{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0001","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[0, 0], [-1, 1], [1, -1]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 假], [0, -2, 2])","explanation_zh":"封閉比較真假與差=([真, 真, 假], [0, -2, 2])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123001,"parameters":{"pairs":[[0,0],[-1,1],[1,-1]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"0\",\"-2\",\"2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0002","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[-11, 0], [0, 2], [0, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [-11, -2, 0])","explanation_zh":"封閉比較真假與差=([真, 真, 真], [-11, -2, 0])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123002,"parameters":{"pairs":[[-11,0],[0,2],[0,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-11\",\"-2\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0003","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[-10, 0], [-1, 2], [1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 假], [-10, -3, 1])","explanation_zh":"封閉比較真假與差=([真, 真, 假], [-10, -3, 1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123003,"parameters":{"pairs":[[-10,0],[-1,2],[1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-10\",\"-3\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0004","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[-9, 0], [-2, 2], [-1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [-9, -4, -1])","explanation_zh":"封閉比較真假與差=([真, 真, 真], [-9, -4, -1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123004,"parameters":{"pairs":[[-9,0],[-2,2],[-1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-9\",\"-4\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0005","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[-8, 0], [-3, 2], [0, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [-8, -5, 0])","explanation_zh":"封閉比較真假與差=([真, 真, 真], [-8, -5, 0])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123005,"parameters":{"pairs":[[-8,0],[-3,2],[0,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-8\",\"-5\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0006","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[-7, 0], [-4, 2], [1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 假], [-7, -6, 1])","explanation_zh":"封閉比較真假與差=([真, 真, 假], [-7, -6, 1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123006,"parameters":{"pairs":[[-7,0],[-4,2],[1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-7\",\"-6\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0007","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[-6, 0], [-5, 2], [-1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [-6, -7, -1])","explanation_zh":"封閉比較真假與差=([真, 真, 真], [-6, -7, -1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123007,"parameters":{"pairs":[[-6,0],[-5,2],[-1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-6\",\"-7\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0008","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[-5, 0], [-6, 2], [0, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [-5, -8, 0])","explanation_zh":"封閉比較真假與差=([真, 真, 真], [-5, -8, 0])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123008,"parameters":{"pairs":[[-5,0],[-6,2],[0,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-5\",\"-8\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0009","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[-4, 0], [-7, 2], [1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 假], [-4, -9, 1])","explanation_zh":"封閉比較真假與差=([真, 真, 假], [-4, -9, 1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123009,"parameters":{"pairs":[[-4,0],[-7,2],[1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-4\",\"-9\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0010","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[-3, 0], [-8, 2], [-1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [-3, -10, -1])","explanation_zh":"封閉比較真假與差=([真, 真, 真], [-3, -10, -1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123010,"parameters":{"pairs":[[-3,0],[-8,2],[-1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-3\",\"-10\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0011","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[-2, 0], [-9, 2], [0, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [-2, -11, 0])","explanation_zh":"封閉比較真假與差=([真, 真, 真], [-2, -11, 0])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123011,"parameters":{"pairs":[[-2,0],[-9,2],[0,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-2\",\"-11\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0012","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[-1, 0], [-10, 2], [1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 假], [-1, -12, 1])","explanation_zh":"封閉比較真假與差=([真, 真, 假], [-1, -12, 1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123012,"parameters":{"pairs":[[-1,0],[-10,2],[1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"-1\",\"-12\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0013","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[0, 0], [-11, 2], [-1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [0, -13, -1])","explanation_zh":"封閉比較真假與差=([真, 真, 真], [0, -13, -1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123013,"parameters":{"pairs":[[0,0],[-11,2],[-1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"0\",\"-13\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0014","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[1, 0], [-12, 2], [0, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [1, -14, 0])","explanation_zh":"封閉比較真假與差=([假, 真, 真], [1, -14, 0])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123014,"parameters":{"pairs":[[1,0],[-12,2],[0,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"1\",\"-14\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0015","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[2, 0], [-13, 2], [1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 假], [2, -15, 1])","explanation_zh":"封閉比較真假與差=([假, 真, 假], [2, -15, 1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123015,"parameters":{"pairs":[[2,0],[-13,2],[1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"2\",\"-15\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0016","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[3, 0], [-14, 2], [-1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [3, -16, -1])","explanation_zh":"封閉比較真假與差=([假, 真, 真], [3, -16, -1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123016,"parameters":{"pairs":[[3,0],[-14,2],[-1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"3\",\"-16\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0017","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[4, 0], [-15, 2], [0, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [4, -17, 0])","explanation_zh":"封閉比較真假與差=([假, 真, 真], [4, -17, 0])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123017,"parameters":{"pairs":[[4,0],[-15,2],[0,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"4\",\"-17\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0018","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[5, 0], [-16, 2], [1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 假], [5, -18, 1])","explanation_zh":"封閉比較真假與差=([假, 真, 假], [5, -18, 1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123018,"parameters":{"pairs":[[5,0],[-16,2],[1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"5\",\"-18\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0019","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[6, 0], [-17, 2], [-1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [6, -19, -1])","explanation_zh":"封閉比較真假與差=([假, 真, 真], [6, -19, -1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123019,"parameters":{"pairs":[[6,0],[-17,2],[-1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"6\",\"-19\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0020","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[7, 0], [-18, 2], [0, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [7, -20, 0])","explanation_zh":"封閉比較真假與差=([假, 真, 真], [7, -20, 0])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123020,"parameters":{"pairs":[[7,0],[-18,2],[0,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"7\",\"-20\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0021","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[8, 0], [-19, 2], [1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 假], [8, -21, 1])","explanation_zh":"封閉比較真假與差=([假, 真, 假], [8, -21, 1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123021,"parameters":{"pairs":[[8,0],[-19,2],[1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"8\",\"-21\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0022","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[9, 0], [-20, 2], [-1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [9, -22, -1])","explanation_zh":"封閉比較真假與差=([假, 真, 真], [9, -22, -1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123022,"parameters":{"pairs":[[9,0],[-20,2],[-1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"9\",\"-22\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0023","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[10, 0], [-21, 2], [0, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [10, -23, 0])","explanation_zh":"封閉比較真假與差=([假, 真, 真], [10, -23, 0])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123023,"parameters":{"pairs":[[10,0],[-21,2],[0,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"10\",\"-23\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0024","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[11, 0], [-22, 2], [1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 假], [11, -24, 1])","explanation_zh":"封閉比較真假與差=([假, 真, 假], [11, -24, 1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123024,"parameters":{"pairs":[[11,0],[-22,2],[1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"11\",\"-24\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0025","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"封閉整數比較的真假與差","stem_zh":"古典整數語境；三列有序pairs=[[12, 0], [-23, 2], [-1, 0]]，第i列命題是a_i≤b_i。依序列 (三個命題真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [12, -25, -1])","explanation_zh":"封閉比較真假與差=([假, 真, 真], [12, -25, -1])。逐列比較並相減，負差及零差通過≤；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["封閉整數比較的真假與差"]},"variant":{"family_id":"qf-density12-proposition-comparison-witnesses","seed":202610123025,"parameters":{"pairs":[[12,0],[-23,2],[-1,0]],"operation":"density12_proposition_comparison_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-comparison-witnesses/[\"12\",\"-25\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0026","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤-2，assignment=null（未綁定）。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(否, 未封閉, 不適用)","explanation_zh":"封閉資格與代入真假=(否, 未封閉, 不適用)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123026,"parameters":{"threshold":-2,"assignment":null,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/[\"open_sentence\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0027","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤-1，assignment=-12。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 真, -11)","explanation_zh":"封閉資格與代入真假=(是, 真, -11)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123027,"parameters":{"threshold":-1,"assignment":-12,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"-11\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0028","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤0，assignment=-10。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 真, -10)","explanation_zh":"封閉資格與代入真假=(是, 真, -10)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123028,"parameters":{"threshold":0,"assignment":-10,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"-10\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0029","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤1，assignment=-8。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 真, -9)","explanation_zh":"封閉資格與代入真假=(是, 真, -9)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123029,"parameters":{"threshold":1,"assignment":-8,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"-9\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0030","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤2，assignment=-6。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 真, -8)","explanation_zh":"封閉資格與代入真假=(是, 真, -8)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123030,"parameters":{"threshold":2,"assignment":-6,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"-8\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0031","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤-2，assignment=-9。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 真, -7)","explanation_zh":"封閉資格與代入真假=(是, 真, -7)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123031,"parameters":{"threshold":-2,"assignment":-9,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"-7\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0032","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤-1，assignment=-7。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 真, -6)","explanation_zh":"封閉資格與代入真假=(是, 真, -6)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123032,"parameters":{"threshold":-1,"assignment":-7,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"-6\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0033","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤0，assignment=-5。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 真, -5)","explanation_zh":"封閉資格與代入真假=(是, 真, -5)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123033,"parameters":{"threshold":0,"assignment":-5,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"-5\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0034","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤1，assignment=-3。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 真, -4)","explanation_zh":"封閉資格與代入真假=(是, 真, -4)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123034,"parameters":{"threshold":1,"assignment":-3,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0035","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤2，assignment=-1。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 真, -3)","explanation_zh":"封閉資格與代入真假=(是, 真, -3)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123035,"parameters":{"threshold":2,"assignment":-1,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0045","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤2，assignment=9。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 假, 7)","explanation_zh":"封閉資格與代入真假=(是, 假, 7)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123045,"parameters":{"threshold":2,"assignment":9,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"7\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0046","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤-2，assignment=6。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 假, 8)","explanation_zh":"封閉資格與代入真假=(是, 假, 8)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123046,"parameters":{"threshold":-2,"assignment":6,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"8\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0047","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤-1，assignment=8。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 假, 9)","explanation_zh":"封閉資格與代入真假=(是, 假, 9)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123047,"parameters":{"threshold":-1,"assignment":8,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"9\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0048","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤0，assignment=10。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 假, 10)","explanation_zh":"封閉資格與代入真假=(是, 假, 10)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123048,"parameters":{"threshold":0,"assignment":10,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"10\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0049","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤1，assignment=12。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 假, 11)","explanation_zh":"封閉資格與代入真假=(是, 假, 11)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123049,"parameters":{"threshold":1,"assignment":12,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"11\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0050","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"自由變數語句的封閉資格","stem_zh":"古典整數語境；語句x≤2，assignment=14。依序答 (是否已封閉, 封閉後真假或未封閉, 代入差或不適用)。","answer":{"kind":"expression","value":"(是, 假, 12)","explanation_zh":"封閉資格與代入真假=(是, 假, 12)。有整數綁定才代入判定，未封閉不給真假，不把未綁定當假；真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["自由變數語句的封閉資格"]},"variant":{"family_id":"qf-density12-proposition-open-binding","seed":202610123050,"parameters":{"threshold":2,"assignment":14,"operation":"density12_proposition_open_binding","domain":"integer_open_sentence_binding","semantic_template":"qf-density12-proposition-open-binding/\"12\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0051","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[0, 0], [-1, 1], [1, -1]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 假, 假], [假, 真, 真], [0, -2, 2])","explanation_zh":"相等命題及否定見證=([真, 假, 假], [假, 真, 真], [0, -2, 2])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123051,"parameters":{"pairs":[[0,0],[-1,1],[1,-1]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"0\",\"-2\",\"2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0052","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[-11, 0], [0, 2], [0, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 真], [真, 真, 假], [-11, -2, 0])","explanation_zh":"相等命題及否定見證=([假, 假, 真], [真, 真, 假], [-11, -2, 0])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123052,"parameters":{"pairs":[[-11,0],[0,2],[0,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-11\",\"-2\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0053","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[-10, 0], [-1, 2], [1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [-10, -3, 1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [-10, -3, 1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123053,"parameters":{"pairs":[[-10,0],[-1,2],[1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-10\",\"-3\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0054","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[-9, 0], [-2, 2], [-1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [-9, -4, -1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [-9, -4, -1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123054,"parameters":{"pairs":[[-9,0],[-2,2],[-1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-9\",\"-4\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0055","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[-8, 0], [-3, 2], [0, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 真], [真, 真, 假], [-8, -5, 0])","explanation_zh":"相等命題及否定見證=([假, 假, 真], [真, 真, 假], [-8, -5, 0])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123055,"parameters":{"pairs":[[-8,0],[-3,2],[0,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-8\",\"-5\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0056","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[-7, 0], [-4, 2], [1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [-7, -6, 1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [-7, -6, 1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123056,"parameters":{"pairs":[[-7,0],[-4,2],[1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-7\",\"-6\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0057","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[-6, 0], [-5, 2], [-1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [-6, -7, -1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [-6, -7, -1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123057,"parameters":{"pairs":[[-6,0],[-5,2],[-1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-6\",\"-7\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0058","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[-5, 0], [-6, 2], [0, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 真], [真, 真, 假], [-5, -8, 0])","explanation_zh":"相等命題及否定見證=([假, 假, 真], [真, 真, 假], [-5, -8, 0])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123058,"parameters":{"pairs":[[-5,0],[-6,2],[0,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-5\",\"-8\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0059","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[-4, 0], [-7, 2], [1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [-4, -9, 1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [-4, -9, 1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123059,"parameters":{"pairs":[[-4,0],[-7,2],[1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-4\",\"-9\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0060","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[-3, 0], [-8, 2], [-1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [-3, -10, -1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [-3, -10, -1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123060,"parameters":{"pairs":[[-3,0],[-8,2],[-1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-3\",\"-10\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0061","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[-2, 0], [-9, 2], [0, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 真], [真, 真, 假], [-2, -11, 0])","explanation_zh":"相等命題及否定見證=([假, 假, 真], [真, 真, 假], [-2, -11, 0])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123061,"parameters":{"pairs":[[-2,0],[-9,2],[0,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-2\",\"-11\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0062","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[-1, 0], [-10, 2], [1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [-1, -12, 1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [-1, -12, 1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123062,"parameters":{"pairs":[[-1,0],[-10,2],[1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"-1\",\"-12\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0063","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[0, 0], [-11, 2], [-1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 假, 假], [假, 真, 真], [0, -13, -1])","explanation_zh":"相等命題及否定見證=([真, 假, 假], [假, 真, 真], [0, -13, -1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123063,"parameters":{"pairs":[[0,0],[-11,2],[-1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"0\",\"-13\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0064","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[1, 0], [-12, 2], [0, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 真], [真, 真, 假], [1, -14, 0])","explanation_zh":"相等命題及否定見證=([假, 假, 真], [真, 真, 假], [1, -14, 0])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123064,"parameters":{"pairs":[[1,0],[-12,2],[0,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"1\",\"-14\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0065","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[2, 0], [-13, 2], [1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [2, -15, 1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [2, -15, 1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123065,"parameters":{"pairs":[[2,0],[-13,2],[1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"2\",\"-15\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0066","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[3, 0], [-14, 2], [-1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [3, -16, -1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [3, -16, -1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123066,"parameters":{"pairs":[[3,0],[-14,2],[-1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"3\",\"-16\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0067","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[4, 0], [-15, 2], [0, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 真], [真, 真, 假], [4, -17, 0])","explanation_zh":"相等命題及否定見證=([假, 假, 真], [真, 真, 假], [4, -17, 0])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123067,"parameters":{"pairs":[[4,0],[-15,2],[0,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"4\",\"-17\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0068","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[5, 0], [-16, 2], [1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [5, -18, 1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [5, -18, 1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123068,"parameters":{"pairs":[[5,0],[-16,2],[1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"5\",\"-18\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0069","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[6, 0], [-17, 2], [-1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [6, -19, -1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [6, -19, -1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123069,"parameters":{"pairs":[[6,0],[-17,2],[-1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"6\",\"-19\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0070","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[7, 0], [-18, 2], [0, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 真], [真, 真, 假], [7, -20, 0])","explanation_zh":"相等命題及否定見證=([假, 假, 真], [真, 真, 假], [7, -20, 0])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123070,"parameters":{"pairs":[[7,0],[-18,2],[0,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"7\",\"-20\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0071","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[8, 0], [-19, 2], [1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [8, -21, 1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [8, -21, 1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123071,"parameters":{"pairs":[[8,0],[-19,2],[1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"8\",\"-21\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0072","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[9, 0], [-20, 2], [-1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [9, -22, -1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [9, -22, -1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123072,"parameters":{"pairs":[[9,0],[-20,2],[-1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"9\",\"-22\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0073","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[10, 0], [-21, 2], [0, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 真], [真, 真, 假], [10, -23, 0])","explanation_zh":"相等命題及否定見證=([假, 假, 真], [真, 真, 假], [10, -23, 0])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123073,"parameters":{"pairs":[[10,0],[-21,2],[0,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"10\",\"-23\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0074","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[11, 0], [-22, 2], [1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [11, -24, 1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [11, -24, 1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123074,"parameters":{"pairs":[[11,0],[-22,2],[1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"11\",\"-24\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0075","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"相等命題與否定的有序見證","stem_zh":"古典整數語境；pairs=[[12, 0], [-23, 2], [-1, 0]]，P_i為a_i=b_i。依序列 (三個P真假, 三個¬P真假, 三個有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 假, 假], [真, 真, 真], [12, -25, -1])","explanation_zh":"相等命題及否定見證=([假, 假, 假], [真, 真, 真], [12, -25, -1])。差零才相等，再逐項取古典否定；假命題仍是命題，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["相等命題與否定的有序見證"]},"variant":{"family_id":"qf-density12-proposition-equality-negation-witnesses","seed":202610123075,"parameters":{"pairs":[[12,0],[-23,2],[-1,0]],"operation":"density12_proposition_equality_negation_witnesses","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-equality-negation-witnesses/[\"12\",\"-25\",\"-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0076","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[0, 0], [-1, 1], [1, -1]]，P_i為a_i≤b_i；候選自報reported=[1, 1, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 假], [是, 是, 是], [0, -2, 2])","explanation_zh":"候選自報與真假核對=([真, 真, 假], [是, 是, 是], [0, -2, 2])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123076,"parameters":{"pairs":[[0,0],[-1,1],[1,-1]],"reported":[1,1,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"0\",\"-2\",\"2\"],[1,1,0]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0077","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[-11, 0], [0, 2], [0, 0]]，P_i為a_i≤b_i；候選自報reported=[1, 0, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [是, 否, 否], [-11, -2, 0])","explanation_zh":"候選自報與真假核對=([真, 真, 真], [是, 否, 否], [-11, -2, 0])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123077,"parameters":{"pairs":[[-11,0],[0,2],[0,0]],"reported":[1,0,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"-11\",\"-2\",\"0\"],[1,0,0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0078","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[-10, 0], [-1, 2], [1, 0]]，P_i為a_i≤b_i；候選自報reported=[0, 1, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 假], [否, 是, 是], [-10, -3, 1])","explanation_zh":"候選自報與真假核對=([真, 真, 假], [否, 是, 是], [-10, -3, 1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123078,"parameters":{"pairs":[[-10,0],[-1,2],[1,0]],"reported":[0,1,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"-10\",\"-3\",\"1\"],[0,1,0]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0079","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[-9, 0], [-2, 2], [-1, 0]]，P_i為a_i≤b_i；候選自報reported=[1, 1, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [是, 是, 否], [-9, -4, -1])","explanation_zh":"候選自報與真假核對=([真, 真, 真], [是, 是, 否], [-9, -4, -1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123079,"parameters":{"pairs":[[-9,0],[-2,2],[-1,0]],"reported":[1,1,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"-9\",\"-4\",\"-1\"],[1,1,0]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0080","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[-8, 0], [-3, 2], [0, 0]]，P_i為a_i≤b_i；候選自報reported=[0, 0, 1]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [否, 否, 是], [-8, -5, 0])","explanation_zh":"候選自報與真假核對=([真, 真, 真], [否, 否, 是], [-8, -5, 0])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123080,"parameters":{"pairs":[[-8,0],[-3,2],[0,0]],"reported":[0,0,1],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"-8\",\"-5\",\"0\"],[0,0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0081","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[-7, 0], [-4, 2], [1, 0]]，P_i為a_i≤b_i；候選自報reported=[1, 0, 1]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 假], [是, 否, 否], [-7, -6, 1])","explanation_zh":"候選自報與真假核對=([真, 真, 假], [是, 否, 否], [-7, -6, 1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123081,"parameters":{"pairs":[[-7,0],[-4,2],[1,0]],"reported":[1,0,1],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"-7\",\"-6\",\"1\"],[1,0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0082","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[-6, 0], [-5, 2], [-1, 0]]，P_i為a_i≤b_i；候選自報reported=[0, 1, 1]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [否, 是, 是], [-6, -7, -1])","explanation_zh":"候選自報與真假核對=([真, 真, 真], [否, 是, 是], [-6, -7, -1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123082,"parameters":{"pairs":[[-6,0],[-5,2],[-1,0]],"reported":[0,1,1],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"-6\",\"-7\",\"-1\"],[0,1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0083","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[-5, 0], [-6, 2], [0, 0]]，P_i為a_i≤b_i；候選自報reported=[1, 1, 1]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [是, 是, 是], [-5, -8, 0])","explanation_zh":"候選自報與真假核對=([真, 真, 真], [是, 是, 是], [-5, -8, 0])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123083,"parameters":{"pairs":[[-5,0],[-6,2],[0,0]],"reported":[1,1,1],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"-5\",\"-8\",\"0\"],[1,1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0084","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[-4, 0], [-7, 2], [1, 0]]，P_i為a_i≤b_i；候選自報reported=[0, 0, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 假], [否, 否, 是], [-4, -9, 1])","explanation_zh":"候選自報與真假核對=([真, 真, 假], [否, 否, 是], [-4, -9, 1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123084,"parameters":{"pairs":[[-4,0],[-7,2],[1,0]],"reported":[0,0,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"-4\",\"-9\",\"1\"],[0,0,0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0085","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[-3, 0], [-8, 2], [-1, 0]]，P_i為a_i≤b_i；候選自報reported=[1, 0, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [是, 否, 否], [-3, -10, -1])","explanation_zh":"候選自報與真假核對=([真, 真, 真], [是, 否, 否], [-3, -10, -1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123085,"parameters":{"pairs":[[-3,0],[-8,2],[-1,0]],"reported":[1,0,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"-3\",\"-10\",\"-1\"],[1,0,0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0086","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[-2, 0], [-9, 2], [0, 0]]，P_i為a_i≤b_i；候選自報reported=[0, 1, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [否, 是, 否], [-2, -11, 0])","explanation_zh":"候選自報與真假核對=([真, 真, 真], [否, 是, 否], [-2, -11, 0])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123086,"parameters":{"pairs":[[-2,0],[-9,2],[0,0]],"reported":[0,1,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"-2\",\"-11\",\"0\"],[0,1,0]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0087","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[-1, 0], [-10, 2], [1, 0]]，P_i為a_i≤b_i；候選自報reported=[1, 1, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 假], [是, 是, 是], [-1, -12, 1])","explanation_zh":"候選自報與真假核對=([真, 真, 假], [是, 是, 是], [-1, -12, 1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123087,"parameters":{"pairs":[[-1,0],[-10,2],[1,0]],"reported":[1,1,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"-1\",\"-12\",\"1\"],[1,1,0]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0088","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[0, 0], [-11, 2], [-1, 0]]，P_i為a_i≤b_i；候選自報reported=[0, 0, 1]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([真, 真, 真], [否, 否, 是], [0, -13, -1])","explanation_zh":"候選自報與真假核對=([真, 真, 真], [否, 否, 是], [0, -13, -1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123088,"parameters":{"pairs":[[0,0],[-11,2],[-1,0]],"reported":[0,0,1],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"0\",\"-13\",\"-1\"],[0,0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0089","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[1, 0], [-12, 2], [0, 0]]，P_i為a_i≤b_i；候選自報reported=[1, 0, 1]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [否, 否, 是], [1, -14, 0])","explanation_zh":"候選自報與真假核對=([假, 真, 真], [否, 否, 是], [1, -14, 0])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123089,"parameters":{"pairs":[[1,0],[-12,2],[0,0]],"reported":[1,0,1],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"1\",\"-14\",\"0\"],[1,0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0090","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[2, 0], [-13, 2], [1, 0]]，P_i為a_i≤b_i；候選自報reported=[0, 1, 1]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 假], [是, 是, 否], [2, -15, 1])","explanation_zh":"候選自報與真假核對=([假, 真, 假], [是, 是, 否], [2, -15, 1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123090,"parameters":{"pairs":[[2,0],[-13,2],[1,0]],"reported":[0,1,1],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"2\",\"-15\",\"1\"],[0,1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0091","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[3, 0], [-14, 2], [-1, 0]]，P_i為a_i≤b_i；候選自報reported=[1, 1, 1]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [否, 是, 是], [3, -16, -1])","explanation_zh":"候選自報與真假核對=([假, 真, 真], [否, 是, 是], [3, -16, -1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123091,"parameters":{"pairs":[[3,0],[-14,2],[-1,0]],"reported":[1,1,1],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"3\",\"-16\",\"-1\"],[1,1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0092","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[4, 0], [-15, 2], [0, 0]]，P_i為a_i≤b_i；候選自報reported=[0, 0, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [是, 否, 否], [4, -17, 0])","explanation_zh":"候選自報與真假核對=([假, 真, 真], [是, 否, 否], [4, -17, 0])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123092,"parameters":{"pairs":[[4,0],[-15,2],[0,0]],"reported":[0,0,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"4\",\"-17\",\"0\"],[0,0,0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0093","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[5, 0], [-16, 2], [1, 0]]，P_i為a_i≤b_i；候選自報reported=[1, 0, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 假], [否, 否, 是], [5, -18, 1])","explanation_zh":"候選自報與真假核對=([假, 真, 假], [否, 否, 是], [5, -18, 1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123093,"parameters":{"pairs":[[5,0],[-16,2],[1,0]],"reported":[1,0,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"5\",\"-18\",\"1\"],[1,0,0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0094","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[6, 0], [-17, 2], [-1, 0]]，P_i為a_i≤b_i；候選自報reported=[0, 1, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [是, 是, 否], [6, -19, -1])","explanation_zh":"候選自報與真假核對=([假, 真, 真], [是, 是, 否], [6, -19, -1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123094,"parameters":{"pairs":[[6,0],[-17,2],[-1,0]],"reported":[0,1,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"6\",\"-19\",\"-1\"],[0,1,0]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0095","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[7, 0], [-18, 2], [0, 0]]，P_i為a_i≤b_i；候選自報reported=[1, 1, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [否, 是, 否], [7, -20, 0])","explanation_zh":"候選自報與真假核對=([假, 真, 真], [否, 是, 否], [7, -20, 0])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123095,"parameters":{"pairs":[[7,0],[-18,2],[0,0]],"reported":[1,1,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"7\",\"-20\",\"0\"],[1,1,0]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0096","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[8, 0], [-19, 2], [1, 0]]，P_i為a_i≤b_i；候選自報reported=[0, 0, 1]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 假], [是, 否, 否], [8, -21, 1])","explanation_zh":"候選自報與真假核對=([假, 真, 假], [是, 否, 否], [8, -21, 1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123096,"parameters":{"pairs":[[8,0],[-19,2],[1,0]],"reported":[0,0,1],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"8\",\"-21\",\"1\"],[0,0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0097","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[9, 0], [-20, 2], [-1, 0]]，P_i為a_i≤b_i；候選自報reported=[1, 0, 1]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [否, 否, 是], [9, -22, -1])","explanation_zh":"候選自報與真假核對=([假, 真, 真], [否, 否, 是], [9, -22, -1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123097,"parameters":{"pairs":[[9,0],[-20,2],[-1,0]],"reported":[1,0,1],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"9\",\"-22\",\"-1\"],[1,0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0098","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[10, 0], [-21, 2], [0, 0]]，P_i為a_i≤b_i；候選自報reported=[0, 1, 1]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [是, 是, 是], [10, -23, 0])","explanation_zh":"候選自報與真假核對=([假, 真, 真], [是, 是, 是], [10, -23, 0])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123098,"parameters":{"pairs":[[10,0],[-21,2],[0,0]],"reported":[0,1,1],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"10\",\"-23\",\"0\"],[0,1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0099","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[11, 0], [-22, 2], [1, 0]]，P_i為a_i≤b_i；候選自報reported=[1, 1, 1]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 假], [否, 是, 否], [11, -24, 1])","explanation_zh":"候選自報與真假核對=([假, 真, 假], [否, 是, 否], [11, -24, 1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123099,"parameters":{"pairs":[[11,0],[-22,2],[1,0]],"reported":[1,1,1],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"11\",\"-24\",\"1\"],[1,1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r03-proposition-0100","batch_id":"batch-20261008-advance-20261012-r03-proposition-foundations","language":"zh-Hant","source_mko_ids":["mko-proposition"],"title_zh":"候選自報真假與實際核對","stem_zh":"古典整數語境；pairs=[[12, 0], [-23, 2], [-1, 0]]，P_i為a_i≤b_i；候選自報reported=[0, 0, 0]，1自報真、0自報假。依序答 (實際真假, 自報是否匹配, 有號a_i−b_i)。","answer":{"kind":"expression","value":"([假, 真, 真], [是, 否, 否], [12, -25, -1])","explanation_zh":"候選自報與真假核對=([假, 真, 真], [是, 否, 否], [12, -25, -1])。由整數比較另算真假再核各報告；候選自報非證據，真假不等於已有證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["候選自報真假與實際核對"]},"variant":{"family_id":"qf-density12-proposition-report-defects","seed":202610123100,"parameters":{"pairs":[[12,0],[-23,2],[-1,0]],"reported":[0,0,0],"operation":"density12_proposition_report_defects","domain":"closed_classical_integer_statements","semantic_template":"qf-density12-proposition-report-defects/[[\"12\",\"-25\",\"-1\"],[0,0,0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}