{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0001","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=1/-2、b=1/3。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(-1/6, -5/6)","explanation_zh":"有理和與差=(-1/6, -5/6)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133001,"parameters":{"a":[1,-2],"b":[1,3],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"-1/2\",\"1/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0002","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=0/-5、b=-2/-3。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(2/3, -2/3)","explanation_zh":"有理和與差=(2/3, -2/3)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133002,"parameters":{"a":[0,-5],"b":[-2,-3],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"0/1\",\"2/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0003","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=-10/3、b=3/5。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(-41/15, -59/15)","explanation_zh":"有理和與差=(-41/15, -59/15)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133003,"parameters":{"a":[-10,3],"b":[3,5],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"-10/3\",\"3/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0004","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=-9/1、b=-4/-4。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(-8/1, -10/1)","explanation_zh":"有理和與差=(-8/1, -10/1)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133004,"parameters":{"a":[-9,1],"b":[-4,-4],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"-9/1\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0005","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=-8/2、b=5/5。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(-3/1, -5/1)","explanation_zh":"有理和與差=(-3/1, -5/1)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133005,"parameters":{"a":[-8,2],"b":[5,5],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"-4/1\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0006","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=-7/3、b=-6/-4。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(-5/6, -23/6)","explanation_zh":"有理和與差=(-5/6, -23/6)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133006,"parameters":{"a":[-7,3],"b":[-6,-4],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"-7/3\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0007","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=-6/1、b=7/5。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(-23/5, -37/5)","explanation_zh":"有理和與差=(-23/5, -37/5)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133007,"parameters":{"a":[-6,1],"b":[7,5],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"-6/1\",\"7/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0008","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=-5/2、b=-8/-4。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(-1/2, -9/2)","explanation_zh":"有理和與差=(-1/2, -9/2)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133008,"parameters":{"a":[-5,2],"b":[-8,-4],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"-5/2\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0009","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=-4/3、b=9/5。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(7/15, -47/15)","explanation_zh":"有理和與差=(7/15, -47/15)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133009,"parameters":{"a":[-4,3],"b":[9,5],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"-4/3\",\"9/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0010","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=-3/1、b=-10/-4。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(-1/2, -11/2)","explanation_zh":"有理和與差=(-1/2, -11/2)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133010,"parameters":{"a":[-3,1],"b":[-10,-4],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"-3/1\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0011","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=-2/2、b=11/5。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(6/5, -16/5)","explanation_zh":"有理和與差=(6/5, -16/5)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133011,"parameters":{"a":[-2,2],"b":[11,5],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"-1/1\",\"11/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0012","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=-1/3、b=-12/-4。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(8/3, -10/3)","explanation_zh":"有理和與差=(8/3, -10/3)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133012,"parameters":{"a":[-1,3],"b":[-12,-4],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"-1/3\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0013","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=0/1、b=13/5。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(13/5, -13/5)","explanation_zh":"有理和與差=(13/5, -13/5)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133013,"parameters":{"a":[0,1],"b":[13,5],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"0/1\",\"13/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0014","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=1/2、b=-14/-4。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(4/1, -3/1)","explanation_zh":"有理和與差=(4/1, -3/1)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133014,"parameters":{"a":[1,2],"b":[-14,-4],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"1/2\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0015","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=2/3、b=15/5。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(11/3, -7/3)","explanation_zh":"有理和與差=(11/3, -7/3)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133015,"parameters":{"a":[2,3],"b":[15,5],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"2/3\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0016","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=3/1、b=-16/-4。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(7/1, -1/1)","explanation_zh":"有理和與差=(7/1, -1/1)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133016,"parameters":{"a":[3,1],"b":[-16,-4],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"3/1\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0017","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=4/2、b=17/5。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(27/5, -7/5)","explanation_zh":"有理和與差=(27/5, -7/5)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133017,"parameters":{"a":[4,2],"b":[17,5],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"2/1\",\"17/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0018","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=5/3、b=-18/-4。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(37/6, -17/6)","explanation_zh":"有理和與差=(37/6, -17/6)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133018,"parameters":{"a":[5,3],"b":[-18,-4],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"5/3\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0019","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=6/1、b=19/5。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(49/5, 11/5)","explanation_zh":"有理和與差=(49/5, 11/5)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133019,"parameters":{"a":[6,1],"b":[19,5],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"6/1\",\"19/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0020","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=7/2、b=-20/-4。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(17/2, -3/2)","explanation_zh":"有理和與差=(17/2, -3/2)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133020,"parameters":{"a":[7,2],"b":[-20,-4],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"7/2\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0021","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=8/3、b=21/5。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(103/15, -23/15)","explanation_zh":"有理和與差=(103/15, -23/15)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133021,"parameters":{"a":[8,3],"b":[21,5],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"8/3\",\"21/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0022","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=9/1、b=-22/-4。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(29/2, 7/2)","explanation_zh":"有理和與差=(29/2, 7/2)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133022,"parameters":{"a":[9,1],"b":[-22,-4],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"9/1\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0023","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=10/2、b=23/5。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(48/5, 2/5)","explanation_zh":"有理和與差=(48/5, 2/5)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133023,"parameters":{"a":[10,2],"b":[23,5],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"5/1\",\"23/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0024","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=11/3、b=-24/-4。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(29/3, -7/3)","explanation_zh":"有理和與差=(29/3, -7/3)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133024,"parameters":{"a":[11,3],"b":[-24,-4],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"11/3\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0025","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理和與有序差","stem_zh":"精確有理數協議；q≠0，a=12/1、b=25/5。依序求 (a+b, a−b)，均寫最簡n/d。","answer":{"kind":"expression","value":"(17/1, 7/1)","explanation_zh":"有理和與差=(17/1, 7/1)。通分後相加與相減；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理和與有序差"]},"variant":{"family_id":"qf-density13-rational-sum-difference-pair","seed":202610133025,"parameters":{"a":[12,1],"b":[25,5],"operation":"density13_rational_sum_difference_pair","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-sum-difference-pair/[\"12/1\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0026","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-32/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(<, -3/1)","explanation_zh":"次序與精確差=(<, -3/1)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133026,"parameters":{"a":[-32,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"-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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0027","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=29/-12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(<, -11/4)","explanation_zh":"次序與精確差=(<, -11/4)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133027,"parameters":{"a":[29,-12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"-11/4\""}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0028","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-26/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(<, -5/2)","explanation_zh":"次序與精確差=(<, -5/2)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133028,"parameters":{"a":[-26,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"-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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0029","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=23/-12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(<, -9/4)","explanation_zh":"次序與精確差=(<, -9/4)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133029,"parameters":{"a":[23,-12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"-9/4\""}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0030","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-20/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(<, -2/1)","explanation_zh":"次序與精確差=(<, -2/1)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133030,"parameters":{"a":[-20,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"-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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0031","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=17/-12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(<, -7/4)","explanation_zh":"次序與精確差=(<, -7/4)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133031,"parameters":{"a":[17,-12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"-7/4\""}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0032","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-14/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(<, -3/2)","explanation_zh":"次序與精確差=(<, -3/2)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133032,"parameters":{"a":[-14,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"-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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0033","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=11/-12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(<, -5/4)","explanation_zh":"次序與精確差=(<, -5/4)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133033,"parameters":{"a":[11,-12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"-5/4\""}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0034","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-8/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(<, -1/1)","explanation_zh":"次序與精確差=(<, -1/1)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133034,"parameters":{"a":[-8,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"-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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0035","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=5/-12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(<, -3/4)","explanation_zh":"次序與精確差=(<, -3/4)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133035,"parameters":{"a":[5,-12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"-3/4\""}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0036","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-2/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(<, -1/2)","explanation_zh":"次序與精確差=(<, -1/2)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133036,"parameters":{"a":[-2,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"-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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0037","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-1/-12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(<, -1/4)","explanation_zh":"次序與精確差=(<, -1/4)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133037,"parameters":{"a":[-1,-12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"-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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0038","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=4/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(=, 0/1)","explanation_zh":"次序與精確差=(=, 0/1)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133038,"parameters":{"a":[4,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0039","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-7/-12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(>, 1/4)","explanation_zh":"次序與精確差=(>, 1/4)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133039,"parameters":{"a":[-7,-12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0040","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=10/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(>, 1/2)","explanation_zh":"次序與精確差=(>, 1/2)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133040,"parameters":{"a":[10,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0041","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-13/-12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(>, 3/4)","explanation_zh":"次序與精確差=(>, 3/4)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133041,"parameters":{"a":[-13,-12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"3/4\""}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0042","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=16/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(>, 1/1)","explanation_zh":"次序與精確差=(>, 1/1)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133042,"parameters":{"a":[16,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0043","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-19/-12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(>, 5/4)","explanation_zh":"次序與精確差=(>, 5/4)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133043,"parameters":{"a":[-19,-12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"5/4\""}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0044","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=22/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(>, 3/2)","explanation_zh":"次序與精確差=(>, 3/2)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133044,"parameters":{"a":[22,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0045","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-25/-12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(>, 7/4)","explanation_zh":"次序與精確差=(>, 7/4)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133045,"parameters":{"a":[-25,-12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"7/4\""}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0046","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=28/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(>, 2/1)","explanation_zh":"次序與精確差=(>, 2/1)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133046,"parameters":{"a":[28,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0047","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-31/-12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(>, 9/4)","explanation_zh":"次序與精確差=(>, 9/4)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133047,"parameters":{"a":[-31,-12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"9/4\""}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0048","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=34/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(>, 5/2)","explanation_zh":"次序與精確差=(>, 5/2)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133048,"parameters":{"a":[34,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0049","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=-37/-12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(>, 11/4)","explanation_zh":"次序與精確差=(>, 11/4)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133049,"parameters":{"a":[-37,-12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"11/4\""}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0050","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理次序與精確差","stem_zh":"精確有理數協議；q≠0，a=40/12、b=1/3。依序答 (a與b的<／=／>關係, a−b的最簡分數)。","answer":{"kind":"expression","value":"(>, 3/1)","explanation_zh":"次序與精確差=(>, 3/1)。以精確差的符號判次序，不使用小數；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理次序與精確差"]},"variant":{"family_id":"qf-density13-rational-ordered-pair-compare","seed":202610133050,"parameters":{"a":[40,12],"b":[1,3],"operation":"density13_rational_ordered_pair_compare","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-ordered-pair-compare/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0051","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=5/-2、b=0/-3。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(否, 未定義, 未定義)","explanation_zh":"除法資格與回乘=(否, 未定義, 未定義)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133051,"parameters":{"a":[5,-2],"b":[0,-3],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"undefined\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0052","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=0/-7、b=2/-3。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, 0/1, 0/1)","explanation_zh":"除法資格與回乘=(是, 0/1, 0/1)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133052,"parameters":{"a":[0,-7],"b":[2,-3],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"zero_quotient\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0053","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=3/-3、b=4/5。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, -5/4, -1/1)","explanation_zh":"除法資格與回乘=(是, -5/4, -1/1)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133053,"parameters":{"a":[3,-3],"b":[4,5],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-1/1\",\"4/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0054","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=4/-3、b=5/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, 16/15, -4/3)","explanation_zh":"除法資格與回乘=(是, 16/15, -4/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133054,"parameters":{"a":[4,-3],"b":[5,-4],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-4/3\",\"-5/4\"]"}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0055","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=5/-3、b=6/5。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, -25/18, -5/3)","explanation_zh":"除法資格與回乘=(是, -25/18, -5/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133055,"parameters":{"a":[5,-3],"b":[6,5],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-5/3\",\"6/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0056","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=6/-3、b=7/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, 8/7, -2/1)","explanation_zh":"除法資格與回乘=(是, 8/7, -2/1)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133056,"parameters":{"a":[6,-3],"b":[7,-4],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-2/1\",\"-7/4\"]"}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0057","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=7/-3、b=8/5。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, -35/24, -7/3)","explanation_zh":"除法資格與回乘=(是, -35/24, -7/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133057,"parameters":{"a":[7,-3],"b":[8,5],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-7/3\",\"8/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0058","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=8/-3、b=9/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, 32/27, -8/3)","explanation_zh":"除法資格與回乘=(是, 32/27, -8/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133058,"parameters":{"a":[8,-3],"b":[9,-4],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-8/3\",\"-9/4\"]"}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0059","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=9/-3、b=10/5。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, -3/2, -3/1)","explanation_zh":"除法資格與回乘=(是, -3/2, -3/1)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133059,"parameters":{"a":[9,-3],"b":[10,5],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-3/1\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0060","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=10/-3、b=11/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, 40/33, -10/3)","explanation_zh":"除法資格與回乘=(是, 40/33, -10/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133060,"parameters":{"a":[10,-3],"b":[11,-4],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-10/3\",\"-11/4\"]"}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0061","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=11/-3、b=12/5。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, -55/36, -11/3)","explanation_zh":"除法資格與回乘=(是, -55/36, -11/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133061,"parameters":{"a":[11,-3],"b":[12,5],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-11/3\",\"12/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0062","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=12/-3、b=13/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, 16/13, -4/1)","explanation_zh":"除法資格與回乘=(是, 16/13, -4/1)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133062,"parameters":{"a":[12,-3],"b":[13,-4],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-4/1\",\"-13/4\"]"}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0063","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=13/-3、b=14/5。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, -65/42, -13/3)","explanation_zh":"除法資格與回乘=(是, -65/42, -13/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133063,"parameters":{"a":[13,-3],"b":[14,5],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-13/3\",\"14/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0064","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=14/-3、b=15/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, 56/45, -14/3)","explanation_zh":"除法資格與回乘=(是, 56/45, -14/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133064,"parameters":{"a":[14,-3],"b":[15,-4],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-14/3\",\"-15/4\"]"}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0065","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=15/-3、b=16/5。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, -25/16, -5/1)","explanation_zh":"除法資格與回乘=(是, -25/16, -5/1)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133065,"parameters":{"a":[15,-3],"b":[16,5],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-5/1\",\"16/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0066","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=16/-3、b=17/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, 64/51, -16/3)","explanation_zh":"除法資格與回乘=(是, 64/51, -16/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133066,"parameters":{"a":[16,-3],"b":[17,-4],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-16/3\",\"-17/4\"]"}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0067","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=17/-3、b=18/5。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, -85/54, -17/3)","explanation_zh":"除法資格與回乘=(是, -85/54, -17/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133067,"parameters":{"a":[17,-3],"b":[18,5],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-17/3\",\"18/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0068","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=18/-3、b=19/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, 24/19, -6/1)","explanation_zh":"除法資格與回乘=(是, 24/19, -6/1)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133068,"parameters":{"a":[18,-3],"b":[19,-4],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-6/1\",\"-19/4\"]"}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0069","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=19/-3、b=20/5。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, -19/12, -19/3)","explanation_zh":"除法資格與回乘=(是, -19/12, -19/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133069,"parameters":{"a":[19,-3],"b":[20,5],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-19/3\",\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0070","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=20/-3、b=21/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, 80/63, -20/3)","explanation_zh":"除法資格與回乘=(是, 80/63, -20/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133070,"parameters":{"a":[20,-3],"b":[21,-4],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-20/3\",\"-21/4\"]"}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0071","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=21/-3、b=22/5。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, -35/22, -7/1)","explanation_zh":"除法資格與回乘=(是, -35/22, -7/1)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133071,"parameters":{"a":[21,-3],"b":[22,5],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-7/1\",\"22/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0072","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=22/-3、b=23/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, 88/69, -22/3)","explanation_zh":"除法資格與回乘=(是, 88/69, -22/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133072,"parameters":{"a":[22,-3],"b":[23,-4],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-22/3\",\"-23/4\"]"}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0073","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=23/-3、b=24/5。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, -115/72, -23/3)","explanation_zh":"除法資格與回乘=(是, -115/72, -23/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133073,"parameters":{"a":[23,-3],"b":[24,5],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-23/3\",\"24/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0074","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=24/-3、b=25/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, 32/25, -8/1)","explanation_zh":"除法資格與回乘=(是, 32/25, -8/1)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133074,"parameters":{"a":[24,-3],"b":[25,-4],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-8/1\",\"-25/4\"]"}},"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0075","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"有理除法資格與回乘","stem_zh":"精確有理數協議；q≠0，a=25/-3、b=26/5。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。","answer":{"kind":"expression","value":"(是, -125/78, -25/3)","explanation_zh":"除法資格與回乘=(是, -125/78, -25/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有理除法資格與回乘"]},"variant":{"family_id":"qf-density13-rational-division-guard","seed":202610133075,"parameters":{"a":[25,-3],"b":[26,5],"operation":"density13_rational_division_guard","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-division-guard/[\"-25/3\",\"26/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0076","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-23/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(-4/1, 0/1)","explanation_zh":"缺失加數與回代差=(-4/1, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133076,"parameters":{"a":[1,2],"b":[-1,3],"target":[-23,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"-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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0077","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=21/-6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(-11/3, 0/1)","explanation_zh":"缺失加數與回代差=(-11/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133077,"parameters":{"a":[1,2],"b":[-1,3],"target":[21,-6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"-11/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0078","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-19/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(-10/3, 0/1)","explanation_zh":"缺失加數與回代差=(-10/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133078,"parameters":{"a":[1,2],"b":[-1,3],"target":[-19,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"-10/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0079","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=17/-6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(-3/1, 0/1)","explanation_zh":"缺失加數與回代差=(-3/1, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133079,"parameters":{"a":[1,2],"b":[-1,3],"target":[17,-6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"-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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0080","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-15/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(-8/3, 0/1)","explanation_zh":"缺失加數與回代差=(-8/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133080,"parameters":{"a":[1,2],"b":[-1,3],"target":[-15,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"-8/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0081","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=13/-6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(-7/3, 0/1)","explanation_zh":"缺失加數與回代差=(-7/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133081,"parameters":{"a":[1,2],"b":[-1,3],"target":[13,-6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"-7/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0082","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-11/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(-2/1, 0/1)","explanation_zh":"缺失加數與回代差=(-2/1, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133082,"parameters":{"a":[1,2],"b":[-1,3],"target":[-11,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"-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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0083","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=9/-6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(-5/3, 0/1)","explanation_zh":"缺失加數與回代差=(-5/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133083,"parameters":{"a":[1,2],"b":[-1,3],"target":[9,-6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"-5/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0084","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-7/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(-4/3, 0/1)","explanation_zh":"缺失加數與回代差=(-4/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133084,"parameters":{"a":[1,2],"b":[-1,3],"target":[-7,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"-4/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0085","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=5/-6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(-1/1, 0/1)","explanation_zh":"缺失加數與回代差=(-1/1, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133085,"parameters":{"a":[1,2],"b":[-1,3],"target":[5,-6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"-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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0086","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-3/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(-2/3, 0/1)","explanation_zh":"缺失加數與回代差=(-2/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133086,"parameters":{"a":[1,2],"b":[-1,3],"target":[-3,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"-2/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0087","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=1/-6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(-1/3, 0/1)","explanation_zh":"缺失加數與回代差=(-1/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133087,"parameters":{"a":[1,2],"b":[-1,3],"target":[1,-6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"-1/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0088","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=1/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(0/1, 0/1)","explanation_zh":"缺失加數與回代差=(0/1, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133088,"parameters":{"a":[1,2],"b":[-1,3],"target":[1,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0089","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-3/-6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(1/3, 0/1)","explanation_zh":"缺失加數與回代差=(1/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133089,"parameters":{"a":[1,2],"b":[-1,3],"target":[-3,-6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"1/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0090","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=5/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(2/3, 0/1)","explanation_zh":"缺失加數與回代差=(2/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133090,"parameters":{"a":[1,2],"b":[-1,3],"target":[5,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"2/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0091","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-7/-6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(1/1, 0/1)","explanation_zh":"缺失加數與回代差=(1/1, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133091,"parameters":{"a":[1,2],"b":[-1,3],"target":[-7,-6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0092","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=9/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(4/3, 0/1)","explanation_zh":"缺失加數與回代差=(4/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133092,"parameters":{"a":[1,2],"b":[-1,3],"target":[9,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"4/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0093","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-11/-6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(5/3, 0/1)","explanation_zh":"缺失加數與回代差=(5/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133093,"parameters":{"a":[1,2],"b":[-1,3],"target":[-11,-6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"5/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0094","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=13/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(2/1, 0/1)","explanation_zh":"缺失加數與回代差=(2/1, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133094,"parameters":{"a":[1,2],"b":[-1,3],"target":[13,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0095","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-15/-6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(7/3, 0/1)","explanation_zh":"缺失加數與回代差=(7/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133095,"parameters":{"a":[1,2],"b":[-1,3],"target":[-15,-6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"7/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0096","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=17/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(8/3, 0/1)","explanation_zh":"缺失加數與回代差=(8/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133096,"parameters":{"a":[1,2],"b":[-1,3],"target":[17,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"8/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0097","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-19/-6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(3/1, 0/1)","explanation_zh":"缺失加數與回代差=(3/1, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133097,"parameters":{"a":[1,2],"b":[-1,3],"target":[-19,-6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0098","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=21/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(10/3, 0/1)","explanation_zh":"缺失加數與回代差=(10/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133098,"parameters":{"a":[1,2],"b":[-1,3],"target":[21,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"10/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0099","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=-23/-6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(11/3, 0/1)","explanation_zh":"缺失加數與回代差=(11/3, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133099,"parameters":{"a":[1,2],"b":[-1,3],"target":[-23,-6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"11/3\""}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r03-rational-number-0100","batch_id":"batch-20261009-advance-20261013-r03-rational-number-foundations","language":"zh-Hant","source_mko_ids":["mko-rational-number"],"title_zh":"缺失加數與精確回代","stem_zh":"精確有理數協議；q≠0，x+a+b=target，a=1/2、b=-1/3、target=25/6。依序求 (x, x+a+b−target)。","answer":{"kind":"expression","value":"(4/1, 0/1)","explanation_zh":"缺失加數與回代差=(4/1, 0/1)。從目標扣兩個已知加數再回代；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["缺失加數與精確回代"]},"variant":{"family_id":"qf-density13-rational-target-missing-addend","seed":202610133100,"parameters":{"a":[1,2],"b":[-1,3],"target":[25,6],"operation":"density13_rational_target_missing_addend","domain":"finite_rational_representation_protocol","semantic_template":"qf-density13-rational-target-missing-addend/\"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":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}