{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0001","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(0)/(1), (0)/(2), (0)/(3)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([0/1, 0/1, 0/1], [0, 1, 2], [0/1, 0/1, 0/1], [=, =])","explanation_zh":"離散結構核對=([0/1, 0/1, 0/1], [0, 1, 2], [0/1, 0/1, 0/1], [=, =])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174001,"parameters":{"values":[[0,1],[0,2],[0,3]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"0/1\",\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0002","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(-1)/(2), (1)/(3), (-2)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([-1/2, 1/3, -1/2], [0, 2, 1], [-1/2, -1/2, 1/3], [=, <])","explanation_zh":"離散結構核對=([-1/2, 1/3, -1/2], [0, 2, 1], [-1/2, -1/2, 1/3], [=, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174002,"parameters":{"values":[[-1,2],[1,3],[-2,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-1/2\",\"1/3\",\"-1/2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0003","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(1)/(2), (-1)/(2), (0)/(1)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([1/2, -1/2, 0/1], [1, 2, 0], [-1/2, 0/1, 1/2], [<, <])","explanation_zh":"離散結構核對=([1/2, -1/2, 0/1], [1, 2, 0], [-1/2, 0/1, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174003,"parameters":{"values":[[1,2],[-1,2],[0,1]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"1/2\",\"-1/2\",\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0004","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(-9)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([-3/1, 1/2, -1/4], [0, 2, 1], [-3/1, -1/4, 1/2], [<, <])","explanation_zh":"離散結構核對=([-3/1, 1/2, -1/4], [0, 2, 1], [-3/1, -1/4, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174004,"parameters":{"values":[[-9,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-3/1\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0005","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(-8)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([-8/3, 1/2, -1/4], [0, 2, 1], [-8/3, -1/4, 1/2], [<, <])","explanation_zh":"離散結構核對=([-8/3, 1/2, -1/4], [0, 2, 1], [-8/3, -1/4, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174005,"parameters":{"values":[[-8,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-8/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0006","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(-7)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([-7/3, 1/2, -1/4], [0, 2, 1], [-7/3, -1/4, 1/2], [<, <])","explanation_zh":"離散結構核對=([-7/3, 1/2, -1/4], [0, 2, 1], [-7/3, -1/4, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174006,"parameters":{"values":[[-7,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-7/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0007","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(-6)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([-2/1, 1/2, -1/4], [0, 2, 1], [-2/1, -1/4, 1/2], [<, <])","explanation_zh":"離散結構核對=([-2/1, 1/2, -1/4], [0, 2, 1], [-2/1, -1/4, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174007,"parameters":{"values":[[-6,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-2/1\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0008","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(-5)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([-5/3, 1/2, -1/4], [0, 2, 1], [-5/3, -1/4, 1/2], [<, <])","explanation_zh":"離散結構核對=([-5/3, 1/2, -1/4], [0, 2, 1], [-5/3, -1/4, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174008,"parameters":{"values":[[-5,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-5/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0009","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(-4)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([-4/3, 1/2, -1/4], [0, 2, 1], [-4/3, -1/4, 1/2], [<, <])","explanation_zh":"離散結構核對=([-4/3, 1/2, -1/4], [0, 2, 1], [-4/3, -1/4, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174009,"parameters":{"values":[[-4,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-4/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0010","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(-3)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([-1/1, 1/2, -1/4], [0, 2, 1], [-1/1, -1/4, 1/2], [<, <])","explanation_zh":"離散結構核對=([-1/1, 1/2, -1/4], [0, 2, 1], [-1/1, -1/4, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174010,"parameters":{"values":[[-3,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-1/1\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0011","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(-2)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <])","explanation_zh":"離散結構核對=([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174011,"parameters":{"values":[[-2,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-2/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0012","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(-1)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([-1/3, 1/2, -1/4], [0, 2, 1], [-1/3, -1/4, 1/2], [<, <])","explanation_zh":"離散結構核對=([-1/3, 1/2, -1/4], [0, 2, 1], [-1/3, -1/4, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174012,"parameters":{"values":[[-1,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-1/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0013","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(0)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([0/1, 1/2, -1/4], [2, 0, 1], [-1/4, 0/1, 1/2], [<, <])","explanation_zh":"離散結構核對=([0/1, 1/2, -1/4], [2, 0, 1], [-1/4, 0/1, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174013,"parameters":{"values":[[0,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"0/1\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0014","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(1)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([1/3, 1/2, -1/4], [2, 0, 1], [-1/4, 1/3, 1/2], [<, <])","explanation_zh":"離散結構核對=([1/3, 1/2, -1/4], [2, 0, 1], [-1/4, 1/3, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174014,"parameters":{"values":[[1,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"1/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0015","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(2)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([2/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/3], [<, <])","explanation_zh":"離散結構核對=([2/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/3], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174015,"parameters":{"values":[[2,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"2/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0016","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(3)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([1/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 1/1], [<, <])","explanation_zh":"離散結構核對=([1/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 1/1], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174016,"parameters":{"values":[[3,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"1/1\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0017","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(4)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([4/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/3], [<, <])","explanation_zh":"離散結構核對=([4/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/3], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174017,"parameters":{"values":[[4,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"4/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0018","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(5)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([5/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 5/3], [<, <])","explanation_zh":"離散結構核對=([5/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 5/3], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174018,"parameters":{"values":[[5,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"5/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0019","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(6)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([2/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/1], [<, <])","explanation_zh":"離散結構核對=([2/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/1], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174019,"parameters":{"values":[[6,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"2/1\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0020","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(7)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([7/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 7/3], [<, <])","explanation_zh":"離散結構核對=([7/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 7/3], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174020,"parameters":{"values":[[7,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"7/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0021","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(8)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([8/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 8/3], [<, <])","explanation_zh":"離散結構核對=([8/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 8/3], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174021,"parameters":{"values":[[8,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"8/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0022","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(9)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([3/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 3/1], [<, <])","explanation_zh":"離散結構核對=([3/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 3/1], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174022,"parameters":{"values":[[9,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"3/1\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0023","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(10)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([10/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 10/3], [<, <])","explanation_zh":"離散結構核對=([10/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 10/3], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174023,"parameters":{"values":[[10,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"10/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0024","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(11)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([11/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 11/3], [<, <])","explanation_zh":"離散結構核對=([11/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 11/3], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174024,"parameters":{"values":[[11,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"11/3\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0025","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"三分數的穩定有序索引","stem_zh":"R通常全序，原分母為正。原序values=[(12)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。","answer":{"kind":"expression","value":"([4/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/1], [<, <])","explanation_zh":"離散結構核對=([4/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/1], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三分數的穩定有序索引"]},"variant":{"family_id":"qf-density17-order-three-rational-sort","seed":202610174025,"parameters":{"values":[[12,3],[1,2],[-1,4]],"domain":"R_positive_denominator_order","operation":"order-three-rational-sort","semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"4/1\",\"1/2\",\"-1/4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0026","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(0)/(1), (1)/(1)]，I2=[(1)/(1), (2)/(1)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([0/1, 1/1], [1/1, 2/1], 1/1, 1/1, 是, [1/1, 1/1])","explanation_zh":"離散結構核對=([0/1, 1/1], [1/1, 2/1], 1/1, 1/1, 是, [1/1, 1/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174026,"parameters":{"l1":[0,1],"u1":[1,1],"l2":[1,1],"u2":[2,1],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"0/1\",\"l2\":\"1/1\",\"u1\":\"1/1\",\"u2\":\"2/1\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0027","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(-2)/(1), (-1)/(1)]，I2=[(0)/(1), (1)/(1)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([-2/1, -1/1], [0/1, 1/1], 0/1, -1/1, 否, 空集)","explanation_zh":"離散結構核對=([-2/1, -1/1], [0/1, 1/1], 0/1, -1/1, 否, 空集)；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174027,"parameters":{"l1":[-2,1],"u1":[-1,1],"l2":[0,1],"u2":[1,1],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-2/1\",\"l2\":\"0/1\",\"u1\":\"-1/1\",\"u2\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0028","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(0)/(1), (0)/(1)]，I2=[(0)/(1), (0)/(1)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([0/1, 0/1], [0/1, 0/1], 0/1, 0/1, 是, [0/1, 0/1])","explanation_zh":"離散結構核對=([0/1, 0/1], [0/1, 0/1], 0/1, 0/1, 是, [0/1, 0/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174028,"parameters":{"l1":[0,1],"u1":[0,1],"l2":[0,1],"u2":[0,1],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"0/1\",\"l2\":\"0/1\",\"u1\":\"0/1\",\"u2\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0029","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(-2)/(1), (2)/(1)]，I2=[(-1)/(1), (3)/(1)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([-2/1, 2/1], [-1/1, 3/1], -1/1, 2/1, 是, [-1/1, 2/1])","explanation_zh":"離散結構核對=([-2/1, 2/1], [-1/1, 3/1], -1/1, 2/1, 是, [-1/1, 2/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174029,"parameters":{"l1":[-2,1],"u1":[2,1],"l2":[-1,1],"u2":[3,1],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-2/1\",\"l2\":\"-1/1\",\"u1\":\"2/1\",\"u2\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0030","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(-8)/(2), (-2)/(2)]，I2=[(-5)/(2), (1)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([-4/1, -1/1], [-5/2, 1/2], -5/2, -1/1, 是, [-5/2, -1/1])","explanation_zh":"離散結構核對=([-4/1, -1/1], [-5/2, 1/2], -5/2, -1/1, 是, [-5/2, -1/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174030,"parameters":{"l1":[-8,2],"u1":[-2,2],"l2":[-5,2],"u2":[1,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-4/1\",\"l2\":\"-5/2\",\"u1\":\"-1/1\",\"u2\":\"1/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0031","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(-7)/(2), (-1)/(2)]，I2=[(-4)/(2), (2)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([-7/2, -1/2], [-2/1, 1/1], -2/1, -1/2, 是, [-2/1, -1/2])","explanation_zh":"離散結構核對=([-7/2, -1/2], [-2/1, 1/1], -2/1, -1/2, 是, [-2/1, -1/2])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174031,"parameters":{"l1":[-7,2],"u1":[-1,2],"l2":[-4,2],"u2":[2,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-7/2\",\"l2\":\"-2/1\",\"u1\":\"-1/2\",\"u2\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0032","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(-6)/(2), (0)/(2)]，I2=[(-3)/(2), (3)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([-3/1, 0/1], [-3/2, 3/2], -3/2, 0/1, 是, [-3/2, 0/1])","explanation_zh":"離散結構核對=([-3/1, 0/1], [-3/2, 3/2], -3/2, 0/1, 是, [-3/2, 0/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174032,"parameters":{"l1":[-6,2],"u1":[0,2],"l2":[-3,2],"u2":[3,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-3/1\",\"l2\":\"-3/2\",\"u1\":\"0/1\",\"u2\":\"3/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0033","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(-5)/(2), (1)/(2)]，I2=[(-2)/(2), (4)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([-5/2, 1/2], [-1/1, 2/1], -1/1, 1/2, 是, [-1/1, 1/2])","explanation_zh":"離散結構核對=([-5/2, 1/2], [-1/1, 2/1], -1/1, 1/2, 是, [-1/1, 1/2])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174033,"parameters":{"l1":[-5,2],"u1":[1,2],"l2":[-2,2],"u2":[4,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-5/2\",\"l2\":\"-1/1\",\"u1\":\"1/2\",\"u2\":\"2/1\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0034","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(-4)/(2), (2)/(2)]，I2=[(-1)/(2), (5)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([-2/1, 1/1], [-1/2, 5/2], -1/2, 1/1, 是, [-1/2, 1/1])","explanation_zh":"離散結構核對=([-2/1, 1/1], [-1/2, 5/2], -1/2, 1/1, 是, [-1/2, 1/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174034,"parameters":{"l1":[-4,2],"u1":[2,2],"l2":[-1,2],"u2":[5,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-2/1\",\"l2\":\"-1/2\",\"u1\":\"1/1\",\"u2\":\"5/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0035","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(-3)/(2), (3)/(2)]，I2=[(0)/(2), (6)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([-3/2, 3/2], [0/1, 3/1], 0/1, 3/2, 是, [0/1, 3/2])","explanation_zh":"離散結構核對=([-3/2, 3/2], [0/1, 3/1], 0/1, 3/2, 是, [0/1, 3/2])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174035,"parameters":{"l1":[-3,2],"u1":[3,2],"l2":[0,2],"u2":[6,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-3/2\",\"l2\":\"0/1\",\"u1\":\"3/2\",\"u2\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0036","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(-2)/(2), (4)/(2)]，I2=[(1)/(2), (7)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([-1/1, 2/1], [1/2, 7/2], 1/2, 2/1, 是, [1/2, 2/1])","explanation_zh":"離散結構核對=([-1/1, 2/1], [1/2, 7/2], 1/2, 2/1, 是, [1/2, 2/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174036,"parameters":{"l1":[-2,2],"u1":[4,2],"l2":[1,2],"u2":[7,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-1/1\",\"l2\":\"1/2\",\"u1\":\"2/1\",\"u2\":\"7/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0037","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(-1)/(2), (5)/(2)]，I2=[(2)/(2), (8)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([-1/2, 5/2], [1/1, 4/1], 1/1, 5/2, 是, [1/1, 5/2])","explanation_zh":"離散結構核對=([-1/2, 5/2], [1/1, 4/1], 1/1, 5/2, 是, [1/1, 5/2])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174037,"parameters":{"l1":[-1,2],"u1":[5,2],"l2":[2,2],"u2":[8,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-1/2\",\"l2\":\"1/1\",\"u1\":\"5/2\",\"u2\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0038","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(0)/(2), (6)/(2)]，I2=[(3)/(2), (9)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([0/1, 3/1], [3/2, 9/2], 3/2, 3/1, 是, [3/2, 3/1])","explanation_zh":"離散結構核對=([0/1, 3/1], [3/2, 9/2], 3/2, 3/1, 是, [3/2, 3/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174038,"parameters":{"l1":[0,2],"u1":[6,2],"l2":[3,2],"u2":[9,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"0/1\",\"l2\":\"3/2\",\"u1\":\"3/1\",\"u2\":\"9/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0039","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(1)/(2), (7)/(2)]，I2=[(4)/(2), (10)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([1/2, 7/2], [2/1, 5/1], 2/1, 7/2, 是, [2/1, 7/2])","explanation_zh":"離散結構核對=([1/2, 7/2], [2/1, 5/1], 2/1, 7/2, 是, [2/1, 7/2])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174039,"parameters":{"l1":[1,2],"u1":[7,2],"l2":[4,2],"u2":[10,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"1/2\",\"l2\":\"2/1\",\"u1\":\"7/2\",\"u2\":\"5/1\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0040","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(2)/(2), (8)/(2)]，I2=[(5)/(2), (11)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([1/1, 4/1], [5/2, 11/2], 5/2, 4/1, 是, [5/2, 4/1])","explanation_zh":"離散結構核對=([1/1, 4/1], [5/2, 11/2], 5/2, 4/1, 是, [5/2, 4/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174040,"parameters":{"l1":[2,2],"u1":[8,2],"l2":[5,2],"u2":[11,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"1/1\",\"l2\":\"5/2\",\"u1\":\"4/1\",\"u2\":\"11/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0041","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(3)/(2), (9)/(2)]，I2=[(6)/(2), (12)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, 是, [3/1, 9/2])","explanation_zh":"離散結構核對=([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, 是, [3/1, 9/2])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174041,"parameters":{"l1":[3,2],"u1":[9,2],"l2":[6,2],"u2":[12,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"3/2\",\"l2\":\"3/1\",\"u1\":\"9/2\",\"u2\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0042","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(4)/(2), (10)/(2)]，I2=[(7)/(2), (13)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([2/1, 5/1], [7/2, 13/2], 7/2, 5/1, 是, [7/2, 5/1])","explanation_zh":"離散結構核對=([2/1, 5/1], [7/2, 13/2], 7/2, 5/1, 是, [7/2, 5/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174042,"parameters":{"l1":[4,2],"u1":[10,2],"l2":[7,2],"u2":[13,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"2/1\",\"l2\":\"7/2\",\"u1\":\"5/1\",\"u2\":\"13/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0043","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(5)/(2), (11)/(2)]，I2=[(8)/(2), (14)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([5/2, 11/2], [4/1, 7/1], 4/1, 11/2, 是, [4/1, 11/2])","explanation_zh":"離散結構核對=([5/2, 11/2], [4/1, 7/1], 4/1, 11/2, 是, [4/1, 11/2])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174043,"parameters":{"l1":[5,2],"u1":[11,2],"l2":[8,2],"u2":[14,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"5/2\",\"l2\":\"4/1\",\"u1\":\"11/2\",\"u2\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0044","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(6)/(2), (12)/(2)]，I2=[(9)/(2), (15)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([3/1, 6/1], [9/2, 15/2], 9/2, 6/1, 是, [9/2, 6/1])","explanation_zh":"離散結構核對=([3/1, 6/1], [9/2, 15/2], 9/2, 6/1, 是, [9/2, 6/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174044,"parameters":{"l1":[6,2],"u1":[12,2],"l2":[9,2],"u2":[15,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"3/1\",\"l2\":\"9/2\",\"u1\":\"6/1\",\"u2\":\"15/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0045","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(7)/(2), (13)/(2)]，I2=[(10)/(2), (16)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([7/2, 13/2], [5/1, 8/1], 5/1, 13/2, 是, [5/1, 13/2])","explanation_zh":"離散結構核對=([7/2, 13/2], [5/1, 8/1], 5/1, 13/2, 是, [5/1, 13/2])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174045,"parameters":{"l1":[7,2],"u1":[13,2],"l2":[10,2],"u2":[16,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"7/2\",\"l2\":\"5/1\",\"u1\":\"13/2\",\"u2\":\"8/1\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0046","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(8)/(2), (14)/(2)]，I2=[(11)/(2), (17)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([4/1, 7/1], [11/2, 17/2], 11/2, 7/1, 是, [11/2, 7/1])","explanation_zh":"離散結構核對=([4/1, 7/1], [11/2, 17/2], 11/2, 7/1, 是, [11/2, 7/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174046,"parameters":{"l1":[8,2],"u1":[14,2],"l2":[11,2],"u2":[17,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"4/1\",\"l2\":\"11/2\",\"u1\":\"7/1\",\"u2\":\"17/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0047","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(9)/(2), (15)/(2)]，I2=[(12)/(2), (18)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([9/2, 15/2], [6/1, 9/1], 6/1, 15/2, 是, [6/1, 15/2])","explanation_zh":"離散結構核對=([9/2, 15/2], [6/1, 9/1], 6/1, 15/2, 是, [6/1, 15/2])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174047,"parameters":{"l1":[9,2],"u1":[15,2],"l2":[12,2],"u2":[18,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"9/2\",\"l2\":\"6/1\",\"u1\":\"15/2\",\"u2\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0048","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(10)/(2), (16)/(2)]，I2=[(13)/(2), (19)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([5/1, 8/1], [13/2, 19/2], 13/2, 8/1, 是, [13/2, 8/1])","explanation_zh":"離散結構核對=([5/1, 8/1], [13/2, 19/2], 13/2, 8/1, 是, [13/2, 8/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174048,"parameters":{"l1":[10,2],"u1":[16,2],"l2":[13,2],"u2":[19,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"5/1\",\"l2\":\"13/2\",\"u1\":\"8/1\",\"u2\":\"19/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0049","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(11)/(2), (17)/(2)]，I2=[(14)/(2), (20)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([11/2, 17/2], [7/1, 10/1], 7/1, 17/2, 是, [7/1, 17/2])","explanation_zh":"離散結構核對=([11/2, 17/2], [7/1, 10/1], 7/1, 17/2, 是, [7/1, 17/2])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174049,"parameters":{"l1":[11,2],"u1":[17,2],"l2":[14,2],"u2":[20,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"11/2\",\"l2\":\"7/1\",\"u1\":\"17/2\",\"u2\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0050","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"兩個閉區間的交集門檻","stem_zh":"R通常全序，原分母為正。閉區間I1=[(12)/(2), (18)/(2)]，I2=[(15)/(2), (21)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。","answer":{"kind":"expression","value":"([6/1, 9/1], [15/2, 21/2], 15/2, 9/1, 是, [15/2, 9/1])","explanation_zh":"離散結構核對=([6/1, 9/1], [15/2, 21/2], 15/2, 9/1, 是, [15/2, 9/1])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["兩個閉區間的交集門檻"]},"variant":{"family_id":"qf-density17-order-interval-intersection","seed":202610174050,"parameters":{"l1":[12,2],"u1":[18,2],"l2":[15,2],"u2":[21,2],"domain":"R_positive_denominator_order","operation":"order-interval-intersection","semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"6/1\",\"l2\":\"15/2\",\"u1\":\"9/1\",\"u2\":\"21/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0051","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(0)/(1), (0)/(1)]，有序三查詢=[(-1)/(1), (0)/(1), (1)/(1)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([0/1, 0/1], [-1/1, 0/1, 1/1], [否, 是, 否], [[-1/1, 1/1], [0/1, 0/1], [1/1, -1/1]])","explanation_zh":"離散結構核對=([0/1, 0/1], [-1/1, 0/1, 1/1], [否, 是, 否], [[-1/1, 1/1], [0/1, 0/1], [1/1, -1/1]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174051,"parameters":{"l":[0,1],"u":[0,1],"queries":[[-1,1],[0,1],[1,1]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"0/1\",\"queries\":[\"-1/1\",\"0/1\",\"1/1\"],\"u\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0052","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-1)/(2), (1)/(2)]，有序三查詢=[(-1)/(2), (1)/(2), (1)/(1)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-1/2, 1/2], [-1/2, 1/2, 1/1], [是, 是, 否], [[0/1, 1/1], [1/1, 0/1], [3/2, -1/2]])","explanation_zh":"離散結構核對=([-1/2, 1/2], [-1/2, 1/2, 1/1], [是, 是, 否], [[0/1, 1/1], [1/1, 0/1], [3/2, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174052,"parameters":{"l":[-1,2],"u":[1,2],"queries":[[-1,2],[1,2],[1,1]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-1/2\",\"queries\":[\"-1/2\",\"1/2\",\"1/1\"],\"u\":\"1/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0053","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-2)/(3), (2)/(2)]，有序三查詢=[(0)/(1), (-2)/(3), (3)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-2/3, 1/1], [0/1, -2/3, 3/2], [是, 是, 否], [[2/3, 1/1], [0/1, 5/3], [13/6, -1/2]])","explanation_zh":"離散結構核對=([-2/3, 1/1], [0/1, -2/3, 3/2], [是, 是, 否], [[2/3, 1/1], [0/1, 5/3], [13/6, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174053,"parameters":{"l":[-2,3],"u":[2,2],"queries":[[0,1],[-2,3],[3,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-2/3\",\"queries\":[\"0/1\",\"-2/3\",\"3/2\"],\"u\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0054","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-3)/(3), (3)/(2)]，有序三查詢=[(0)/(1), (-3)/(3), (4)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-1/1, 3/2], [0/1, -1/1, 2/1], [是, 是, 否], [[1/1, 3/2], [0/1, 5/2], [3/1, -1/2]])","explanation_zh":"離散結構核對=([-1/1, 3/2], [0/1, -1/1, 2/1], [是, 是, 否], [[1/1, 3/2], [0/1, 5/2], [3/1, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174054,"parameters":{"l":[-3,3],"u":[3,2],"queries":[[0,1],[-3,3],[4,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-1/1\",\"queries\":[\"0/1\",\"-1/1\",\"2/1\"],\"u\":\"3/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0055","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-4)/(3), (4)/(2)]，有序三查詢=[(0)/(1), (-4)/(3), (5)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-4/3, 2/1], [0/1, -4/3, 5/2], [是, 是, 否], [[4/3, 2/1], [0/1, 10/3], [23/6, -1/2]])","explanation_zh":"離散結構核對=([-4/3, 2/1], [0/1, -4/3, 5/2], [是, 是, 否], [[4/3, 2/1], [0/1, 10/3], [23/6, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174055,"parameters":{"l":[-4,3],"u":[4,2],"queries":[[0,1],[-4,3],[5,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-4/3\",\"queries\":[\"0/1\",\"-4/3\",\"5/2\"],\"u\":\"2/1\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0056","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-5)/(3), (5)/(2)]，有序三查詢=[(0)/(1), (-5)/(3), (6)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-5/3, 5/2], [0/1, -5/3, 3/1], [是, 是, 否], [[5/3, 5/2], [0/1, 25/6], [14/3, -1/2]])","explanation_zh":"離散結構核對=([-5/3, 5/2], [0/1, -5/3, 3/1], [是, 是, 否], [[5/3, 5/2], [0/1, 25/6], [14/3, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174056,"parameters":{"l":[-5,3],"u":[5,2],"queries":[[0,1],[-5,3],[6,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-5/3\",\"queries\":[\"0/1\",\"-5/3\",\"3/1\"],\"u\":\"5/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0057","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-6)/(3), (6)/(2)]，有序三查詢=[(0)/(1), (-6)/(3), (7)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-2/1, 3/1], [0/1, -2/1, 7/2], [是, 是, 否], [[2/1, 3/1], [0/1, 5/1], [11/2, -1/2]])","explanation_zh":"離散結構核對=([-2/1, 3/1], [0/1, -2/1, 7/2], [是, 是, 否], [[2/1, 3/1], [0/1, 5/1], [11/2, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174057,"parameters":{"l":[-6,3],"u":[6,2],"queries":[[0,1],[-6,3],[7,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-2/1\",\"queries\":[\"0/1\",\"-2/1\",\"7/2\"],\"u\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0058","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-7)/(3), (7)/(2)]，有序三查詢=[(0)/(1), (-7)/(3), (8)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-7/3, 7/2], [0/1, -7/3, 4/1], [是, 是, 否], [[7/3, 7/2], [0/1, 35/6], [19/3, -1/2]])","explanation_zh":"離散結構核對=([-7/3, 7/2], [0/1, -7/3, 4/1], [是, 是, 否], [[7/3, 7/2], [0/1, 35/6], [19/3, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174058,"parameters":{"l":[-7,3],"u":[7,2],"queries":[[0,1],[-7,3],[8,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-7/3\",\"queries\":[\"0/1\",\"-7/3\",\"4/1\"],\"u\":\"7/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0059","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-8)/(3), (8)/(2)]，有序三查詢=[(0)/(1), (-8)/(3), (9)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-8/3, 4/1], [0/1, -8/3, 9/2], [是, 是, 否], [[8/3, 4/1], [0/1, 20/3], [43/6, -1/2]])","explanation_zh":"離散結構核對=([-8/3, 4/1], [0/1, -8/3, 9/2], [是, 是, 否], [[8/3, 4/1], [0/1, 20/3], [43/6, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174059,"parameters":{"l":[-8,3],"u":[8,2],"queries":[[0,1],[-8,3],[9,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-8/3\",\"queries\":[\"0/1\",\"-8/3\",\"9/2\"],\"u\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0060","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-9)/(3), (9)/(2)]，有序三查詢=[(0)/(1), (-9)/(3), (10)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-3/1, 9/2], [0/1, -3/1, 5/1], [是, 是, 否], [[3/1, 9/2], [0/1, 15/2], [8/1, -1/2]])","explanation_zh":"離散結構核對=([-3/1, 9/2], [0/1, -3/1, 5/1], [是, 是, 否], [[3/1, 9/2], [0/1, 15/2], [8/1, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174060,"parameters":{"l":[-9,3],"u":[9,2],"queries":[[0,1],[-9,3],[10,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-3/1\",\"queries\":[\"0/1\",\"-3/1\",\"5/1\"],\"u\":\"9/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0061","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-10)/(3), (10)/(2)]，有序三查詢=[(0)/(1), (-10)/(3), (11)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-10/3, 5/1], [0/1, -10/3, 11/2], [是, 是, 否], [[10/3, 5/1], [0/1, 25/3], [53/6, -1/2]])","explanation_zh":"離散結構核對=([-10/3, 5/1], [0/1, -10/3, 11/2], [是, 是, 否], [[10/3, 5/1], [0/1, 25/3], [53/6, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174061,"parameters":{"l":[-10,3],"u":[10,2],"queries":[[0,1],[-10,3],[11,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-10/3\",\"queries\":[\"0/1\",\"-10/3\",\"11/2\"],\"u\":\"5/1\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0062","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-11)/(3), (11)/(2)]，有序三查詢=[(0)/(1), (-11)/(3), (12)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-11/3, 11/2], [0/1, -11/3, 6/1], [是, 是, 否], [[11/3, 11/2], [0/1, 55/6], [29/3, -1/2]])","explanation_zh":"離散結構核對=([-11/3, 11/2], [0/1, -11/3, 6/1], [是, 是, 否], [[11/3, 11/2], [0/1, 55/6], [29/3, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174062,"parameters":{"l":[-11,3],"u":[11,2],"queries":[[0,1],[-11,3],[12,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-11/3\",\"queries\":[\"0/1\",\"-11/3\",\"6/1\"],\"u\":\"11/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0063","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-12)/(3), (12)/(2)]，有序三查詢=[(0)/(1), (-12)/(3), (13)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-4/1, 6/1], [0/1, -4/1, 13/2], [是, 是, 否], [[4/1, 6/1], [0/1, 10/1], [21/2, -1/2]])","explanation_zh":"離散結構核對=([-4/1, 6/1], [0/1, -4/1, 13/2], [是, 是, 否], [[4/1, 6/1], [0/1, 10/1], [21/2, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174063,"parameters":{"l":[-12,3],"u":[12,2],"queries":[[0,1],[-12,3],[13,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-4/1\",\"queries\":[\"0/1\",\"-4/1\",\"13/2\"],\"u\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0064","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-13)/(3), (13)/(2)]，有序三查詢=[(0)/(1), (-13)/(3), (14)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-13/3, 13/2], [0/1, -13/3, 7/1], [是, 是, 否], [[13/3, 13/2], [0/1, 65/6], [34/3, -1/2]])","explanation_zh":"離散結構核對=([-13/3, 13/2], [0/1, -13/3, 7/1], [是, 是, 否], [[13/3, 13/2], [0/1, 65/6], [34/3, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174064,"parameters":{"l":[-13,3],"u":[13,2],"queries":[[0,1],[-13,3],[14,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-13/3\",\"queries\":[\"0/1\",\"-13/3\",\"7/1\"],\"u\":\"13/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0065","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-14)/(3), (14)/(2)]，有序三查詢=[(0)/(1), (-14)/(3), (15)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-14/3, 7/1], [0/1, -14/3, 15/2], [是, 是, 否], [[14/3, 7/1], [0/1, 35/3], [73/6, -1/2]])","explanation_zh":"離散結構核對=([-14/3, 7/1], [0/1, -14/3, 15/2], [是, 是, 否], [[14/3, 7/1], [0/1, 35/3], [73/6, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174065,"parameters":{"l":[-14,3],"u":[14,2],"queries":[[0,1],[-14,3],[15,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-14/3\",\"queries\":[\"0/1\",\"-14/3\",\"15/2\"],\"u\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0066","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-15)/(3), (15)/(2)]，有序三查詢=[(0)/(1), (-15)/(3), (16)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-5/1, 15/2], [0/1, -5/1, 8/1], [是, 是, 否], [[5/1, 15/2], [0/1, 25/2], [13/1, -1/2]])","explanation_zh":"離散結構核對=([-5/1, 15/2], [0/1, -5/1, 8/1], [是, 是, 否], [[5/1, 15/2], [0/1, 25/2], [13/1, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174066,"parameters":{"l":[-15,3],"u":[15,2],"queries":[[0,1],[-15,3],[16,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-5/1\",\"queries\":[\"0/1\",\"-5/1\",\"8/1\"],\"u\":\"15/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0067","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-16)/(3), (16)/(2)]，有序三查詢=[(0)/(1), (-16)/(3), (17)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-16/3, 8/1], [0/1, -16/3, 17/2], [是, 是, 否], [[16/3, 8/1], [0/1, 40/3], [83/6, -1/2]])","explanation_zh":"離散結構核對=([-16/3, 8/1], [0/1, -16/3, 17/2], [是, 是, 否], [[16/3, 8/1], [0/1, 40/3], [83/6, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174067,"parameters":{"l":[-16,3],"u":[16,2],"queries":[[0,1],[-16,3],[17,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-16/3\",\"queries\":[\"0/1\",\"-16/3\",\"17/2\"],\"u\":\"8/1\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0068","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-17)/(3), (17)/(2)]，有序三查詢=[(0)/(1), (-17)/(3), (18)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-17/3, 17/2], [0/1, -17/3, 9/1], [是, 是, 否], [[17/3, 17/2], [0/1, 85/6], [44/3, -1/2]])","explanation_zh":"離散結構核對=([-17/3, 17/2], [0/1, -17/3, 9/1], [是, 是, 否], [[17/3, 17/2], [0/1, 85/6], [44/3, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174068,"parameters":{"l":[-17,3],"u":[17,2],"queries":[[0,1],[-17,3],[18,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-17/3\",\"queries\":[\"0/1\",\"-17/3\",\"9/1\"],\"u\":\"17/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0069","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-18)/(3), (18)/(2)]，有序三查詢=[(0)/(1), (-18)/(3), (19)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-6/1, 9/1], [0/1, -6/1, 19/2], [是, 是, 否], [[6/1, 9/1], [0/1, 15/1], [31/2, -1/2]])","explanation_zh":"離散結構核對=([-6/1, 9/1], [0/1, -6/1, 19/2], [是, 是, 否], [[6/1, 9/1], [0/1, 15/1], [31/2, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174069,"parameters":{"l":[-18,3],"u":[18,2],"queries":[[0,1],[-18,3],[19,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-6/1\",\"queries\":[\"0/1\",\"-6/1\",\"19/2\"],\"u\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0070","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-19)/(3), (19)/(2)]，有序三查詢=[(0)/(1), (-19)/(3), (20)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-19/3, 19/2], [0/1, -19/3, 10/1], [是, 是, 否], [[19/3, 19/2], [0/1, 95/6], [49/3, -1/2]])","explanation_zh":"離散結構核對=([-19/3, 19/2], [0/1, -19/3, 10/1], [是, 是, 否], [[19/3, 19/2], [0/1, 95/6], [49/3, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174070,"parameters":{"l":[-19,3],"u":[19,2],"queries":[[0,1],[-19,3],[20,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-19/3\",\"queries\":[\"0/1\",\"-19/3\",\"10/1\"],\"u\":\"19/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0071","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-20)/(3), (20)/(2)]，有序三查詢=[(0)/(1), (-20)/(3), (21)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-20/3, 10/1], [0/1, -20/3, 21/2], [是, 是, 否], [[20/3, 10/1], [0/1, 50/3], [103/6, -1/2]])","explanation_zh":"離散結構核對=([-20/3, 10/1], [0/1, -20/3, 21/2], [是, 是, 否], [[20/3, 10/1], [0/1, 50/3], [103/6, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174071,"parameters":{"l":[-20,3],"u":[20,2],"queries":[[0,1],[-20,3],[21,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-20/3\",\"queries\":[\"0/1\",\"-20/3\",\"21/2\"],\"u\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0072","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-21)/(3), (21)/(2)]，有序三查詢=[(0)/(1), (-21)/(3), (22)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-7/1, 21/2], [0/1, -7/1, 11/1], [是, 是, 否], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]])","explanation_zh":"離散結構核對=([-7/1, 21/2], [0/1, -7/1, 11/1], [是, 是, 否], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174072,"parameters":{"l":[-21,3],"u":[21,2],"queries":[[0,1],[-21,3],[22,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-7/1\",\"queries\":[\"0/1\",\"-7/1\",\"11/1\"],\"u\":\"21/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0073","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-22)/(3), (22)/(2)]，有序三查詢=[(0)/(1), (-22)/(3), (23)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-22/3, 11/1], [0/1, -22/3, 23/2], [是, 是, 否], [[22/3, 11/1], [0/1, 55/3], [113/6, -1/2]])","explanation_zh":"離散結構核對=([-22/3, 11/1], [0/1, -22/3, 23/2], [是, 是, 否], [[22/3, 11/1], [0/1, 55/3], [113/6, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174073,"parameters":{"l":[-22,3],"u":[22,2],"queries":[[0,1],[-22,3],[23,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-22/3\",\"queries\":[\"0/1\",\"-22/3\",\"23/2\"],\"u\":\"11/1\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0074","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-23)/(3), (23)/(2)]，有序三查詢=[(0)/(1), (-23)/(3), (24)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-23/3, 23/2], [0/1, -23/3, 12/1], [是, 是, 否], [[23/3, 23/2], [0/1, 115/6], [59/3, -1/2]])","explanation_zh":"離散結構核對=([-23/3, 23/2], [0/1, -23/3, 12/1], [是, 是, 否], [[23/3, 23/2], [0/1, 115/6], [59/3, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174074,"parameters":{"l":[-23,3],"u":[23,2],"queries":[[0,1],[-23,3],[24,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-23/3\",\"queries\":[\"0/1\",\"-23/3\",\"12/1\"],\"u\":\"23/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0075","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"閉區間會員與兩側鬆弛量","stem_zh":"R通常全序，原分母為正。閉區間=[(-24)/(3), (24)/(2)]，有序三查詢=[(0)/(1), (-24)/(3), (25)/(2)]。答 (區間兩界, 三查詢既約值, 三會員資格, 每查詢的[query−l,u−query])。最後兩數是有號差。","answer":{"kind":"expression","value":"([-8/1, 12/1], [0/1, -8/1, 25/2], [是, 是, 否], [[8/1, 12/1], [0/1, 20/1], [41/2, -1/2]])","explanation_zh":"離散結構核對=([-8/1, 12/1], [0/1, -8/1, 25/2], [是, 是, 否], [[8/1, 12/1], [0/1, 20/1], [41/2, -1/2]])；分別算距上下界的有號差；兩差皆非負即含端點會員。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["閉區間會員與兩側鬆弛量"]},"variant":{"family_id":"qf-density17-order-three-membership-slacks","seed":202610174075,"parameters":{"l":[-24,3],"u":[24,2],"queries":[[0,1],[-24,3],[25,2]],"domain":"R_positive_denominator_order","operation":"order-three-membership-slacks","semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-8/1\",\"queries\":[\"0/1\",\"-8/1\",\"25/2\"],\"u\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0076","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(-1)/(2)、b=(1)/(3)、t=(0)/(1)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(-1/2, 1/3, 0/1, 0/1, 0/1, <, =, 合併)","explanation_zh":"離散結構核對=(-1/2, 1/3, 0/1, 0/1, 0/1, <, =, 合併)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174076,"parameters":{"a":[-1,2],"b":[1,3],"t":[0,1],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-1/2\",\"b\":\"1/3\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0077","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(0)/(1)、b=(0)/(1)、t=(-2)/(1)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(0/1, 0/1, -2/1, 0/1, 0/1, =, =, 反向)","explanation_zh":"離散結構核對=(0/1, 0/1, -2/1, 0/1, 0/1, =, =, 反向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174077,"parameters":{"a":[0,1],"b":[0,1],"t":[-2,1],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"0/1\",\"b\":\"0/1\",\"t\":\"-2/1\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0078","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(-10)/(3)、b=(-9)/(2)、t=(3)/(3)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(-10/3, -9/2, 1/1, -10/3, -9/2, >, >, 正向)","explanation_zh":"離散結構核對=(-10/3, -9/2, 1/1, -10/3, -9/2, >, >, 正向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174078,"parameters":{"a":[-10,3],"b":[-9,2],"t":[3,3],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-10/3\",\"b\":\"-9/2\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0079","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(-9)/(3)、b=(-8)/(2)、t=(0)/(4)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(-3/1, -4/1, 0/1, 0/1, 0/1, >, =, 合併)","explanation_zh":"離散結構核對=(-3/1, -4/1, 0/1, 0/1, 0/1, >, =, 合併)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174079,"parameters":{"a":[-9,3],"b":[-8,2],"t":[0,4],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-3/1\",\"b\":\"-4/1\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0080","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(-8)/(3)、b=(-7)/(2)、t=(3)/(1)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(-8/3, -7/2, 3/1, -8/1, -21/2, >, >, 正向)","explanation_zh":"離散結構核對=(-8/3, -7/2, 3/1, -8/1, -21/2, >, >, 正向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174080,"parameters":{"a":[-8,3],"b":[-7,2],"t":[3,1],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-8/3\",\"b\":\"-7/2\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0081","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(-7)/(3)、b=(-6)/(2)、t=(-2)/(2)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(-7/3, -3/1, -1/1, 7/3, 3/1, >, <, 反向)","explanation_zh":"離散結構核對=(-7/3, -3/1, -1/1, 7/3, 3/1, >, <, 反向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174081,"parameters":{"a":[-7,3],"b":[-6,2],"t":[-2,2],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-7/3\",\"b\":\"-3/1\",\"t\":\"-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0082","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(-6)/(3)、b=(-5)/(2)、t=(0)/(3)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(-2/1, -5/2, 0/1, 0/1, 0/1, >, =, 合併)","explanation_zh":"離散結構核對=(-2/1, -5/2, 0/1, 0/1, 0/1, >, =, 合併)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174082,"parameters":{"a":[-6,3],"b":[-5,2],"t":[0,3],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-2/1\",\"b\":\"-5/2\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0083","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(-5)/(3)、b=(-4)/(2)、t=(-2)/(4)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(-5/3, -2/1, -1/2, 5/6, 1/1, >, <, 反向)","explanation_zh":"離散結構核對=(-5/3, -2/1, -1/2, 5/6, 1/1, >, <, 反向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174083,"parameters":{"a":[-5,3],"b":[-4,2],"t":[-2,4],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-5/3\",\"b\":\"-2/1\",\"t\":\"-1/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0084","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(-4)/(3)、b=(-3)/(2)、t=(3)/(1)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(-4/3, -3/2, 3/1, -4/1, -9/2, >, >, 正向)","explanation_zh":"離散結構核對=(-4/3, -3/2, 3/1, -4/1, -9/2, >, >, 正向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174084,"parameters":{"a":[-4,3],"b":[-3,2],"t":[3,1],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-4/3\",\"b\":\"-3/2\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0085","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(-3)/(3)、b=(-2)/(2)、t=(0)/(2)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(-1/1, -1/1, 0/1, 0/1, 0/1, =, =, 合併)","explanation_zh":"離散結構核對=(-1/1, -1/1, 0/1, 0/1, 0/1, =, =, 合併)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174085,"parameters":{"a":[-3,3],"b":[-2,2],"t":[0,2],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-1/1\",\"b\":\"-1/1\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0086","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(-2)/(3)、b=(-1)/(2)、t=(3)/(3)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(-2/3, -1/2, 1/1, -2/3, -1/2, <, <, 正向)","explanation_zh":"離散結構核對=(-2/3, -1/2, 1/1, -2/3, -1/2, <, <, 正向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174086,"parameters":{"a":[-2,3],"b":[-1,2],"t":[3,3],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-2/3\",\"b\":\"-1/2\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0087","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(-1)/(3)、b=(0)/(2)、t=(-2)/(4)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(-1/3, 0/1, -1/2, 1/6, 0/1, <, >, 反向)","explanation_zh":"離散結構核對=(-1/3, 0/1, -1/2, 1/6, 0/1, <, >, 反向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174087,"parameters":{"a":[-1,3],"b":[0,2],"t":[-2,4],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-1/3\",\"b\":\"0/1\",\"t\":\"-1/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0088","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(0)/(3)、b=(1)/(2)、t=(0)/(1)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(0/1, 1/2, 0/1, 0/1, 0/1, <, =, 合併)","explanation_zh":"離散結構核對=(0/1, 1/2, 0/1, 0/1, 0/1, <, =, 合併)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174088,"parameters":{"a":[0,3],"b":[1,2],"t":[0,1],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"0/1\",\"b\":\"1/2\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0089","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(1)/(3)、b=(2)/(2)、t=(-2)/(2)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(1/3, 1/1, -1/1, -1/3, -1/1, <, >, 反向)","explanation_zh":"離散結構核對=(1/3, 1/1, -1/1, -1/3, -1/1, <, >, 反向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174089,"parameters":{"a":[1,3],"b":[2,2],"t":[-2,2],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"1/3\",\"b\":\"1/1\",\"t\":\"-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0090","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(2)/(3)、b=(3)/(2)、t=(3)/(3)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(2/3, 3/2, 1/1, 2/3, 3/2, <, <, 正向)","explanation_zh":"離散結構核對=(2/3, 3/2, 1/1, 2/3, 3/2, <, <, 正向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174090,"parameters":{"a":[2,3],"b":[3,2],"t":[3,3],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"2/3\",\"b\":\"3/2\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0091","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(3)/(3)、b=(4)/(2)、t=(0)/(4)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(1/1, 2/1, 0/1, 0/1, 0/1, <, =, 合併)","explanation_zh":"離散結構核對=(1/1, 2/1, 0/1, 0/1, 0/1, <, =, 合併)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174091,"parameters":{"a":[3,3],"b":[4,2],"t":[0,4],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"1/1\",\"b\":\"2/1\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0092","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(4)/(3)、b=(5)/(2)、t=(3)/(1)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(4/3, 5/2, 3/1, 4/1, 15/2, <, <, 正向)","explanation_zh":"離散結構核對=(4/3, 5/2, 3/1, 4/1, 15/2, <, <, 正向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174092,"parameters":{"a":[4,3],"b":[5,2],"t":[3,1],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"4/3\",\"b\":\"5/2\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0093","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(5)/(3)、b=(6)/(2)、t=(-2)/(2)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(5/3, 3/1, -1/1, -5/3, -3/1, <, >, 反向)","explanation_zh":"離散結構核對=(5/3, 3/1, -1/1, -5/3, -3/1, <, >, 反向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174093,"parameters":{"a":[5,3],"b":[6,2],"t":[-2,2],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"5/3\",\"b\":\"3/1\",\"t\":\"-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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0094","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(6)/(3)、b=(7)/(2)、t=(0)/(3)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(2/1, 7/2, 0/1, 0/1, 0/1, <, =, 合併)","explanation_zh":"離散結構核對=(2/1, 7/2, 0/1, 0/1, 0/1, <, =, 合併)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174094,"parameters":{"a":[6,3],"b":[7,2],"t":[0,3],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"2/1\",\"b\":\"7/2\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0095","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(7)/(3)、b=(8)/(2)、t=(-2)/(4)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(7/3, 4/1, -1/2, -7/6, -2/1, <, >, 反向)","explanation_zh":"離散結構核對=(7/3, 4/1, -1/2, -7/6, -2/1, <, >, 反向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174095,"parameters":{"a":[7,3],"b":[8,2],"t":[-2,4],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"7/3\",\"b\":\"4/1\",\"t\":\"-1/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0096","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(8)/(3)、b=(9)/(2)、t=(3)/(1)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(8/3, 9/2, 3/1, 8/1, 27/2, <, <, 正向)","explanation_zh":"離散結構核對=(8/3, 9/2, 3/1, 8/1, 27/2, <, <, 正向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174096,"parameters":{"a":[8,3],"b":[9,2],"t":[3,1],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"8/3\",\"b\":\"9/2\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0097","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(9)/(3)、b=(10)/(2)、t=(0)/(2)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(3/1, 5/1, 0/1, 0/1, 0/1, <, =, 合併)","explanation_zh":"離散結構核對=(3/1, 5/1, 0/1, 0/1, 0/1, <, =, 合併)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174097,"parameters":{"a":[9,3],"b":[10,2],"t":[0,2],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"3/1\",\"b\":\"5/1\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0098","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(10)/(3)、b=(11)/(2)、t=(3)/(3)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(10/3, 11/2, 1/1, 10/3, 11/2, <, <, 正向)","explanation_zh":"離散結構核對=(10/3, 11/2, 1/1, 10/3, 11/2, <, <, 正向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174098,"parameters":{"a":[10,3],"b":[11,2],"t":[3,3],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"10/3\",\"b\":\"11/2\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0099","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(11)/(3)、b=(12)/(2)、t=(-2)/(4)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(11/3, 6/1, -1/2, -11/6, -3/1, <, >, 反向)","explanation_zh":"離散結構核對=(11/3, 6/1, -1/2, -11/6, -3/1, <, >, 反向)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174099,"parameters":{"a":[11,3],"b":[12,2],"t":[-2,4],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"11/3\",\"b\":\"6/1\",\"t\":\"-1/2\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261012-advance-20261017-r04-order-relation-0100","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-order-relation"],"title_zh":"同乘純量後的大小方向","stem_zh":"R通常全序，原分母為正。a=(12)/(3)、b=(13)/(2)、t=(0)/(1)。答 (a,b,t,t*a,t*b, 原比較, 新比較, 正向或反向或合併)。","answer":{"kind":"expression","value":"(4/1, 13/2, 0/1, 0/1, 0/1, <, =, 合併)","explanation_zh":"離散結構核對=(4/1, 13/2, 0/1, 0/1, 0/1, <, =, 合併)；精確算兩個乘積再比較；t的正、負、零各給指定方向標籤。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同乘純量後的大小方向"]},"variant":{"family_id":"qf-density17-order-scaling-direction","seed":202610174100,"parameters":{"a":[12,3],"b":[13,2],"t":[0,1],"domain":"R_positive_denominator_order","operation":"order-scaling-direction","semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"4/1\",\"b\":\"13/2\",\"t\":\"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":"2026-10-12","contract_version":"ocme-question-factory-v0.1"}}