{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0001","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(0x+(0))/(1)、R(x)=0x+(0)，q≠0、x=0/1。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(0/1, 0/1, 0/1, 是)","explanation_zh":"原式雙側值與殘差=(0/1, 0/1, 0/1, 是)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143001,"parameters":{"a":0,"b":0,"q":1,"c":0,"d":0,"x":[0,1],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0002","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(2x+(1))/(-3)、R(x)=1x+(0)，q≠0、x=1/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(-2/3, 1/2, -7/6, 否)","explanation_zh":"原式雙側值與殘差=(-2/3, 1/2, -7/6, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143002,"parameters":{"a":2,"b":1,"q":-3,"c":1,"d":0,"x":[1,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"-2/3\",\"1/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0003","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(-10x+(3))/(3)、R(x)=1x+(-2)，q≠0、x=3/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(-4/1, -1/2, -7/2, 否)","explanation_zh":"原式雙側值與殘差=(-4/1, -1/2, -7/2, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143003,"parameters":{"a":-10,"b":3,"q":3,"c":1,"d":-2,"x":[3,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"-4/1\",\"-1/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0004","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(-9x+(4))/(-2)、R(x)=-1x+(-3)，q≠0、x=4/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(7/1, -5/1, 12/1, 否)","explanation_zh":"原式雙側值與殘差=(7/1, -5/1, 12/1, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143004,"parameters":{"a":-9,"b":4,"q":-2,"c":-1,"d":-3,"x":[4,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"7/1\",\"-5/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0005","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(-8x+(5))/(3)、R(x)=0x+(-4)，q≠0、x=5/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(-5/1, -4/1, -1/1, 否)","explanation_zh":"原式雙側值與殘差=(-5/1, -4/1, -1/1, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143005,"parameters":{"a":-8,"b":5,"q":3,"c":0,"d":-4,"x":[5,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"-5/1\",\"-4/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0006","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(-7x+(6))/(-2)、R(x)=1x+(-5)，q≠0、x=6/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(15/2, -2/1, 19/2, 否)","explanation_zh":"原式雙側值與殘差=(15/2, -2/1, 19/2, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143006,"parameters":{"a":-7,"b":6,"q":-2,"c":1,"d":-5,"x":[6,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"15/2\",\"-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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0007","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(-6x+(7))/(3)、R(x)=-1x+(-6)，q≠0、x=7/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(-14/3, -19/2, 29/6, 否)","explanation_zh":"原式雙側值與殘差=(-14/3, -19/2, 29/6, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143007,"parameters":{"a":-6,"b":7,"q":3,"c":-1,"d":-6,"x":[7,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"-14/3\",\"-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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0008","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(-5x+(8))/(-2)、R(x)=0x+(-7)，q≠0、x=8/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(6/1, -7/1, 13/1, 否)","explanation_zh":"原式雙側值與殘差=(6/1, -7/1, 13/1, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143008,"parameters":{"a":-5,"b":8,"q":-2,"c":0,"d":-7,"x":[8,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"6/1\",\"-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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0009","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(-4x+(9))/(3)、R(x)=1x+(-8)，q≠0、x=9/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(-3/1, -7/2, 1/2, 否)","explanation_zh":"原式雙側值與殘差=(-3/1, -7/2, 1/2, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143009,"parameters":{"a":-4,"b":9,"q":3,"c":1,"d":-8,"x":[9,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"-3/1\",\"-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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0010","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(-3x+(10))/(-2)、R(x)=-1x+(-9)，q≠0、x=10/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(5/2, -14/1, 33/2, 否)","explanation_zh":"原式雙側值與殘差=(5/2, -14/1, 33/2, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143010,"parameters":{"a":-3,"b":10,"q":-2,"c":-1,"d":-9,"x":[10,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"5/2\",\"-14/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0011","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(-2x+(11))/(3)、R(x)=0x+(-10)，q≠0、x=11/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(0/1, -10/1, 10/1, 否)","explanation_zh":"原式雙側值與殘差=(0/1, -10/1, 10/1, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143011,"parameters":{"a":-2,"b":11,"q":3,"c":0,"d":-10,"x":[11,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"0/1\",\"-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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0012","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(-1x+(12))/(-2)、R(x)=1x+(-11)，q≠0、x=12/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(-3/1, -5/1, 2/1, 否)","explanation_zh":"原式雙側值與殘差=(-3/1, -5/1, 2/1, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143012,"parameters":{"a":-1,"b":12,"q":-2,"c":1,"d":-11,"x":[12,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"-3/1\",\"-5/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0013","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(0x+(13))/(3)、R(x)=-1x+(-12)，q≠0、x=13/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(13/3, -37/2, 137/6, 否)","explanation_zh":"原式雙側值與殘差=(13/3, -37/2, 137/6, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143013,"parameters":{"a":0,"b":13,"q":3,"c":-1,"d":-12,"x":[13,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"13/3\",\"-37/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0014","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(1x+(14))/(-2)、R(x)=0x+(-13)，q≠0、x=14/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(-21/2, -13/1, 5/2, 否)","explanation_zh":"原式雙側值與殘差=(-21/2, -13/1, 5/2, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143014,"parameters":{"a":1,"b":14,"q":-2,"c":0,"d":-13,"x":[14,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"-21/2\",\"-13/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0015","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(2x+(15))/(3)、R(x)=1x+(-14)，q≠0、x=15/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(10/1, -13/2, 33/2, 否)","explanation_zh":"原式雙側值與殘差=(10/1, -13/2, 33/2, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143015,"parameters":{"a":2,"b":15,"q":3,"c":1,"d":-14,"x":[15,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"10/1\",\"-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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0016","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(3x+(16))/(-2)、R(x)=-1x+(-15)，q≠0、x=16/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(-20/1, -23/1, 3/1, 否)","explanation_zh":"原式雙側值與殘差=(-20/1, -23/1, 3/1, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143016,"parameters":{"a":3,"b":16,"q":-2,"c":-1,"d":-15,"x":[16,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"-20/1\",\"-23/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0017","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(4x+(17))/(3)、R(x)=0x+(-16)，q≠0、x=17/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(17/1, -16/1, 33/1, 否)","explanation_zh":"原式雙側值與殘差=(17/1, -16/1, 33/1, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143017,"parameters":{"a":4,"b":17,"q":3,"c":0,"d":-16,"x":[17,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"17/1\",\"-16/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0018","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(5x+(18))/(-2)、R(x)=1x+(-17)，q≠0、x=18/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(-63/2, -8/1, -47/2, 否)","explanation_zh":"原式雙側值與殘差=(-63/2, -8/1, -47/2, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143018,"parameters":{"a":5,"b":18,"q":-2,"c":1,"d":-17,"x":[18,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"-63/2\",\"-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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0019","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(6x+(19))/(3)、R(x)=-1x+(-18)，q≠0、x=19/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(76/3, -55/2, 317/6, 否)","explanation_zh":"原式雙側值與殘差=(76/3, -55/2, 317/6, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143019,"parameters":{"a":6,"b":19,"q":3,"c":-1,"d":-18,"x":[19,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"76/3\",\"-55/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0020","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(7x+(20))/(-2)、R(x)=0x+(-19)，q≠0、x=20/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(-45/1, -19/1, -26/1, 否)","explanation_zh":"原式雙側值與殘差=(-45/1, -19/1, -26/1, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143020,"parameters":{"a":7,"b":20,"q":-2,"c":0,"d":-19,"x":[20,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"-45/1\",\"-19/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0021","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(8x+(21))/(3)、R(x)=1x+(-20)，q≠0、x=21/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(35/1, -19/2, 89/2, 否)","explanation_zh":"原式雙側值與殘差=(35/1, -19/2, 89/2, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143021,"parameters":{"a":8,"b":21,"q":3,"c":1,"d":-20,"x":[21,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"35/1\",\"-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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0022","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(9x+(22))/(-2)、R(x)=-1x+(-21)，q≠0、x=22/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(-121/2, -32/1, -57/2, 否)","explanation_zh":"原式雙側值與殘差=(-121/2, -32/1, -57/2, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143022,"parameters":{"a":9,"b":22,"q":-2,"c":-1,"d":-21,"x":[22,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"-121/2\",\"-32/1\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0023","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(10x+(23))/(3)、R(x)=0x+(-22)，q≠0、x=23/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(46/1, -22/1, 68/1, 否)","explanation_zh":"原式雙側值與殘差=(46/1, -22/1, 68/1, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143023,"parameters":{"a":10,"b":23,"q":3,"c":0,"d":-22,"x":[23,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"46/1\",\"-22/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0024","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(11x+(24))/(-2)、R(x)=1x+(-23)，q≠0、x=24/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(-78/1, -11/1, -67/1, 否)","explanation_zh":"原式雙側值與殘差=(-78/1, -11/1, -67/1, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143024,"parameters":{"a":11,"b":24,"q":-2,"c":1,"d":-23,"x":[24,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"-78/1\",\"-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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0025","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式分數雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L(x)=(12x+(25))/(3)、R(x)=-1x+(-24)，q≠0、x=25/2。依序求 (L, R, L−R, 原式成立是否)。","answer":{"kind":"expression","value":"(175/3, -73/2, 569/6, 否)","explanation_zh":"原式雙側值與殘差=(175/3, -73/2, 569/6, 否)。分別代入原式的合法兩側，再核精確差是否零。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式分數雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-fraction-side-check","seed":202610143025,"parameters":{"a":12,"b":25,"q":3,"c":-1,"d":-24,"x":[25,2],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_fraction_side_check","semantic_template":"qf-density14-equation-fraction-side-check/[\"175/3\",\"-73/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0026","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式0x+(0)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(全Q, 全Q, 是)","explanation_zh":"乘數前後解集與等價性=(全Q, 全Q, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143026,"parameters":{"a":0,"b":0,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"全Q\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0027","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式0x+(1)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(無解, 全Q, 否)","explanation_zh":"乘數前後解集與等價性=(無解, 全Q, 否)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143027,"parameters":{"a":0,"b":1,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"無解\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0028","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式0x+(1)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(無解, 無解, 是)","explanation_zh":"乘數前後解集與等價性=(無解, 無解, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143028,"parameters":{"a":0,"b":1,"k":2,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"無解\",\"無解\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0029","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(0)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 0/1, 全Q, 否)","explanation_zh":"乘數前後解集與等價性=(單解 0/1, 全Q, 否)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143029,"parameters":{"a":2,"b":0,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 0/1\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0030","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-1)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 1/2, 單解 1/2, 是)","explanation_zh":"乘數前後解集與等價性=(單解 1/2, 單解 1/2, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143030,"parameters":{"a":2,"b":-1,"k":2,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 1/2\",\"單解 1/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0031","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-2)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 1/1, 全Q, 否)","explanation_zh":"乘數前後解集與等價性=(單解 1/1, 全Q, 否)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143031,"parameters":{"a":2,"b":-2,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 1/1\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0032","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-3)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 3/2, 單解 3/2, 是)","explanation_zh":"乘數前後解集與等價性=(單解 3/2, 單解 3/2, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143032,"parameters":{"a":2,"b":-3,"k":2,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 3/2\",\"單解 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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0033","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-4)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 2/1, 全Q, 否)","explanation_zh":"乘數前後解集與等價性=(單解 2/1, 全Q, 否)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143033,"parameters":{"a":2,"b":-4,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 2/1\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0034","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-5)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 5/2, 單解 5/2, 是)","explanation_zh":"乘數前後解集與等價性=(單解 5/2, 單解 5/2, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143034,"parameters":{"a":2,"b":-5,"k":2,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 5/2\",\"單解 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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0035","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-6)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 3/1, 全Q, 否)","explanation_zh":"乘數前後解集與等價性=(單解 3/1, 全Q, 否)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143035,"parameters":{"a":2,"b":-6,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 3/1\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0036","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-7)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 7/2, 單解 7/2, 是)","explanation_zh":"乘數前後解集與等價性=(單解 7/2, 單解 7/2, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143036,"parameters":{"a":2,"b":-7,"k":2,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 7/2\",\"單解 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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0037","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-8)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 4/1, 全Q, 否)","explanation_zh":"乘數前後解集與等價性=(單解 4/1, 全Q, 否)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143037,"parameters":{"a":2,"b":-8,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 4/1\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0038","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-9)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 9/2, 單解 9/2, 是)","explanation_zh":"乘數前後解集與等價性=(單解 9/2, 單解 9/2, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143038,"parameters":{"a":2,"b":-9,"k":2,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 9/2\",\"單解 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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0039","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-10)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 5/1, 全Q, 否)","explanation_zh":"乘數前後解集與等價性=(單解 5/1, 全Q, 否)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143039,"parameters":{"a":2,"b":-10,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 5/1\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0040","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-11)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 11/2, 單解 11/2, 是)","explanation_zh":"乘數前後解集與等價性=(單解 11/2, 單解 11/2, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143040,"parameters":{"a":2,"b":-11,"k":2,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 11/2\",\"單解 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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0041","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-12)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 6/1, 全Q, 否)","explanation_zh":"乘數前後解集與等價性=(單解 6/1, 全Q, 否)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143041,"parameters":{"a":2,"b":-12,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 6/1\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0042","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-13)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 13/2, 單解 13/2, 是)","explanation_zh":"乘數前後解集與等價性=(單解 13/2, 單解 13/2, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143042,"parameters":{"a":2,"b":-13,"k":2,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 13/2\",\"單解 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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0043","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-14)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 7/1, 全Q, 否)","explanation_zh":"乘數前後解集與等價性=(單解 7/1, 全Q, 否)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143043,"parameters":{"a":2,"b":-14,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 7/1\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0044","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-15)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 15/2, 單解 15/2, 是)","explanation_zh":"乘數前後解集與等價性=(單解 15/2, 單解 15/2, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143044,"parameters":{"a":2,"b":-15,"k":2,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 15/2\",\"單解 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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0045","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-16)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 8/1, 全Q, 否)","explanation_zh":"乘數前後解集與等價性=(單解 8/1, 全Q, 否)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143045,"parameters":{"a":2,"b":-16,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 8/1\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0046","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-17)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 17/2, 單解 17/2, 是)","explanation_zh":"乘數前後解集與等價性=(單解 17/2, 單解 17/2, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143046,"parameters":{"a":2,"b":-17,"k":2,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 17/2\",\"單解 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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0047","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-18)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 9/1, 全Q, 否)","explanation_zh":"乘數前後解集與等價性=(單解 9/1, 全Q, 否)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143047,"parameters":{"a":2,"b":-18,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 9/1\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0048","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-19)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 19/2, 單解 19/2, 是)","explanation_zh":"乘數前後解集與等價性=(單解 19/2, 單解 19/2, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143048,"parameters":{"a":2,"b":-19,"k":2,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 19/2\",\"單解 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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0049","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-20)=0，兩側乘k=0。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 10/1, 全Q, 否)","explanation_zh":"乘數前後解集與等價性=(單解 10/1, 全Q, 否)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143049,"parameters":{"a":2,"b":-20,"k":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 10/1\",\"全Q\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0050","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"乘數前後的解集分類","stem_zh":"共同有理論域；依原式定義資格；原式2x+(-21)=0，兩側乘k=2。依序答 (原解集class, 乘後class, 兩class是否相同)，class用全Q／無解／單解 n/d。","answer":{"kind":"expression","value":"(單解 21/2, 單解 21/2, 是)","explanation_zh":"乘數前後解集與等價性=(單解 21/2, 單解 21/2, 是)。分別按一次或退化常數式分類；乘零不能一般保持解集。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘數前後的解集分類"]},"variant":{"family_id":"qf-density14-equation-zero-scale-solution-loss","seed":202610143050,"parameters":{"a":2,"b":-21,"k":2,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_zero_scale_solution_loss","semantic_template":"qf-density14-equation-zero-scale-solution-loss/[\"單解 21/2\",\"單解 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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0051","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(0x+(0))/(x−(0))=0；原式排除x=h，h=0，queries=[0/1, 1/1, -1/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 0/1, 0/1], [0/1, 0/1, 0/1], [否, 是, 是])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 0/1, 0/1], [0/1, 0/1, 0/1], [否, 是, 是])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143051,"parameters":{"a":0,"b":0,"c":0,"h":0,"queries":[[0,1],[1,1],[-1,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"0/1\"],[true,\"0/1\",\"0/1\"],[true,\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0052","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(2x+(-2))/(x−(1))=0；原式排除x=h，h=1，queries=[1/1, 2/1, 0/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 2/1, 2/1], [0/1, 2/1, -2/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 2/1, 2/1], [0/1, 2/1, -2/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143052,"parameters":{"a":2,"b":-2,"c":0,"h":1,"queries":[[1,1],[2,1],[0,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"0/1\"],[true,\"2/1\",\"2/1\"],[true,\"2/1\",\"-2/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0053","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(-10x+(3))/(x−(0))=1；原式排除x=h，h=0，queries=[0/1, 1/2, 2/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, -5/1, -19/2], [3/1, -5/2, -19/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, -5/1, -19/2], [3/1, -5/2, -19/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143053,"parameters":{"a":-10,"b":3,"c":1,"h":0,"queries":[[0,1],[1,2],[2,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"3/1\"],[true,\"-5/1\",\"-5/2\"],[true,\"-19/2\",\"-19/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0054","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(-9x+(4))/(x−(1))=-1；原式排除x=h，h=1，queries=[1/1, 3/2, 3/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, -18/1, -21/2], [-5/1, -9/1, -21/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, -18/1, -21/2], [-5/1, -9/1, -21/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143054,"parameters":{"a":-9,"b":4,"c":-1,"h":1,"queries":[[1,1],[3,2],[3,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"-5/1\"],[true,\"-18/1\",\"-9/1\"],[true,\"-21/2\",\"-21/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0055","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(-8x+(5))/(x−(2))=0；原式排除x=h，h=2，queries=[2/1, 5/2, 4/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, -30/1, -27/2], [-11/1, -15/1, -27/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, -30/1, -27/2], [-11/1, -15/1, -27/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143055,"parameters":{"a":-8,"b":5,"c":0,"h":2,"queries":[[2,1],[5,2],[4,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"-11/1\"],[true,\"-30/1\",\"-15/1\"],[true,\"-27/2\",\"-27/1\"]]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0056","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(-7x+(6))/(x−(-2))=1；原式排除x=h，h=-2，queries=[-2/1, -3/2, 0/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 32/1, 2/1], [20/1, 16/1, 4/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 32/1, 2/1], [20/1, 16/1, 4/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143056,"parameters":{"a":-7,"b":6,"c":1,"h":-2,"queries":[[-2,1],[-3,2],[0,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"20/1\"],[true,\"32/1\",\"16/1\"],[true,\"2/1\",\"4/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0057","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(-6x+(7))/(x−(-1))=-1；原式排除x=h，h=-1，queries=[-1/1, -1/2, 1/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 21/1, 3/2], [13/1, 21/2, 3/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 21/1, 3/2], [13/1, 21/2, 3/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143057,"parameters":{"a":-6,"b":7,"c":-1,"h":-1,"queries":[[-1,1],[-1,2],[1,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"13/1\"],[true,\"21/1\",\"21/2\"],[true,\"3/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0058","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(-5x+(8))/(x−(0))=0；原式排除x=h，h=0，queries=[0/1, 1/2, 2/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 11/1, -1/1], [8/1, 11/2, -2/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 11/1, -1/1], [8/1, 11/2, -2/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143058,"parameters":{"a":-5,"b":8,"c":0,"h":0,"queries":[[0,1],[1,2],[2,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"8/1\"],[true,\"11/1\",\"11/2\"],[true,\"-1/1\",\"-2/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0059","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(-4x+(9))/(x−(1))=1；原式排除x=h，h=1，queries=[1/1, 3/2, 3/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 5/1, -5/2], [5/1, 5/2, -5/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 5/1, -5/2], [5/1, 5/2, -5/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143059,"parameters":{"a":-4,"b":9,"c":1,"h":1,"queries":[[1,1],[3,2],[3,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"5/1\"],[true,\"5/1\",\"5/2\"],[true,\"-5/2\",\"-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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0060","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(-3x+(10))/(x−(2))=-1；原式排除x=h，h=2，queries=[2/1, 5/2, 4/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 6/1, 0/1], [4/1, 3/1, 0/1], [否, 否, 是])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 6/1, 0/1], [4/1, 3/1, 0/1], [否, 否, 是])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143060,"parameters":{"a":-3,"b":10,"c":-1,"h":2,"queries":[[2,1],[5,2],[4,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"4/1\"],[true,\"6/1\",\"3/1\"],[true,\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0061","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(-2x+(11))/(x−(-2))=0；原式排除x=h，h=-2，queries=[-2/1, -3/2, 0/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 28/1, 11/2], [15/1, 14/1, 11/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 28/1, 11/2], [15/1, 14/1, 11/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143061,"parameters":{"a":-2,"b":11,"c":0,"h":-2,"queries":[[-2,1],[-3,2],[0,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"15/1\"],[true,\"28/1\",\"14/1\"],[true,\"11/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0062","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(-1x+(12))/(x−(-1))=1；原式排除x=h，h=-1，queries=[-1/1, -1/2, 1/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 24/1, 9/2], [13/1, 12/1, 9/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 24/1, 9/2], [13/1, 12/1, 9/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143062,"parameters":{"a":-1,"b":12,"c":1,"h":-1,"queries":[[-1,1],[-1,2],[1,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"13/1\"],[true,\"24/1\",\"12/1\"],[true,\"9/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0063","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(0x+(13))/(x−(0))=-1；原式排除x=h，h=0，queries=[0/1, 1/2, 2/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 27/1, 15/2], [13/1, 27/2, 15/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 27/1, 15/2], [13/1, 27/2, 15/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143063,"parameters":{"a":0,"b":13,"c":-1,"h":0,"queries":[[0,1],[1,2],[2,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"13/1\"],[true,\"27/1\",\"27/2\"],[true,\"15/2\",\"15/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0064","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(1x+(14))/(x−(1))=0；原式排除x=h，h=1，queries=[1/1, 3/2, 3/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 31/1, 17/2], [15/1, 31/2, 17/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 31/1, 17/2], [15/1, 31/2, 17/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143064,"parameters":{"a":1,"b":14,"c":0,"h":1,"queries":[[1,1],[3,2],[3,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"15/1\"],[true,\"31/1\",\"31/2\"],[true,\"17/2\",\"17/1\"]]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0065","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(2x+(15))/(x−(2))=1；原式排除x=h，h=2，queries=[2/1, 5/2, 4/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 39/1, 21/2], [19/1, 39/2, 21/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 39/1, 21/2], [19/1, 39/2, 21/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143065,"parameters":{"a":2,"b":15,"c":1,"h":2,"queries":[[2,1],[5,2],[4,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"19/1\"],[true,\"39/1\",\"39/2\"],[true,\"21/2\",\"21/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0066","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(3x+(16))/(x−(-2))=-1；原式排除x=h，h=-2，queries=[-2/1, -3/2, 0/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 24/1, 9/1], [10/1, 12/1, 18/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 24/1, 9/1], [10/1, 12/1, 18/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143066,"parameters":{"a":3,"b":16,"c":-1,"h":-2,"queries":[[-2,1],[-3,2],[0,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"10/1\"],[true,\"24/1\",\"12/1\"],[true,\"9/1\",\"18/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0067","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(4x+(17))/(x−(-1))=0；原式排除x=h，h=-1，queries=[-1/1, -1/2, 1/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 30/1, 21/2], [13/1, 15/1, 21/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 30/1, 21/2], [13/1, 15/1, 21/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143067,"parameters":{"a":4,"b":17,"c":0,"h":-1,"queries":[[-1,1],[-1,2],[1,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"13/1\"],[true,\"30/1\",\"15/1\"],[true,\"21/2\",\"21/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0068","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(5x+(18))/(x−(0))=1；原式排除x=h，h=0，queries=[0/1, 1/2, 2/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 40/1, 13/1], [18/1, 20/1, 26/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 40/1, 13/1], [18/1, 20/1, 26/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143068,"parameters":{"a":5,"b":18,"c":1,"h":0,"queries":[[0,1],[1,2],[2,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"18/1\"],[true,\"40/1\",\"20/1\"],[true,\"13/1\",\"26/1\"]]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0069","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(6x+(19))/(x−(1))=-1；原式排除x=h，h=1，queries=[1/1, 3/2, 3/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 57/1, 39/2], [25/1, 57/2, 39/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 57/1, 39/2], [25/1, 57/2, 39/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143069,"parameters":{"a":6,"b":19,"c":-1,"h":1,"queries":[[1,1],[3,2],[3,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"25/1\"],[true,\"57/1\",\"57/2\"],[true,\"39/2\",\"39/1\"]]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0070","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(7x+(20))/(x−(2))=0；原式排除x=h，h=2，queries=[2/1, 5/2, 4/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 75/1, 24/1], [34/1, 75/2, 48/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 75/1, 24/1], [34/1, 75/2, 48/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143070,"parameters":{"a":7,"b":20,"c":0,"h":2,"queries":[[2,1],[5,2],[4,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"34/1\"],[true,\"75/1\",\"75/2\"],[true,\"24/1\",\"48/1\"]]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0071","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(8x+(21))/(x−(-2))=1；原式排除x=h，h=-2，queries=[-2/1, -3/2, 0/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 17/1, 19/2], [5/1, 17/2, 19/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 17/1, 19/2], [5/1, 17/2, 19/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143071,"parameters":{"a":8,"b":21,"c":1,"h":-2,"queries":[[-2,1],[-3,2],[0,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"5/1\"],[true,\"17/1\",\"17/2\"],[true,\"19/2\",\"19/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0072","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(9x+(22))/(x−(-1))=-1；原式排除x=h，h=-1，queries=[-1/1, -1/2, 1/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 36/1, 33/2], [13/1, 18/1, 33/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 36/1, 33/2], [13/1, 18/1, 33/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143072,"parameters":{"a":9,"b":22,"c":-1,"h":-1,"queries":[[-1,1],[-1,2],[1,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"13/1\"],[true,\"36/1\",\"18/1\"],[true,\"33/2\",\"33/1\"]]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0073","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(10x+(23))/(x−(0))=0；原式排除x=h，h=0，queries=[0/1, 1/2, 2/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 56/1, 43/2], [23/1, 28/1, 43/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 56/1, 43/2], [23/1, 28/1, 43/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143073,"parameters":{"a":10,"b":23,"c":0,"h":0,"queries":[[0,1],[1,2],[2,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"23/1\"],[true,\"56/1\",\"28/1\"],[true,\"43/2\",\"43/1\"]]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0074","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(11x+(24))/(x−(1))=1；原式排除x=h，h=1，queries=[1/1, 3/2, 3/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 80/1, 55/2], [35/1, 40/1, 55/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 80/1, 55/2], [35/1, 40/1, 55/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143074,"parameters":{"a":11,"b":24,"c":1,"h":1,"queries":[[1,1],[3,2],[3,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"35/1\"],[true,\"80/1\",\"40/1\"],[true,\"55/2\",\"55/1\"]]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0075","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"原式pole資格與交乘殘差","stem_zh":"共同有理論域；依原式定義資格；(12x+(25))/(x−(2))=-1；原式排除x=h，h=2，queries=[2/1, 5/2, 4/1]。依序列 (定義資格, 原式殘差或未定義, 交乘殘差, 原式解資格)，四個列表保留原序。","answer":{"kind":"expression","value":"([否, 是, 是], [未定義, 111/1, 75/2], [49/1, 111/2, 75/1], [否, 否, 否])","explanation_zh":"原式定義資格與交乘殘差=([否, 是, 是], [未定義, 111/1, 75/2], [49/1, 111/2, 75/1], [否, 否, 否])。先檢原分母，再代原式；交乘值另列，不替代定義資格；交叉相乘不恢復排除點。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["原式pole資格與交乘殘差"]},"variant":{"family_id":"qf-density14-equation-denominator-exclusion-check","seed":202610143075,"parameters":{"a":12,"b":25,"c":-1,"h":2,"queries":[[2,1],[5,2],[4,1]],"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_denominator_exclusion_check","semantic_template":"qf-density14-equation-denominator-exclusion-check/[[false,\"undefined\",\"49/1\"],[true,\"111/1\",\"111/2\"],[true,\"75/2\",\"75/1\"]]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0076","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=0x+(0)、R=0x+(0)，x=0/1，同加e=1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(0/1, 1/1, 1/1, 0/1, 是)","explanation_zh":"同量平移雙側與殘差=(0/1, 1/1, 1/1, 0/1, 是)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143076,"parameters":{"a":0,"b":0,"c":0,"d":0,"x":[0,1],"e":1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0077","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-1)、R=-1x+(2)，x=1/2，同加e=0。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-2/1, -1/2, 3/2, -2/1, 否)","explanation_zh":"同量平移雙側與殘差=(-2/1, -1/2, 3/2, -2/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143077,"parameters":{"a":1,"b":-1,"c":-1,"d":2,"x":[1,2],"e":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0078","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-2)、R=-1x+(3)，x=2/2，同加e=1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-3/1, 0/1, 3/1, -3/1, 否)","explanation_zh":"同量平移雙側與殘差=(-3/1, 0/1, 3/1, -3/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143078,"parameters":{"a":1,"b":-2,"c":-1,"d":3,"x":[2,2],"e":1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"0/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0079","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-3)、R=-1x+(4)，x=3/2，同加e=-1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-4/1, -5/2, 3/2, -4/1, 否)","explanation_zh":"同量平移雙側與殘差=(-4/1, -5/2, 3/2, -4/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143079,"parameters":{"a":1,"b":-3,"c":-1,"d":4,"x":[3,2],"e":-1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-5/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0080","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-4)、R=-1x+(5)，x=4/2，同加e=0。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-5/1, -2/1, 3/1, -5/1, 否)","explanation_zh":"同量平移雙側與殘差=(-5/1, -2/1, 3/1, -5/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143080,"parameters":{"a":1,"b":-4,"c":-1,"d":5,"x":[4,2],"e":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-2/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0081","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-5)、R=-1x+(6)，x=5/2，同加e=1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-6/1, -3/2, 9/2, -6/1, 否)","explanation_zh":"同量平移雙側與殘差=(-6/1, -3/2, 9/2, -6/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143081,"parameters":{"a":1,"b":-5,"c":-1,"d":6,"x":[5,2],"e":1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-3/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0082","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-6)、R=-1x+(7)，x=6/2，同加e=-1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-7/1, -4/1, 3/1, -7/1, 否)","explanation_zh":"同量平移雙側與殘差=(-7/1, -4/1, 3/1, -7/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143082,"parameters":{"a":1,"b":-6,"c":-1,"d":7,"x":[6,2],"e":-1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-4/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0083","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-7)、R=-1x+(8)，x=7/2，同加e=0。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-8/1, -7/2, 9/2, -8/1, 否)","explanation_zh":"同量平移雙側與殘差=(-8/1, -7/2, 9/2, -8/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143083,"parameters":{"a":1,"b":-7,"c":-1,"d":8,"x":[7,2],"e":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-7/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0084","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-8)、R=-1x+(9)，x=8/2，同加e=1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-9/1, -3/1, 6/1, -9/1, 否)","explanation_zh":"同量平移雙側與殘差=(-9/1, -3/1, 6/1, -9/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143084,"parameters":{"a":1,"b":-8,"c":-1,"d":9,"x":[8,2],"e":1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-3/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0085","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-9)、R=-1x+(10)，x=9/2，同加e=-1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-10/1, -11/2, 9/2, -10/1, 否)","explanation_zh":"同量平移雙側與殘差=(-10/1, -11/2, 9/2, -10/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143085,"parameters":{"a":1,"b":-9,"c":-1,"d":10,"x":[9,2],"e":-1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-11/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0086","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-10)、R=-1x+(11)，x=10/2，同加e=0。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-11/1, -5/1, 6/1, -11/1, 否)","explanation_zh":"同量平移雙側與殘差=(-11/1, -5/1, 6/1, -11/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143086,"parameters":{"a":1,"b":-10,"c":-1,"d":11,"x":[10,2],"e":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-5/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0087","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-11)、R=-1x+(12)，x=11/2，同加e=1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-12/1, -9/2, 15/2, -12/1, 否)","explanation_zh":"同量平移雙側與殘差=(-12/1, -9/2, 15/2, -12/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143087,"parameters":{"a":1,"b":-11,"c":-1,"d":12,"x":[11,2],"e":1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-9/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0088","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-12)、R=-1x+(13)，x=12/2，同加e=-1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-13/1, -7/1, 6/1, -13/1, 否)","explanation_zh":"同量平移雙側與殘差=(-13/1, -7/1, 6/1, -13/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143088,"parameters":{"a":1,"b":-12,"c":-1,"d":13,"x":[12,2],"e":-1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-7/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0089","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-13)、R=-1x+(14)，x=13/2，同加e=0。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-14/1, -13/2, 15/2, -14/1, 否)","explanation_zh":"同量平移雙側與殘差=(-14/1, -13/2, 15/2, -14/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143089,"parameters":{"a":1,"b":-13,"c":-1,"d":14,"x":[13,2],"e":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-13/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0090","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-14)、R=-1x+(15)，x=14/2，同加e=1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-15/1, -6/1, 9/1, -15/1, 否)","explanation_zh":"同量平移雙側與殘差=(-15/1, -6/1, 9/1, -15/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143090,"parameters":{"a":1,"b":-14,"c":-1,"d":15,"x":[14,2],"e":1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-6/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0091","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-15)、R=-1x+(16)，x=15/2，同加e=-1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-16/1, -17/2, 15/2, -16/1, 否)","explanation_zh":"同量平移雙側與殘差=(-16/1, -17/2, 15/2, -16/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143091,"parameters":{"a":1,"b":-15,"c":-1,"d":16,"x":[15,2],"e":-1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-17/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0092","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-16)、R=-1x+(17)，x=16/2，同加e=0。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-17/1, -8/1, 9/1, -17/1, 否)","explanation_zh":"同量平移雙側與殘差=(-17/1, -8/1, 9/1, -17/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143092,"parameters":{"a":1,"b":-16,"c":-1,"d":17,"x":[16,2],"e":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-8/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0093","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-17)、R=-1x+(18)，x=17/2，同加e=1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-18/1, -15/2, 21/2, -18/1, 否)","explanation_zh":"同量平移雙側與殘差=(-18/1, -15/2, 21/2, -18/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143093,"parameters":{"a":1,"b":-17,"c":-1,"d":18,"x":[17,2],"e":1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-15/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0094","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-18)、R=-1x+(19)，x=18/2，同加e=-1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-19/1, -10/1, 9/1, -19/1, 否)","explanation_zh":"同量平移雙側與殘差=(-19/1, -10/1, 9/1, -19/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143094,"parameters":{"a":1,"b":-18,"c":-1,"d":19,"x":[18,2],"e":-1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-10/1\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0095","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-19)、R=-1x+(20)，x=19/2，同加e=0。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-20/1, -19/2, 21/2, -20/1, 否)","explanation_zh":"同量平移雙側與殘差=(-20/1, -19/2, 21/2, -20/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143095,"parameters":{"a":1,"b":-19,"c":-1,"d":20,"x":[19,2],"e":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-19/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0096","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-20)、R=-1x+(21)，x=20/2，同加e=1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-21/1, -9/1, 12/1, -21/1, 否)","explanation_zh":"同量平移雙側與殘差=(-21/1, -9/1, 12/1, -21/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143096,"parameters":{"a":1,"b":-20,"c":-1,"d":21,"x":[20,2],"e":1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-9/1\",\"12/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0097","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-21)、R=-1x+(22)，x=21/2，同加e=-1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-22/1, -23/2, 21/2, -22/1, 否)","explanation_zh":"同量平移雙側與殘差=(-22/1, -23/2, 21/2, -22/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143097,"parameters":{"a":1,"b":-21,"c":-1,"d":22,"x":[21,2],"e":-1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-23/2\",\"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0098","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-22)、R=-1x+(23)，x=22/2，同加e=0。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-23/1, -11/1, 12/1, -23/1, 否)","explanation_zh":"同量平移雙側與殘差=(-23/1, -11/1, 12/1, -23/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143098,"parameters":{"a":1,"b":-22,"c":-1,"d":23,"x":[22,2],"e":0,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-11/1\",\"12/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0099","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-23)、R=-1x+(24)，x=23/2，同加e=1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-24/1, -21/2, 27/2, -24/1, 否)","explanation_zh":"同量平移雙側與殘差=(-24/1, -21/2, 27/2, -24/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143099,"parameters":{"a":1,"b":-23,"c":-1,"d":24,"x":[23,2],"e":1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-21/2\",\"27/2\"]"}},"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":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r03-equation-0100","batch_id":"batch-20261010-advance-20261014-r03-equation-foundations","language":"zh-Hant","source_mko_ids":["mko-equation"],"title_zh":"同量平移的雙側與殘差","stem_zh":"共同有理論域；依原式定義資格；L=1x+(-24)、R=-1x+(25)，x=24/2，同加e=-1。依序列 (原L−R, L+e, R+e, 平移後差, 平移後相等是否)。","answer":{"kind":"expression","value":"(-25/1, -13/1, 12/1, -25/1, 否)","explanation_zh":"同量平移雙側與殘差=(-25/1, -13/1, 12/1, -25/1, 否)。兩側各加同一有理量，再直接相減核對，最後資格是雙側相等。有限雙側核對非一般方程求解證明。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["同量平移的雙側與殘差"]},"variant":{"family_id":"qf-density14-equation-common-shift-check","seed":202610143100,"parameters":{"a":1,"b":-24,"c":-1,"d":25,"x":[24,2],"e":-1,"domain":"finite_Q_original_equation_protocol","operation":"density14_equation_common_shift_check","semantic_template":"qf-density14-equation-common-shift-check/[\"-13/1\",\"12/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}