf-density17-counter-three-candidate-guards","seed":202610177050,"parameters":{"n":2,"truth":[true,false],"queries":[1,0,1],"domain":"Fin_predicate_table","operation":"counter-three-candidate-guards","semantic_template":"qf-density17-counter-three-candidate-guards/{\"n\":2,\"queries\":[1,0,1],\"truth\":[true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0051","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(0)=∅，依索引列P的完整布林述詞表=[]，是表示真、否表示假。來源子集列項=[]，原候選見證=0。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([], [], [], 否, 否, 是)","explanation_zh":"離散結構核對=([], [], [], 否, 否, 是)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177051,"parameters":{"n":0,"truth":[],"subset":[],"witness":0,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":0,\"subset\":[],\"truth\":[],\"witness\":\"untyped\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0052","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(3)=[0, 1, 2]，依索引列P的完整布林述詞表=[否, 是, 否]，是表示真、否表示假。來源子集列項=[]，原候選見證=0。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 2], [], [], 是, 否, 是)","explanation_zh":"離散結構核對=([0, 2], [], [], 是, 否, 是)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177052,"parameters":{"n":3,"truth":[false,true,false],"subset":[],"witness":0,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":3,\"subset\":[],\"truth\":[false,true,false],\"witness\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0053","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(3)=[0, 1, 2]，依索引列P的完整布林述詞表=[否, 是, 否]，是表示真、否表示假。來源子集列項=[0]，原候選見證=0。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 2], [0], [0], 是, 是, 否)","explanation_zh":"離散結構核對=([0, 2], [0], [0], 是, 是, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177053,"parameters":{"n":3,"truth":[false,true,false],"subset":[0],"witness":0,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":3,\"subset\":[0],\"truth\":[false,true,false],\"witness\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0054","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(3)=[0, 1, 2]，依索引列P的完整布林述詞表=[否, 是, 否]，是表示真、否表示假。來源子集列項=[1]，原候選見證=2。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 2], [1], [], 是, 否, 是)","explanation_zh":"離散結構核對=([0, 2], [1], [], 是, 否, 是)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177054,"parameters":{"n":3,"truth":[false,true,false],"subset":[1],"witness":2,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":3,\"subset\":[1],\"truth\":[false,true,false],\"witness\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0055","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(3)=[0, 1, 2]，依索引列P的完整布林述詞表=[是, 是, 是]，是表示真、否表示假。來源子集列項=[2, 0]，原候選見證=-1。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([], [0, 2], [], 否, 否, 是)","explanation_zh":"離散結構核對=([], [0, 2], [], 否, 否, 是)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177055,"parameters":{"n":3,"truth":[true,true,true],"subset":[2,0],"witness":-1,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":3,\"subset\":[0,2],\"truth\":[true,true,true],\"witness\":\"untyped\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0056","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 否, 否, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=4。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4], [0, 2, 4], [0, 2, 4], 是, 是, 否)","explanation_zh":"離散結構核對=([0, 1, 2, 3, 4], [0, 2, 4], [0, 2, 4], 是, 是, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177056,"parameters":{"n":5,"truth":[false,false,false,false,false],"subset":[4,2,0],"witness":4,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,false,false,false,false],\"witness\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0057","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 否, 否, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=5。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([1, 2, 3, 4], [0, 2, 4], [2, 4], 否, 否, 否)","explanation_zh":"離散結構核對=([1, 2, 3, 4], [0, 2, 4], [2, 4], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177057,"parameters":{"n":5,"truth":[true,false,false,false,false],"subset":[4,2,0],"witness":5,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,false,false,false,false],\"witness\":\"untyped\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0058","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 否, 否, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=-1。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 2, 3, 4], [0, 2, 4], [0, 2, 4], 否, 否, 否)","explanation_zh":"離散結構核對=([0, 2, 3, 4], [0, 2, 4], [0, 2, 4], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177058,"parameters":{"n":5,"truth":[false,true,false,false,false],"subset":[4,2,0],"witness":-1,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,true,false,false,false],\"witness\":\"untyped\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0059","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 否, 否, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=0。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([2, 3, 4], [0, 2, 4], [2, 4], 否, 否, 否)","explanation_zh":"離散結構核對=([2, 3, 4], [0, 2, 4], [2, 4], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177059,"parameters":{"n":5,"truth":[true,true,false,false,false],"subset":[4,2,0],"witness":0,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,true,false,false,false],\"witness\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0060","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 是, 否, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=1。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 1, 3, 4], [0, 2, 4], [0, 4], 是, 否, 否)","explanation_zh":"離散結構核對=([0, 1, 3, 4], [0, 2, 4], [0, 4], 是, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177060,"parameters":{"n":5,"truth":[false,false,true,false,false],"subset":[4,2,0],"witness":1,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,false,true,false,false],\"witness\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0061","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 是, 否, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=2。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([1, 3, 4], [0, 2, 4], [4], 否, 否, 否)","explanation_zh":"離散結構核對=([1, 3, 4], [0, 2, 4], [4], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177061,"parameters":{"n":5,"truth":[true,false,true,false,false],"subset":[4,2,0],"witness":2,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,false,true,false,false],\"witness\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0062","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 是, 否, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=3。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 3, 4], [0, 2, 4], [0, 4], 是, 否, 否)","explanation_zh":"離散結構核對=([0, 3, 4], [0, 2, 4], [0, 4], 是, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177062,"parameters":{"n":5,"truth":[false,true,true,false,false],"subset":[4,2,0],"witness":3,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,true,true,false,false],\"witness\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0063","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 是, 否, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=4。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([3, 4], [0, 2, 4], [4], 是, 是, 否)","explanation_zh":"離散結構核對=([3, 4], [0, 2, 4], [4], 是, 是, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177063,"parameters":{"n":5,"truth":[true,true,true,false,false],"subset":[4,2,0],"witness":4,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,true,true,false,false],\"witness\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0064","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 否, 是, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=5。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 1, 2, 4], [0, 2, 4], [0, 2, 4], 否, 否, 否)","explanation_zh":"離散結構核對=([0, 1, 2, 4], [0, 2, 4], [0, 2, 4], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177064,"parameters":{"n":5,"truth":[false,false,false,true,false],"subset":[4,2,0],"witness":5,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,false,false,true,false],\"witness\":\"untyped\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0065","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 否, 是, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=-1。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([1, 2, 4], [0, 2, 4], [2, 4], 否, 否, 否)","explanation_zh":"離散結構核對=([1, 2, 4], [0, 2, 4], [2, 4], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177065,"parameters":{"n":5,"truth":[true,false,false,true,false],"subset":[4,2,0],"witness":-1,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,false,false,true,false],\"witness\":\"untyped\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0066","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 否, 是, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=0。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 2, 4], [0, 2, 4], [0, 2, 4], 是, 是, 否)","explanation_zh":"離散結構核對=([0, 2, 4], [0, 2, 4], [0, 2, 4], 是, 是, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177066,"parameters":{"n":5,"truth":[false,true,false,true,false],"subset":[4,2,0],"witness":0,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,true,false,true,false],\"witness\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0067","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 否, 是, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=1。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([2, 4], [0, 2, 4], [2, 4], 否, 否, 否)","explanation_zh":"離散結構核對=([2, 4], [0, 2, 4], [2, 4], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177067,"parameters":{"n":5,"truth":[true,true,false,true,false],"subset":[4,2,0],"witness":1,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,true,false,true,false],\"witness\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0068","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 是, 是, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=2。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 1, 4], [0, 2, 4], [0, 4], 否, 否, 否)","explanation_zh":"離散結構核對=([0, 1, 4], [0, 2, 4], [0, 4], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177068,"parameters":{"n":5,"truth":[false,false,true,true,false],"subset":[4,2,0],"witness":2,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,false,true,true,false],\"witness\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0069","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 是, 是, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=3。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([1, 4], [0, 2, 4], [4], 否, 否, 否)","explanation_zh":"離散結構核對=([1, 4], [0, 2, 4], [4], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177069,"parameters":{"n":5,"truth":[true,false,true,true,false],"subset":[4,2,0],"witness":3,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,false,true,true,false],\"witness\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0070","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 是, 是, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=4。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 4], [0, 2, 4], [0, 4], 是, 是, 否)","explanation_zh":"離散結構核對=([0, 4], [0, 2, 4], [0, 4], 是, 是, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177070,"parameters":{"n":5,"truth":[false,true,true,true,false],"subset":[4,2,0],"witness":4,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,true,true,true,false],\"witness\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0071","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 是, 是, 否]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=5。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([4], [0, 2, 4], [4], 否, 否, 否)","explanation_zh":"離散結構核對=([4], [0, 2, 4], [4], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177071,"parameters":{"n":5,"truth":[true,true,true,true,false],"subset":[4,2,0],"witness":5,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,true,true,true,false],\"witness\":\"untyped\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0072","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 否, 否, 是]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=-1。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 1, 2, 3], [0, 2, 4], [0, 2], 否, 否, 否)","explanation_zh":"離散結構核對=([0, 1, 2, 3], [0, 2, 4], [0, 2], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177072,"parameters":{"n":5,"truth":[false,false,false,false,true],"subset":[4,2,0],"witness":-1,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,false,false,false,true],\"witness\":\"untyped\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0073","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 否, 否, 是]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=0。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([1, 2, 3], [0, 2, 4], [2], 否, 否, 否)","explanation_zh":"離散結構核對=([1, 2, 3], [0, 2, 4], [2], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177073,"parameters":{"n":5,"truth":[true,false,false,false,true],"subset":[4,2,0],"witness":0,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,false,false,false,true],\"witness\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0074","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 否, 否, 是]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=1。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([0, 2, 3], [0, 2, 4], [0, 2], 否, 否, 否)","explanation_zh":"離散結構核對=([0, 2, 3], [0, 2, 4], [0, 2], 否, 否, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177074,"parameters":{"n":5,"truth":[false,true,false,false,true],"subset":[4,2,0],"witness":1,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[false,true,false,false,true],\"witness\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0075","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"限制論域後的原見證資格","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 否, 否, 是]，是表示真、否表示假。來源子集列項=[4, 2, 0]，原候選見證=2。答 (原全部失敗集, 正規子集, 限制後失敗集, 原見證合法資格, 限制後見證合法資格, 限制後全稱為真資格)。","answer":{"kind":"expression","value":"([2, 3], [0, 2, 4], [2], 是, 是, 否)","explanation_zh":"離散結構核對=([2, 3], [0, 2, 4], [2], 是, 是, 否)；先收原失敗，再與子集交集；見證是否仍在子集另判。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["限制論域後的原見證資格"]},"variant":{"family_id":"qf-density17-counter-restriction-witness","seed":202610177075,"parameters":{"n":5,"truth":[true,true,false,false,true],"subset":[4,2,0],"witness":2,"domain":"Fin_predicate_table","operation":"counter-restriction-witness","semantic_template":"qf-density17-counter-restriction-witness/{\"n\":5,\"subset\":[0,2,4],\"truth\":[true,true,false,false,true],\"witness\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0076","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(0)=∅，依索引列P的完整布林述詞表=[]，是表示真、否表示假。新增標籤n=0，新值extra_truth=是，延伸論域為Fin(1)=[0]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([], [[0, 是]], [], 是, 是, 0)","explanation_zh":"離散結構核對=([], [[0, 是]], [], 是, 是, 0)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177076,"parameters":{"n":0,"truth":[],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":0,\"truth\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0077","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(0)=∅，依索引列P的完整布林述詞表=[]，是表示真、否表示假。新增標籤n=0，新值extra_truth=否，延伸論域為Fin(1)=[0]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([], [[0, 否]], [0], 是, 否, 0)","explanation_zh":"離散結構核對=([], [[0, 否]], [0], 是, 否, 0)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177077,"parameters":{"n":0,"truth":[],"extra_truth":false,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":0,\"truth\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0078","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，依索引列P的完整布林述詞表=[是, 是, 是, 是, 是, 是, 是, 是]，是表示真、否表示假。新增標籤n=8，新值extra_truth=否，延伸論域為Fin(9)=[0, 1, 2, 3, 4, 5, 6, 7, 8]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([], [[0, 是], [1, 是], [2, 是], [3, 是], [4, 是], [5, 是], [6, 是], [7, 是], [8, 否]], [8], 是, 否, 8)","explanation_zh":"離散結構核對=([], [[0, 是], [1, 是], [2, 是], [3, 是], [4, 是], [5, 是], [6, 是], [7, 是], [8, 否]], [8], 是, 否, 8)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177078,"parameters":{"n":8,"truth":[true,true,true,true,true,true,true,true],"extra_truth":false,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":8,\"truth\":[true,true,true,true,true,true,true,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0079","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7]，依索引列P的完整布林述詞表=[否, 否, 否, 否, 否, 否, 否, 否]，是表示真、否表示假。新增標籤n=8，新值extra_truth=是，延伸論域為Fin(9)=[0, 1, 2, 3, 4, 5, 6, 7, 8]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], [[0, 否], [1, 否], [2, 否], [3, 否], [4, 否], [5, 否], [6, 否], [7, 否], [8, 是]], [0, 1, 2, 3, 4, 5, 6, 7], 否, 否, 8)","explanation_zh":"離散結構核對=([0, 1, 2, 3, 4, 5, 6, 7], [[0, 否], [1, 否], [2, 否], [3, 否], [4, 否], [5, 否], [6, 否], [7, 否], [8, 是]], [0, 1, 2, 3, 4, 5, 6, 7], 否, 否, 8)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177079,"parameters":{"n":8,"truth":[false,false,false,false,false,false,false,false],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":8,\"truth\":[false,false,false,false,false,false,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0080","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 否, 否, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=是，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4], [[0, 否], [1, 否], [2, 否], [3, 否], [4, 否], [5, 是]], [0, 1, 2, 3, 4], 否, 否, 5)","explanation_zh":"離散結構核對=([0, 1, 2, 3, 4], [[0, 否], [1, 否], [2, 否], [3, 否], [4, 否], [5, 是]], [0, 1, 2, 3, 4], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177080,"parameters":{"n":5,"truth":[false,false,false,false,false],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,false,false,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0081","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 否, 否, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=否，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([1, 2, 3, 4], [[0, 是], [1, 否], [2, 否], [3, 否], [4, 否], [5, 否]], [1, 2, 3, 4, 5], 否, 否, 5)","explanation_zh":"離散結構核對=([1, 2, 3, 4], [[0, 是], [1, 否], [2, 否], [3, 否], [4, 否], [5, 否]], [1, 2, 3, 4, 5], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177081,"parameters":{"n":5,"truth":[true,false,false,false,false],"extra_truth":false,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,false,false,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0082","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 否, 否, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=是，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([0, 2, 3, 4], [[0, 否], [1, 是], [2, 否], [3, 否], [4, 否], [5, 是]], [0, 2, 3, 4], 否, 否, 5)","explanation_zh":"離散結構核對=([0, 2, 3, 4], [[0, 否], [1, 是], [2, 否], [3, 否], [4, 否], [5, 是]], [0, 2, 3, 4], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177082,"parameters":{"n":5,"truth":[false,true,false,false,false],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,true,false,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0083","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 否, 否, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=否，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([2, 3, 4], [[0, 是], [1, 是], [2, 否], [3, 否], [4, 否], [5, 否]], [2, 3, 4, 5], 否, 否, 5)","explanation_zh":"離散結構核對=([2, 3, 4], [[0, 是], [1, 是], [2, 否], [3, 否], [4, 否], [5, 否]], [2, 3, 4, 5], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177083,"parameters":{"n":5,"truth":[true,true,false,false,false],"extra_truth":false,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,true,false,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0084","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 是, 否, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=是，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([0, 1, 3, 4], [[0, 否], [1, 否], [2, 是], [3, 否], [4, 否], [5, 是]], [0, 1, 3, 4], 否, 否, 5)","explanation_zh":"離散結構核對=([0, 1, 3, 4], [[0, 否], [1, 否], [2, 是], [3, 否], [4, 否], [5, 是]], [0, 1, 3, 4], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177084,"parameters":{"n":5,"truth":[false,false,true,false,false],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,false,true,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0085","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 是, 否, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=否，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([1, 3, 4], [[0, 是], [1, 否], [2, 是], [3, 否], [4, 否], [5, 否]], [1, 3, 4, 5], 否, 否, 5)","explanation_zh":"離散結構核對=([1, 3, 4], [[0, 是], [1, 否], [2, 是], [3, 否], [4, 否], [5, 否]], [1, 3, 4, 5], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177085,"parameters":{"n":5,"truth":[true,false,true,false,false],"extra_truth":false,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,false,true,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0086","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 是, 否, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=是，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([0, 3, 4], [[0, 否], [1, 是], [2, 是], [3, 否], [4, 否], [5, 是]], [0, 3, 4], 否, 否, 5)","explanation_zh":"離散結構核對=([0, 3, 4], [[0, 否], [1, 是], [2, 是], [3, 否], [4, 否], [5, 是]], [0, 3, 4], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177086,"parameters":{"n":5,"truth":[false,true,true,false,false],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,true,true,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0087","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 是, 否, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=否，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([3, 4], [[0, 是], [1, 是], [2, 是], [3, 否], [4, 否], [5, 否]], [3, 4, 5], 否, 否, 5)","explanation_zh":"離散結構核對=([3, 4], [[0, 是], [1, 是], [2, 是], [3, 否], [4, 否], [5, 否]], [3, 4, 5], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177087,"parameters":{"n":5,"truth":[true,true,true,false,false],"extra_truth":false,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,true,true,false,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0088","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 否, 是, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=是，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([0, 1, 2, 4], [[0, 否], [1, 否], [2, 否], [3, 是], [4, 否], [5, 是]], [0, 1, 2, 4], 否, 否, 5)","explanation_zh":"離散結構核對=([0, 1, 2, 4], [[0, 否], [1, 否], [2, 否], [3, 是], [4, 否], [5, 是]], [0, 1, 2, 4], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177088,"parameters":{"n":5,"truth":[false,false,false,true,false],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,false,false,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0089","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 否, 是, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=否，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([1, 2, 4], [[0, 是], [1, 否], [2, 否], [3, 是], [4, 否], [5, 否]], [1, 2, 4, 5], 否, 否, 5)","explanation_zh":"離散結構核對=([1, 2, 4], [[0, 是], [1, 否], [2, 否], [3, 是], [4, 否], [5, 否]], [1, 2, 4, 5], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177089,"parameters":{"n":5,"truth":[true,false,false,true,false],"extra_truth":false,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,false,false,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0090","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 否, 是, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=是，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([0, 2, 4], [[0, 否], [1, 是], [2, 否], [3, 是], [4, 否], [5, 是]], [0, 2, 4], 否, 否, 5)","explanation_zh":"離散結構核對=([0, 2, 4], [[0, 否], [1, 是], [2, 否], [3, 是], [4, 否], [5, 是]], [0, 2, 4], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177090,"parameters":{"n":5,"truth":[false,true,false,true,false],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,true,false,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0091","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 否, 是, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=否，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([2, 4], [[0, 是], [1, 是], [2, 否], [3, 是], [4, 否], [5, 否]], [2, 4, 5], 否, 否, 5)","explanation_zh":"離散結構核對=([2, 4], [[0, 是], [1, 是], [2, 否], [3, 是], [4, 否], [5, 否]], [2, 4, 5], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177091,"parameters":{"n":5,"truth":[true,true,false,true,false],"extra_truth":false,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,true,false,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0092","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 是, 是, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=是，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([0, 1, 4], [[0, 否], [1, 否], [2, 是], [3, 是], [4, 否], [5, 是]], [0, 1, 4], 否, 否, 5)","explanation_zh":"離散結構核對=([0, 1, 4], [[0, 否], [1, 否], [2, 是], [3, 是], [4, 否], [5, 是]], [0, 1, 4], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177092,"parameters":{"n":5,"truth":[false,false,true,true,false],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,false,true,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0093","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 是, 是, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=否，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([1, 4], [[0, 是], [1, 否], [2, 是], [3, 是], [4, 否], [5, 否]], [1, 4, 5], 否, 否, 5)","explanation_zh":"離散結構核對=([1, 4], [[0, 是], [1, 否], [2, 是], [3, 是], [4, 否], [5, 否]], [1, 4, 5], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177093,"parameters":{"n":5,"truth":[true,false,true,true,false],"extra_truth":false,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,false,true,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0094","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 是, 是, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=是，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([0, 4], [[0, 否], [1, 是], [2, 是], [3, 是], [4, 否], [5, 是]], [0, 4], 否, 否, 5)","explanation_zh":"離散結構核對=([0, 4], [[0, 否], [1, 是], [2, 是], [3, 是], [4, 否], [5, 是]], [0, 4], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177094,"parameters":{"n":5,"truth":[false,true,true,true,false],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,true,true,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0095","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 是, 是, 否]，是表示真、否表示假。新增標籤n=5，新值extra_truth=否，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([4], [[0, 是], [1, 是], [2, 是], [3, 是], [4, 否], [5, 否]], [4, 5], 否, 否, 5)","explanation_zh":"離散結構核對=([4], [[0, 是], [1, 是], [2, 是], [3, 是], [4, 否], [5, 否]], [4, 5], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177095,"parameters":{"n":5,"truth":[true,true,true,true,false],"extra_truth":false,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,true,true,true,false]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0096","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 否, 否, 是]，是表示真、否表示假。新增標籤n=5，新值extra_truth=是，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([0, 1, 2, 3], [[0, 否], [1, 否], [2, 否], [3, 否], [4, 是], [5, 是]], [0, 1, 2, 3], 否, 否, 5)","explanation_zh":"離散結構核對=([0, 1, 2, 3], [[0, 否], [1, 否], [2, 否], [3, 否], [4, 是], [5, 是]], [0, 1, 2, 3], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177096,"parameters":{"n":5,"truth":[false,false,false,false,true],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,false,false,false,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0097","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 否, 否, 否, 是]，是表示真、否表示假。新增標籤n=5，新值extra_truth=否，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([1, 2, 3], [[0, 是], [1, 否], [2, 否], [3, 否], [4, 是], [5, 否]], [1, 2, 3, 5], 否, 否, 5)","explanation_zh":"離散結構核對=([1, 2, 3], [[0, 是], [1, 否], [2, 否], [3, 否], [4, 是], [5, 否]], [1, 2, 3, 5], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177097,"parameters":{"n":5,"truth":[true,false,false,false,true],"extra_truth":false,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,false,false,false,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0098","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 是, 否, 否, 是]，是表示真、否表示假。新增標籤n=5，新值extra_truth=是，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([0, 2, 3], [[0, 否], [1, 是], [2, 否], [3, 否], [4, 是], [5, 是]], [0, 2, 3], 否, 否, 5)","explanation_zh":"離散結構核對=([0, 2, 3], [[0, 否], [1, 是], [2, 否], [3, 否], [4, 是], [5, 是]], [0, 2, 3], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177098,"parameters":{"n":5,"truth":[false,true,false,false,true],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,true,false,false,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0099","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[是, 是, 否, 否, 是]，是表示真、否表示假。新增標籤n=5，新值extra_truth=否，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([2, 3], [[0, 是], [1, 是], [2, 否], [3, 否], [4, 是], [5, 否]], [2, 3, 5], 否, 否, 5)","explanation_zh":"離散結構核對=([2, 3], [[0, 是], [1, 是], [2, 否], [3, 否], [4, 是], [5, 否]], [2, 3, 5], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177099,"parameters":{"n":5,"truth":[true,true,false,false,true],"extra_truth":false,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":false,\"n\":5,\"truth\":[true,true,false,false,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r07-counterexample-0100","batch_id":"batch-20261012-advance-20261017-r07-counterexample-foundations","language":"zh-Hant","source_mko_ids":["mko-counterexample"],"title_zh":"增加一個明示述詞位置","stem_zh":"明示有限論域，完整布林述詞表。U=Fin(5)=[0, 1, 2, 3, 4]，依索引列P的完整布林述詞表=[否, 否, 是, 否, 是]，是表示真、否表示假。新增標籤n=5，新值extra_truth=是，延伸論域為Fin(6)=[0, 1, 2, 3, 4, 5]。答 (舊失敗集, 按索引列[i,P(i)]的延伸完整表, 新失敗集, 舊全稱資格, 新全稱資格, 新標籤n)。","answer":{"kind":"expression","value":"([0, 1, 3], [[0, 否], [1, 否], [2, 是], [3, 否], [4, 是], [5, 是]], [0, 1, 3], 否, 否, 5)","explanation_zh":"離散結構核對=([0, 1, 3], [[0, 否], [1, 否], [2, 是], [3, 否], [4, 是], [5, 是]], [0, 1, 3], 否, 否, 5)；只把新增布林值接到表尾，再完整枚舉新失敗集。；有限精確核對不是一般形式證明；域外失敗不是反例；有限枚舉不當無限域證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["增加一個明示述詞位置"]},"variant":{"family_id":"qf-density17-counter-domain-extension","seed":202610177100,"parameters":{"n":5,"truth":[false,false,true,false,true],"extra_truth":true,"domain":"Fin_predicate_table","operation":"counter-domain-extension","semantic_template":"qf-density17-counter-domain-extension/{\"extra_truth\":true,\"n\":5,\"truth\":[false,false,true,false,true]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}