{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0001","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(1)/(2)、b=(1)/(-2)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(0/1, 否, 未定義, 未定義, 否)","explanation_zh":"分數真和與錯加候選稽核=(0/1, 否, 未定義, 未定義, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151001,"parameters":{"a":[1,2],"b":[1,-2],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"0/1\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0002","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(1)/(2)、b=(-1)/(-2)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(1/1, 否, 未定義, 未定義, 否)","explanation_zh":"分數真和與錯加候選稽核=(1/1, 否, 未定義, 未定義, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151002,"parameters":{"a":[1,2],"b":[-1,-2],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"1/1\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0003","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(0)/(2)、b=(0)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(0/1, 是, 0/1, 0/1, 是)","explanation_zh":"分數真和與錯加候選稽核=(0/1, 是, 0/1, 0/1, 是)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151003,"parameters":{"a":[0,2],"b":[0,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"0/1\",\"0/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0004","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(1)/(2)、b=(1)/(4)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(3/4, 是, 1/3, -5/12, 否)","explanation_zh":"分數真和與錯加候選稽核=(3/4, 是, 1/3, -5/12, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151004,"parameters":{"a":[1,2],"b":[1,4],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"3/4\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0005","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(-8)/(2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(-11/3, 是, -7/5, 34/15, 否)","explanation_zh":"分數真和與錯加候選稽核=(-11/3, 是, -7/5, 34/15, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151005,"parameters":{"a":[-8,2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-11/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0006","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(7)/(-2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(-19/6, 是, 8/1, 67/6, 否)","explanation_zh":"分數真和與錯加候選稽核=(-19/6, 是, 8/1, 67/6, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151006,"parameters":{"a":[7,-2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-19/6\",\"8/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0007","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(-6)/(2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(-8/3, 是, -1/1, 5/3, 否)","explanation_zh":"分數真和與錯加候選稽核=(-8/3, 是, -1/1, 5/3, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151007,"parameters":{"a":[-6,2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-8/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0008","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(5)/(-2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(-13/6, 是, 6/1, 49/6, 否)","explanation_zh":"分數真和與錯加候選稽核=(-13/6, 是, 6/1, 49/6, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151008,"parameters":{"a":[5,-2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-13/6\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0009","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(-4)/(2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(-5/3, 是, -3/5, 16/15, 否)","explanation_zh":"分數真和與錯加候選稽核=(-5/3, 是, -3/5, 16/15, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151009,"parameters":{"a":[-4,2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-5/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0010","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(3)/(-2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(-7/6, 是, 4/1, 31/6, 否)","explanation_zh":"分數真和與錯加候選稽核=(-7/6, 是, 4/1, 31/6, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151010,"parameters":{"a":[3,-2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-7/6\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0011","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(-2)/(2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(-2/3, 是, -1/5, 7/15, 否)","explanation_zh":"分數真和與錯加候選稽核=(-2/3, 是, -1/5, 7/15, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151011,"parameters":{"a":[-2,2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-2/3\",\"-1/5\"]"}},"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0012","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(1)/(-2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(-1/6, 是, 2/1, 13/6, 否)","explanation_zh":"分數真和與錯加候選稽核=(-1/6, 是, 2/1, 13/6, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151012,"parameters":{"a":[1,-2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-1/6\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0013","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(0)/(2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(1/3, 是, 1/5, -2/15, 否)","explanation_zh":"分數真和與錯加候選稽核=(1/3, 是, 1/5, -2/15, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151013,"parameters":{"a":[0,2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"1/3\",\"1/5\"]"}},"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0014","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(-1)/(-2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(5/6, 是, 0/1, -5/6, 否)","explanation_zh":"分數真和與錯加候選稽核=(5/6, 是, 0/1, -5/6, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151014,"parameters":{"a":[-1,-2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"5/6\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0015","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(2)/(2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(4/3, 是, 3/5, -11/15, 否)","explanation_zh":"分數真和與錯加候選稽核=(4/3, 是, 3/5, -11/15, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151015,"parameters":{"a":[2,2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"4/3\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0016","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(-3)/(-2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(11/6, 是, -2/1, -23/6, 否)","explanation_zh":"分數真和與錯加候選稽核=(11/6, 是, -2/1, -23/6, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151016,"parameters":{"a":[-3,-2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"11/6\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0017","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(4)/(2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(7/3, 是, 1/1, -4/3, 否)","explanation_zh":"分數真和與錯加候選稽核=(7/3, 是, 1/1, -4/3, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151017,"parameters":{"a":[4,2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"7/3\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0018","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(-5)/(-2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(17/6, 是, -4/1, -41/6, 否)","explanation_zh":"分數真和與錯加候選稽核=(17/6, 是, -4/1, -41/6, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151018,"parameters":{"a":[-5,-2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"17/6\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0019","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(6)/(2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(10/3, 是, 7/5, -29/15, 否)","explanation_zh":"分數真和與錯加候選稽核=(10/3, 是, 7/5, -29/15, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151019,"parameters":{"a":[6,2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"10/3\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0020","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(-7)/(-2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(23/6, 是, -6/1, -59/6, 否)","explanation_zh":"分數真和與錯加候選稽核=(23/6, 是, -6/1, -59/6, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151020,"parameters":{"a":[-7,-2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"23/6\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0021","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(8)/(2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(13/3, 是, 9/5, -38/15, 否)","explanation_zh":"分數真和與錯加候選稽核=(13/3, 是, 9/5, -38/15, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151021,"parameters":{"a":[8,2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"13/3\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0022","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(-9)/(-2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(29/6, 是, -8/1, -77/6, 否)","explanation_zh":"分數真和與錯加候選稽核=(29/6, 是, -8/1, -77/6, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151022,"parameters":{"a":[-9,-2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"29/6\",\"-8/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0023","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(10)/(2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(16/3, 是, 11/5, -47/15, 否)","explanation_zh":"分數真和與錯加候選稽核=(16/3, 是, 11/5, -47/15, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151023,"parameters":{"a":[10,2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"16/3\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0024","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(-11)/(-2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(35/6, 是, -10/1, -95/6, 否)","explanation_zh":"分數真和與錯加候選稽核=(35/6, 是, -10/1, -95/6, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151024,"parameters":{"a":[-11,-2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"35/6\",\"-10/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0025","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"真分數和與字面錯加","stem_zh":"合法分數表示，原分母非零。a=(12)/(2)、b=(1)/(3)。錯加候選直接以原分子相加除以原分母相加，不先約分。依序答 (真和, 錯加定義資格, 錯加值或未定義, 錯加減真和或未定義, 錯加正確資格)。","answer":{"kind":"expression","value":"(19/3, 是, 13/5, -56/15, 否)","explanation_zh":"分數真和與錯加候選稽核=(19/3, 是, 13/5, -56/15, 否)。真和先通分；錯加另核原分母和，再比較精確差。分子分母分別加不是分數加法。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["真分數和與字面錯加"]},"variant":{"family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151025,"parameters":{"a":[12,2],"b":[1,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_sum_naive_audit","semantic_template":"qf-density15-fraction-sum-naive-audit/[\"19/3\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0026","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(0)/(1), (0)/(1), (0)/(1), (0)/(1)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([0/1, 0/1, 0/1], 0/1, 0/1, 是)","explanation_zh":"相鄰分數差鏈與端點差=([0/1, 0/1, 0/1], 0/1, 0/1, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151026,"parameters":{"values":[[0,1],[0,1],[0,1],[0,1]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"0/1\",\"0/1\",\"0/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0027","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(0)/(1), (1)/(1), (0)/(1), (1)/(1)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([1/1, -1/1, 1/1], 1/1, 1/1, 是)","explanation_zh":"相鄰分數差鏈與端點差=([1/1, -1/1, 1/1], 1/1, 1/1, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151027,"parameters":{"values":[[0,1],[1,1],[0,1],[1,1]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"1/1\",\"-1/1\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0028","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(-10)/(3), (0)/(1), (1)/(2), (3)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([10/3, 1/2, 1/1], 29/6, 29/6, 是)","explanation_zh":"相鄰分數差鏈與端點差=([10/3, 1/2, 1/1], 29/6, 29/6, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151028,"parameters":{"values":[[-10,3],[0,1],[1,2],[3,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"10/3\",\"1/2\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0029","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(9)/(-3), (0)/(1), (1)/(2), (4)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([3/1, 1/2, 3/2], 5/1, 5/1, 是)","explanation_zh":"相鄰分數差鏈與端點差=([3/1, 1/2, 3/2], 5/1, 5/1, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151029,"parameters":{"values":[[9,-3],[0,1],[1,2],[4,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"3/1\",\"1/2\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0030","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(-8)/(3), (0)/(1), (1)/(2), (5)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([8/3, 1/2, 2/1], 31/6, 31/6, 是)","explanation_zh":"相鄰分數差鏈與端點差=([8/3, 1/2, 2/1], 31/6, 31/6, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151030,"parameters":{"values":[[-8,3],[0,1],[1,2],[5,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"8/3\",\"1/2\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0031","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(7)/(-3), (0)/(1), (1)/(2), (6)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([7/3, 1/2, 5/2], 16/3, 16/3, 是)","explanation_zh":"相鄰分數差鏈與端點差=([7/3, 1/2, 5/2], 16/3, 16/3, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151031,"parameters":{"values":[[7,-3],[0,1],[1,2],[6,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"7/3\",\"1/2\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0032","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(-6)/(3), (0)/(1), (1)/(2), (7)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([2/1, 1/2, 3/1], 11/2, 11/2, 是)","explanation_zh":"相鄰分數差鏈與端點差=([2/1, 1/2, 3/1], 11/2, 11/2, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151032,"parameters":{"values":[[-6,3],[0,1],[1,2],[7,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"2/1\",\"1/2\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0033","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(5)/(-3), (0)/(1), (1)/(2), (8)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([5/3, 1/2, 7/2], 17/3, 17/3, 是)","explanation_zh":"相鄰分數差鏈與端點差=([5/3, 1/2, 7/2], 17/3, 17/3, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151033,"parameters":{"values":[[5,-3],[0,1],[1,2],[8,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"5/3\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0034","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(-4)/(3), (0)/(1), (1)/(2), (9)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([4/3, 1/2, 4/1], 35/6, 35/6, 是)","explanation_zh":"相鄰分數差鏈與端點差=([4/3, 1/2, 4/1], 35/6, 35/6, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151034,"parameters":{"values":[[-4,3],[0,1],[1,2],[9,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"4/3\",\"1/2\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0035","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(3)/(-3), (0)/(1), (1)/(2), (10)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([1/1, 1/2, 9/2], 6/1, 6/1, 是)","explanation_zh":"相鄰分數差鏈與端點差=([1/1, 1/2, 9/2], 6/1, 6/1, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151035,"parameters":{"values":[[3,-3],[0,1],[1,2],[10,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"1/1\",\"1/2\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0036","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(-2)/(3), (0)/(1), (1)/(2), (11)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([2/3, 1/2, 5/1], 37/6, 37/6, 是)","explanation_zh":"相鄰分數差鏈與端點差=([2/3, 1/2, 5/1], 37/6, 37/6, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151036,"parameters":{"values":[[-2,3],[0,1],[1,2],[11,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"2/3\",\"1/2\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0037","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(1)/(-3), (0)/(1), (1)/(2), (12)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([1/3, 1/2, 11/2], 19/3, 19/3, 是)","explanation_zh":"相鄰分數差鏈與端點差=([1/3, 1/2, 11/2], 19/3, 19/3, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151037,"parameters":{"values":[[1,-3],[0,1],[1,2],[12,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"1/3\",\"1/2\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0038","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(0)/(3), (0)/(1), (1)/(2), (13)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([0/1, 1/2, 6/1], 13/2, 13/2, 是)","explanation_zh":"相鄰分數差鏈與端點差=([0/1, 1/2, 6/1], 13/2, 13/2, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151038,"parameters":{"values":[[0,3],[0,1],[1,2],[13,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"0/1\",\"1/2\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0039","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(-1)/(-3), (0)/(1), (1)/(2), (14)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([-1/3, 1/2, 13/2], 20/3, 20/3, 是)","explanation_zh":"相鄰分數差鏈與端點差=([-1/3, 1/2, 13/2], 20/3, 20/3, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151039,"parameters":{"values":[[-1,-3],[0,1],[1,2],[14,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-1/3\",\"1/2\",\"13/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0040","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(2)/(3), (0)/(1), (1)/(2), (15)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([-2/3, 1/2, 7/1], 41/6, 41/6, 是)","explanation_zh":"相鄰分數差鏈與端點差=([-2/3, 1/2, 7/1], 41/6, 41/6, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151040,"parameters":{"values":[[2,3],[0,1],[1,2],[15,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-2/3\",\"1/2\",\"7/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0041","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(-3)/(-3), (0)/(1), (1)/(2), (16)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([-1/1, 1/2, 15/2], 7/1, 7/1, 是)","explanation_zh":"相鄰分數差鏈與端點差=([-1/1, 1/2, 15/2], 7/1, 7/1, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151041,"parameters":{"values":[[-3,-3],[0,1],[1,2],[16,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-1/1\",\"1/2\",\"15/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0042","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(4)/(3), (0)/(1), (1)/(2), (17)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([-4/3, 1/2, 8/1], 43/6, 43/6, 是)","explanation_zh":"相鄰分數差鏈與端點差=([-4/3, 1/2, 8/1], 43/6, 43/6, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151042,"parameters":{"values":[[4,3],[0,1],[1,2],[17,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-4/3\",\"1/2\",\"8/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0043","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(-5)/(-3), (0)/(1), (1)/(2), (18)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([-5/3, 1/2, 17/2], 22/3, 22/3, 是)","explanation_zh":"相鄰分數差鏈與端點差=([-5/3, 1/2, 17/2], 22/3, 22/3, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151043,"parameters":{"values":[[-5,-3],[0,1],[1,2],[18,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-5/3\",\"1/2\",\"17/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0044","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(6)/(3), (0)/(1), (1)/(2), (19)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([-2/1, 1/2, 9/1], 15/2, 15/2, 是)","explanation_zh":"相鄰分數差鏈與端點差=([-2/1, 1/2, 9/1], 15/2, 15/2, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151044,"parameters":{"values":[[6,3],[0,1],[1,2],[19,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-2/1\",\"1/2\",\"9/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0045","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(-7)/(-3), (0)/(1), (1)/(2), (20)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([-7/3, 1/2, 19/2], 23/3, 23/3, 是)","explanation_zh":"相鄰分數差鏈與端點差=([-7/3, 1/2, 19/2], 23/3, 23/3, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151045,"parameters":{"values":[[-7,-3],[0,1],[1,2],[20,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-7/3\",\"1/2\",\"19/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0046","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(8)/(3), (0)/(1), (1)/(2), (21)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([-8/3, 1/2, 10/1], 47/6, 47/6, 是)","explanation_zh":"相鄰分數差鏈與端點差=([-8/3, 1/2, 10/1], 47/6, 47/6, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151046,"parameters":{"values":[[8,3],[0,1],[1,2],[21,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-8/3\",\"1/2\",\"10/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0047","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(-9)/(-3), (0)/(1), (1)/(2), (22)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([-3/1, 1/2, 21/2], 8/1, 8/1, 是)","explanation_zh":"相鄰分數差鏈與端點差=([-3/1, 1/2, 21/2], 8/1, 8/1, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151047,"parameters":{"values":[[-9,-3],[0,1],[1,2],[22,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-3/1\",\"1/2\",\"21/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0048","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(10)/(3), (0)/(1), (1)/(2), (23)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([-10/3, 1/2, 11/1], 49/6, 49/6, 是)","explanation_zh":"相鄰分數差鏈與端點差=([-10/3, 1/2, 11/1], 49/6, 49/6, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151048,"parameters":{"values":[[10,3],[0,1],[1,2],[23,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-10/3\",\"1/2\",\"11/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0049","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(-11)/(-3), (0)/(1), (1)/(2), (24)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([-11/3, 1/2, 23/2], 25/3, 25/3, 是)","explanation_zh":"相鄰分數差鏈與端點差=([-11/3, 1/2, 23/2], 25/3, 25/3, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151049,"parameters":{"values":[[-11,-3],[0,1],[1,2],[24,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-11/3\",\"1/2\",\"23/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0050","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"相鄰差鏈與端點差","stem_zh":"合法分數表示，原分母非零。原序四值=[(12)/(3), (0)/(1), (1)/(2), (25)/(2)]。求 (三個後值減前值, 差鏈和, 最後減起始, 後兩者相同資格)，不可排序。","answer":{"kind":"expression","value":"([-4/1, 1/2, 12/1], 17/2, 17/2, 是)","explanation_zh":"相鄰分數差鏈與端點差=([-4/1, 1/2, 12/1], 17/2, 17/2, 是)。依原序相減，再加總差列並單獨核端點差。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["相鄰差鏈與端點差"]},"variant":{"family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151050,"parameters":{"values":[[12,3],[0,1],[1,2],[25,2]],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_adjacent_difference_chain","semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-4/1\",\"1/2\",\"12/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0051","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(0)/(-2)、b=(0)/(3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(0/1, [否, 否], [未定義, 未定義], 未定義)","explanation_zh":"分數乘積與倒數資格=(0/1, [否, 否], [未定義, 未定義], 未定義)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151051,"parameters":{"a":[0,-2],"b":[0,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"0/1\",\"0/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0052","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(0)/(1)、b=(2)/(3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(0/1, [否, 是], [未定義, 3/2], 未定義)","explanation_zh":"分數乘積與倒數資格=(0/1, [否, 是], [未定義, 3/2], 未定義)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151052,"parameters":{"a":[0,1],"b":[2,3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0053","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(-1)/(2)、b=(0)/(-5)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(0/1, [是, 否], [-2/1, 未定義], 未定義)","explanation_zh":"分數乘積與倒數資格=(0/1, [是, 否], [-2/1, 未定義], 未定義)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151053,"parameters":{"a":[-1,2],"b":[0,-5],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-1/2\",\"0/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0054","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(-9)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(2/1, [是, 是], [-1/3, -3/2], 1/2)","explanation_zh":"分數乘積與倒數資格=(2/1, [是, 是], [-1/3, -3/2], 1/2)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151054,"parameters":{"a":[-9,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-3/1\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0055","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(-8)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(16/9, [是, 是], [-3/8, -3/2], 9/16)","explanation_zh":"分數乘積與倒數資格=(16/9, [是, 是], [-3/8, -3/2], 9/16)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151055,"parameters":{"a":[-8,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-8/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0056","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(-7)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(14/9, [是, 是], [-3/7, -3/2], 9/14)","explanation_zh":"分數乘積與倒數資格=(14/9, [是, 是], [-3/7, -3/2], 9/14)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151056,"parameters":{"a":[-7,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-7/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0057","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(-6)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(4/3, [是, 是], [-1/2, -3/2], 3/4)","explanation_zh":"分數乘積與倒數資格=(4/3, [是, 是], [-1/2, -3/2], 3/4)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151057,"parameters":{"a":[-6,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-2/1\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0058","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(-5)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(10/9, [是, 是], [-3/5, -3/2], 9/10)","explanation_zh":"分數乘積與倒數資格=(10/9, [是, 是], [-3/5, -3/2], 9/10)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151058,"parameters":{"a":[-5,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-5/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0059","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(-4)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(8/9, [是, 是], [-3/4, -3/2], 9/8)","explanation_zh":"分數乘積與倒數資格=(8/9, [是, 是], [-3/4, -3/2], 9/8)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151059,"parameters":{"a":[-4,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-4/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0060","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(-3)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(2/3, [是, 是], [-1/1, -3/2], 3/2)","explanation_zh":"分數乘積與倒數資格=(2/3, [是, 是], [-1/1, -3/2], 3/2)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151060,"parameters":{"a":[-3,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-1/1\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0061","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(-2)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(4/9, [是, 是], [-3/2, -3/2], 9/4)","explanation_zh":"分數乘積與倒數資格=(4/9, [是, 是], [-3/2, -3/2], 9/4)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151061,"parameters":{"a":[-2,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-2/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0062","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(-1)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(2/9, [是, 是], [-3/1, -3/2], 9/2)","explanation_zh":"分數乘積與倒數資格=(2/9, [是, 是], [-3/1, -3/2], 9/2)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151062,"parameters":{"a":[-1,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-1/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0063","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(0)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(0/1, [否, 是], [未定義, -3/2], 未定義)","explanation_zh":"分數乘積與倒數資格=(0/1, [否, 是], [未定義, -3/2], 未定義)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151063,"parameters":{"a":[0,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0064","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(1)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(-2/9, [是, 是], [3/1, -3/2], -9/2)","explanation_zh":"分數乘積與倒數資格=(-2/9, [是, 是], [3/1, -3/2], -9/2)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151064,"parameters":{"a":[1,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"1/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0065","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(2)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(-4/9, [是, 是], [3/2, -3/2], -9/4)","explanation_zh":"分數乘積與倒數資格=(-4/9, [是, 是], [3/2, -3/2], -9/4)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151065,"parameters":{"a":[2,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"2/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0066","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(3)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(-2/3, [是, 是], [1/1, -3/2], -3/2)","explanation_zh":"分數乘積與倒數資格=(-2/3, [是, 是], [1/1, -3/2], -3/2)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151066,"parameters":{"a":[3,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"1/1\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0067","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(4)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(-8/9, [是, 是], [3/4, -3/2], -9/8)","explanation_zh":"分數乘積與倒數資格=(-8/9, [是, 是], [3/4, -3/2], -9/8)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151067,"parameters":{"a":[4,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"4/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0068","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(5)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(-10/9, [是, 是], [3/5, -3/2], -9/10)","explanation_zh":"分數乘積與倒數資格=(-10/9, [是, 是], [3/5, -3/2], -9/10)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151068,"parameters":{"a":[5,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"5/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0069","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(6)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(-4/3, [是, 是], [1/2, -3/2], -3/4)","explanation_zh":"分數乘積與倒數資格=(-4/3, [是, 是], [1/2, -3/2], -3/4)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151069,"parameters":{"a":[6,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"2/1\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0070","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(7)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(-14/9, [是, 是], [3/7, -3/2], -9/14)","explanation_zh":"分數乘積與倒數資格=(-14/9, [是, 是], [3/7, -3/2], -9/14)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151070,"parameters":{"a":[7,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"7/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0071","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(8)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(-16/9, [是, 是], [3/8, -3/2], -9/16)","explanation_zh":"分數乘積與倒數資格=(-16/9, [是, 是], [3/8, -3/2], -9/16)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151071,"parameters":{"a":[8,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"8/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0072","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(9)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(-2/1, [是, 是], [1/3, -3/2], -1/2)","explanation_zh":"分數乘積與倒數資格=(-2/1, [是, 是], [1/3, -3/2], -1/2)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151072,"parameters":{"a":[9,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"3/1\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0073","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(10)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(-20/9, [是, 是], [3/10, -3/2], -9/20)","explanation_zh":"分數乘積與倒數資格=(-20/9, [是, 是], [3/10, -3/2], -9/20)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151073,"parameters":{"a":[10,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"10/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0074","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(11)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(-22/9, [是, 是], [3/11, -3/2], -9/22)","explanation_zh":"分數乘積與倒數資格=(-22/9, [是, 是], [3/11, -3/2], -9/22)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151074,"parameters":{"a":[11,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"11/3\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0075","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"分數乘積與倒數門檻","stem_zh":"合法分數表示，原分母非零。a=(12)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。","answer":{"kind":"expression","value":"(-8/3, [是, 是], [1/4, -3/2], -3/8)","explanation_zh":"分數乘積與倒數資格=(-8/3, [是, 是], [1/4, -3/2], -3/8)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["分數乘積與倒數門檻"]},"variant":{"family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151075,"parameters":{"a":[12,3],"b":[2,-3],"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_inverse_pair_guard","semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"4/1\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0076","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=0、d=1，原序縮放因子[s,t]=[0, 2]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(0/1, [[0, 0], [0, 2]], [否, 是], [未定義, 0/1])","explanation_zh":"兩個縮放表示與合法性=(0/1, [[0, 0], [0, 2]], [否, 是], [未定義, 0/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151076,"parameters":{"n":0,"d":1,"s":0,"t":2,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"0/1\",[[\"0\",\"0\"],[\"0\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0077","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=2、d=-3，原序縮放因子[s,t]=[0, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(-2/3, [[0, 0], [-2, 3]], [否, 是], [未定義, -2/3])","explanation_zh":"兩個縮放表示與合法性=(-2/3, [[0, 0], [-2, 3]], [否, 是], [未定義, -2/3])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151077,"parameters":{"n":2,"d":-3,"s":0,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-2/3\",[[\"0\",\"0\"],[\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0078","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=1、d=2，原序縮放因子[s,t]=[1, 0]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(1/2, [[1, 2], [0, 0]], [是, 否], [1/2, 未定義])","explanation_zh":"兩個縮放表示與合法性=(1/2, [[1, 2], [0, 0]], [是, 否], [1/2, 未定義])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151078,"parameters":{"n":1,"d":2,"s":1,"t":0,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"1/2\",[[\"1\",\"2\"],[\"0\",\"0\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0079","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=9007199254740991、d=-7，原序縮放因子[s,t]=[9007199254740991, -2]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(-9007199254740991/7, [[81129638414606663681390495662081, -63050394783186937], [-18014398509481982, 14]], [是, 是], [-9007199254740991/7, -9007199254740991/7])","explanation_zh":"兩個縮放表示與合法性=(-9007199254740991/7, [[81129638414606663681390495662081, -63050394783186937], [-18014398509481982, 14]], [是, 是], [-9007199254740991/7, -9007199254740991/7])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151079,"parameters":{"n":9007199254740991,"d":-7,"s":9007199254740991,"t":-2,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-9007199254740991/7\",[[\"81129638414606663681390495662081\",\"-63050394783186937\"],[\"-18014398509481982\",\"14\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0080","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=-8、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(-4/1, [[-16, 4], [8, -2]], [是, 是], [-4/1, -4/1])","explanation_zh":"兩個縮放表示與合法性=(-4/1, [[-16, 4], [8, -2]], [是, 是], [-4/1, -4/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151080,"parameters":{"n":-8,"d":2,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-4/1\",[[\"-16\",\"4\"],[\"8\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0081","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=-7、d=-3，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(7/3, [[-14, -6], [7, 3]], [是, 是], [7/3, 7/3])","explanation_zh":"兩個縮放表示與合法性=(7/3, [[-14, -6], [7, 3]], [是, 是], [7/3, 7/3])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151081,"parameters":{"n":-7,"d":-3,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"7/3\",[[\"-14\",\"-6\"],[\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0082","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=-6、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(-3/1, [[-12, 4], [6, -2]], [是, 是], [-3/1, -3/1])","explanation_zh":"兩個縮放表示與合法性=(-3/1, [[-12, 4], [6, -2]], [是, 是], [-3/1, -3/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151082,"parameters":{"n":-6,"d":2,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-3/1\",[[\"-12\",\"4\"],[\"6\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0083","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=-5、d=-3，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(5/3, [[-10, -6], [5, 3]], [是, 是], [5/3, 5/3])","explanation_zh":"兩個縮放表示與合法性=(5/3, [[-10, -6], [5, 3]], [是, 是], [5/3, 5/3])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":1,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151083,"parameters":{"n":-5,"d":-3,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"5/3\",[[\"-10\",\"-6\"],[\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0084","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=-4、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(-2/1, [[-8, 4], [4, -2]], [是, 是], [-2/1, -2/1])","explanation_zh":"兩個縮放表示與合法性=(-2/1, [[-8, 4], [4, -2]], [是, 是], [-2/1, -2/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151084,"parameters":{"n":-4,"d":2,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-2/1\",[[\"-8\",\"4\"],[\"4\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0085","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=-3、d=-3，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(1/1, [[-6, -6], [3, 3]], [是, 是], [1/1, 1/1])","explanation_zh":"兩個縮放表示與合法性=(1/1, [[-6, -6], [3, 3]], [是, 是], [1/1, 1/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151085,"parameters":{"n":-3,"d":-3,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"1/1\",[[\"-6\",\"-6\"],[\"3\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0086","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=-2、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(-1/1, [[-4, 4], [2, -2]], [是, 是], [-1/1, -1/1])","explanation_zh":"兩個縮放表示與合法性=(-1/1, [[-4, 4], [2, -2]], [是, 是], [-1/1, -1/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151086,"parameters":{"n":-2,"d":2,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-1/1\",[[\"-4\",\"4\"],[\"2\",\"-2\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0087","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=-1、d=-3，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(1/3, [[-2, -6], [1, 3]], [是, 是], [1/3, 1/3])","explanation_zh":"兩個縮放表示與合法性=(1/3, [[-2, -6], [1, 3]], [是, 是], [1/3, 1/3])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151087,"parameters":{"n":-1,"d":-3,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"1/3\",[[\"-2\",\"-6\"],[\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0088","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=0、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(0/1, [[0, 4], [0, -2]], [是, 是], [0/1, 0/1])","explanation_zh":"兩個縮放表示與合法性=(0/1, [[0, 4], [0, -2]], [是, 是], [0/1, 0/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151088,"parameters":{"n":0,"d":2,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"0/1\",[[\"0\",\"4\"],[\"0\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0089","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=1、d=-3，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(-1/3, [[2, -6], [-1, 3]], [是, 是], [-1/3, -1/3])","explanation_zh":"兩個縮放表示與合法性=(-1/3, [[2, -6], [-1, 3]], [是, 是], [-1/3, -1/3])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151089,"parameters":{"n":1,"d":-3,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-1/3\",[[\"2\",\"-6\"],[\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0090","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=2、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(1/1, [[4, 4], [-2, -2]], [是, 是], [1/1, 1/1])","explanation_zh":"兩個縮放表示與合法性=(1/1, [[4, 4], [-2, -2]], [是, 是], [1/1, 1/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151090,"parameters":{"n":2,"d":2,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"1/1\",[[\"4\",\"4\"],[\"-2\",\"-2\"]]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0091","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=3、d=-3，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(-1/1, [[6, -6], [-3, 3]], [是, 是], [-1/1, -1/1])","explanation_zh":"兩個縮放表示與合法性=(-1/1, [[6, -6], [-3, 3]], [是, 是], [-1/1, -1/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151091,"parameters":{"n":3,"d":-3,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-1/1\",[[\"6\",\"-6\"],[\"-3\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0092","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=4、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(2/1, [[8, 4], [-4, -2]], [是, 是], [2/1, 2/1])","explanation_zh":"兩個縮放表示與合法性=(2/1, [[8, 4], [-4, -2]], [是, 是], [2/1, 2/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":2,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151092,"parameters":{"n":4,"d":2,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"2/1\",[[\"8\",\"4\"],[\"-4\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0093","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=5、d=-3，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(-5/3, [[10, -6], [-5, 3]], [是, 是], [-5/3, -5/3])","explanation_zh":"兩個縮放表示與合法性=(-5/3, [[10, -6], [-5, 3]], [是, 是], [-5/3, -5/3])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151093,"parameters":{"n":5,"d":-3,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-5/3\",[[\"10\",\"-6\"],[\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0094","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=6、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(3/1, [[12, 4], [-6, -2]], [是, 是], [3/1, 3/1])","explanation_zh":"兩個縮放表示與合法性=(3/1, [[12, 4], [-6, -2]], [是, 是], [3/1, 3/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151094,"parameters":{"n":6,"d":2,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"3/1\",[[\"12\",\"4\"],[\"-6\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0095","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=7、d=-3，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(-7/3, [[14, -6], [-7, 3]], [是, 是], [-7/3, -7/3])","explanation_zh":"兩個縮放表示與合法性=(-7/3, [[14, -6], [-7, 3]], [是, 是], [-7/3, -7/3])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151095,"parameters":{"n":7,"d":-3,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-7/3\",[[\"14\",\"-6\"],[\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0096","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=8、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(4/1, [[16, 4], [-8, -2]], [是, 是], [4/1, 4/1])","explanation_zh":"兩個縮放表示與合法性=(4/1, [[16, 4], [-8, -2]], [是, 是], [4/1, 4/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151096,"parameters":{"n":8,"d":2,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"4/1\",[[\"16\",\"4\"],[\"-8\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0097","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=9、d=-3，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(-3/1, [[18, -6], [-9, 3]], [是, 是], [-3/1, -3/1])","explanation_zh":"兩個縮放表示與合法性=(-3/1, [[18, -6], [-9, 3]], [是, 是], [-3/1, -3/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151097,"parameters":{"n":9,"d":-3,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-3/1\",[[\"18\",\"-6\"],[\"-9\",\"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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0098","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=10、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(5/1, [[20, 4], [-10, -2]], [是, 是], [5/1, 5/1])","explanation_zh":"兩個縮放表示與合法性=(5/1, [[20, 4], [-10, -2]], [是, 是], [5/1, 5/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151098,"parameters":{"n":10,"d":2,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"5/1\",[[\"20\",\"4\"],[\"-10\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0099","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=11、d=-3，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(-11/3, [[22, -6], [-11, 3]], [是, 是], [-11/3, -11/3])","explanation_zh":"兩個縮放表示與合法性=(-11/3, [[22, -6], [-11, 3]], [是, 是], [-11/3, -11/3])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151099,"parameters":{"n":11,"d":-3,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-11/3\",[[\"22\",\"-6\"],[\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261011-advance-20261015-r01-fraction-0100","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","language":"zh-Hant","source_mko_ids":["mko-fraction"],"title_zh":"兩種字面縮放表示","stem_zh":"合法分數表示，原分母非零。原n/d中n=12、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。","answer":{"kind":"expression","value":"(6/1, [[24, 4], [-12, -2]], [是, 是], [6/1, 6/1])","explanation_zh":"兩個縮放表示與合法性=(6/1, [[24, 4], [-12, -2]], [是, 是], [6/1, 6/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。表示與數值分開，有限分數核對非商集良定性證明。"},"difficulty":{"band":"primary_foundation","level":3,"skills":["兩種字面縮放表示"]},"variant":{"family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151100,"parameters":{"n":12,"d":2,"s":2,"t":-1,"domain":"finite_fraction_representation_protocol","operation":"density15_fraction_two_scaled_representations","semantic_template":"qf-density15-fraction-two-scaled-representations/[\"6/1\",[[\"24\",\"4\"],[\"-12\",\"-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":"2026-10-11","contract_version":"ocme-question-factory-v0.1"}}