{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0001","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, 0]、r=0，points=[[0, 0], [1, 0], [0, -1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 1, 1], [0, 1, 1], [圓上, 外, 外], [是, 否, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 1, 1], [0, 1, 1], [圓上, 外, 外], [是, 否, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147001,"parameters":{"O":[0,0],"r":0,"points":[[0,0],[1,0],[0,-1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"0\",[\"0\",\"1\",\"1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0002","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, 0]、r=1，points=[[0, 0], [1, 0], [2, 1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 1, 5], [-1, 0, 4], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 1, 5], [-1, 0, 4], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147002,"parameters":{"O":[0,0],"r":1,"points":[[0,0],[1,0],[2,1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"1\",[\"0\",\"1\",\"5\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0003","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, -1]、r=2，points=[[1, -1], [3, -1], [4, 0]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 4, 10], [-4, 0, 6], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 4, 10], [-4, 0, 6], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147003,"parameters":{"O":[1,-1],"r":2,"points":[[1,-1],[3,-1],[4,0]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"4\",[\"0\",\"4\",\"10\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0004","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, -2]、r=3，points=[[-1, -2], [2, -2], [3, -1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 9, 17], [-9, 0, 8], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 9, 17], [-9, 0, 8], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147004,"parameters":{"O":[-1,-2],"r":3,"points":[[-1,-2],[2,-2],[3,-1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"9\",[\"0\",\"9\",\"17\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0005","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, 1]、r=4，points=[[0, 1], [4, 1], [5, 2]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 16, 26], [-16, 0, 10], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 16, 26], [-16, 0, 10], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147005,"parameters":{"O":[0,1],"r":4,"points":[[0,1],[4,1],[5,2]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"16\",[\"0\",\"16\",\"26\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0006","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, 0]、r=5，points=[[1, 0], [6, 0], [7, 1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 25, 37], [-25, 0, 12], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 25, 37], [-25, 0, 12], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147006,"parameters":{"O":[1,0],"r":5,"points":[[1,0],[6,0],[7,1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"25\",[\"0\",\"25\",\"37\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0007","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, -1]、r=6，points=[[-1, -1], [5, -1], [6, 0]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 36, 50], [-36, 0, 14], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 36, 50], [-36, 0, 14], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147007,"parameters":{"O":[-1,-1],"r":6,"points":[[-1,-1],[5,-1],[6,0]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"36\",[\"0\",\"36\",\"50\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0008","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, -2]、r=7，points=[[0, -2], [7, -2], [8, -1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 49, 65], [-49, 0, 16], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 49, 65], [-49, 0, 16], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147008,"parameters":{"O":[0,-2],"r":7,"points":[[0,-2],[7,-2],[8,-1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"49\",[\"0\",\"49\",\"65\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0009","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, 1]、r=8，points=[[1, 1], [9, 1], [10, 2]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 64, 82], [-64, 0, 18], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 64, 82], [-64, 0, 18], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147009,"parameters":{"O":[1,1],"r":8,"points":[[1,1],[9,1],[10,2]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"64\",[\"0\",\"64\",\"82\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0010","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, 0]、r=9，points=[[-1, 0], [8, 0], [9, 1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 81, 101], [-81, 0, 20], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 81, 101], [-81, 0, 20], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147010,"parameters":{"O":[-1,0],"r":9,"points":[[-1,0],[8,0],[9,1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"81\",[\"0\",\"81\",\"101\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0011","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, -1]、r=10，points=[[0, -1], [10, -1], [11, 0]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 100, 122], [-100, 0, 22], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 100, 122], [-100, 0, 22], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147011,"parameters":{"O":[0,-1],"r":10,"points":[[0,-1],[10,-1],[11,0]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"100\",[\"0\",\"100\",\"122\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0012","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, -2]、r=11，points=[[1, -2], [12, -2], [13, -1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 121, 145], [-121, 0, 24], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 121, 145], [-121, 0, 24], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147012,"parameters":{"O":[1,-2],"r":11,"points":[[1,-2],[12,-2],[13,-1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"121\",[\"0\",\"121\",\"145\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0013","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, 1]、r=12，points=[[-1, 1], [11, 1], [12, 2]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 144, 170], [-144, 0, 26], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 144, 170], [-144, 0, 26], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147013,"parameters":{"O":[-1,1],"r":12,"points":[[-1,1],[11,1],[12,2]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"144\",[\"0\",\"144\",\"170\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0014","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, 0]、r=13，points=[[0, 0], [13, 0], [14, 1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 169, 197], [-169, 0, 28], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 169, 197], [-169, 0, 28], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147014,"parameters":{"O":[0,0],"r":13,"points":[[0,0],[13,0],[14,1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"169\",[\"0\",\"169\",\"197\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0015","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, -1]、r=14，points=[[1, -1], [15, -1], [16, 0]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 196, 226], [-196, 0, 30], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 196, 226], [-196, 0, 30], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147015,"parameters":{"O":[1,-1],"r":14,"points":[[1,-1],[15,-1],[16,0]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"196\",[\"0\",\"196\",\"226\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0016","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, -2]、r=15，points=[[-1, -2], [14, -2], [15, -1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 225, 257], [-225, 0, 32], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 225, 257], [-225, 0, 32], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147016,"parameters":{"O":[-1,-2],"r":15,"points":[[-1,-2],[14,-2],[15,-1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"225\",[\"0\",\"225\",\"257\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0017","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, 1]、r=16，points=[[0, 1], [16, 1], [17, 2]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 256, 290], [-256, 0, 34], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 256, 290], [-256, 0, 34], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147017,"parameters":{"O":[0,1],"r":16,"points":[[0,1],[16,1],[17,2]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"256\",[\"0\",\"256\",\"290\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0018","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, 0]、r=17，points=[[1, 0], [18, 0], [19, 1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 289, 325], [-289, 0, 36], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 289, 325], [-289, 0, 36], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147018,"parameters":{"O":[1,0],"r":17,"points":[[1,0],[18,0],[19,1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"289\",[\"0\",\"289\",\"325\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0019","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, -1]、r=18，points=[[-1, -1], [17, -1], [18, 0]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 324, 362], [-324, 0, 38], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 324, 362], [-324, 0, 38], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147019,"parameters":{"O":[-1,-1],"r":18,"points":[[-1,-1],[17,-1],[18,0]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"324\",[\"0\",\"324\",\"362\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0020","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, -2]、r=19，points=[[0, -2], [19, -2], [20, -1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 361, 401], [-361, 0, 40], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 361, 401], [-361, 0, 40], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147020,"parameters":{"O":[0,-2],"r":19,"points":[[0,-2],[19,-2],[20,-1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"361\",[\"0\",\"361\",\"401\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0021","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, 1]、r=20，points=[[1, 1], [21, 1], [22, 2]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 400, 442], [-400, 0, 42], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 400, 442], [-400, 0, 42], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147021,"parameters":{"O":[1,1],"r":20,"points":[[1,1],[21,1],[22,2]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"400\",[\"0\",\"400\",\"442\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0022","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, 0]、r=21，points=[[-1, 0], [20, 0], [21, 1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 441, 485], [-441, 0, 44], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 441, 485], [-441, 0, 44], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147022,"parameters":{"O":[-1,0],"r":21,"points":[[-1,0],[20,0],[21,1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"441\",[\"0\",\"441\",\"485\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0023","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, -1]、r=22，points=[[0, -1], [22, -1], [23, 0]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 484, 530], [-484, 0, 46], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 484, 530], [-484, 0, 46], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147023,"parameters":{"O":[0,-1],"r":22,"points":[[0,-1],[22,-1],[23,0]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"484\",[\"0\",\"484\",\"530\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0024","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, -2]、r=23，points=[[1, -2], [24, -2], [25, -1]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 529, 577], [-529, 0, 48], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 529, 577], [-529, 0, 48], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147024,"parameters":{"O":[1,-2],"r":23,"points":[[1,-2],[24,-2],[25,-1]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"529\",[\"0\",\"529\",\"577\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0025","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"三點殘差與圓邊界資格","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, 1]、r=24，points=[[-1, 1], [23, 1], [24, 2]]。依序列 (三個平方距離D, 三個D−r², 內／圓上／外, 圓邊界資格)。","answer":{"kind":"expression","value":"([0, 576, 626], [-576, 0, 50], [內, 圓上, 外], [否, 是, 否])","explanation_zh":"三點平方距離殘差與圓資格=([0, 576, 626], [-576, 0, 50], [內, 圓上, 外], [否, 是, 否])。用整數座標差平方比較半徑平方；內點不算圓邊界成員。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["三點殘差與圓邊界資格"]},"variant":{"family_id":"qf-density14-circle-offset-residual-triple","seed":202610147025,"parameters":{"O":[-1,1],"r":24,"points":[[-1,1],[23,1],[24,2]],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_offset_residual_triple","semantic_template":"qf-density14-circle-offset-residual-triple/[\"576\",[\"0\",\"576\",\"626\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0026","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, 0]、r=0、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([0, 0], [0, 0], 0, 0)","explanation_zh":"軸向對徑端點平方與直徑=([0, 0], [0, 0], 0, 0)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147026,"parameters":{"O":[0,0],"r":0,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,0],0,\"collapsed\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0027","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, 0]、r=1、axis=y。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([0, 1], [0, -1], 4, 2)","explanation_zh":"軸向對徑端點平方與直徑=([0, 1], [0, -1], 4, 2)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147027,"parameters":{"O":[0,0],"r":1,"axis":"y","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,0],1,\"y\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0028","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, -10]、r=2、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([2, -10], [-2, -10], 16, 4)","explanation_zh":"軸向對徑端點平方與直徑=([2, -10], [-2, -10], 16, 4)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147028,"parameters":{"O":[0,-10],"r":2,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,-10],2,\"x\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0029","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, -9]、r=3、axis=y。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([1, -6], [1, -12], 36, 6)","explanation_zh":"軸向對徑端點平方與直徑=([1, -6], [1, -12], 36, 6)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147029,"parameters":{"O":[1,-9],"r":3,"axis":"y","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[1,-9],3,\"y\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0030","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-2, -8]、r=4、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([2, -8], [-6, -8], 64, 8)","explanation_zh":"軸向對徑端點平方與直徑=([2, -8], [-6, -8], 64, 8)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147030,"parameters":{"O":[-2,-8],"r":4,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-2,-8],4,\"x\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0031","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, -7]、r=5、axis=y。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([-1, -2], [-1, -12], 100, 10)","explanation_zh":"軸向對徑端點平方與直徑=([-1, -2], [-1, -12], 100, 10)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147031,"parameters":{"O":[-1,-7],"r":5,"axis":"y","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-1,-7],5,\"y\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0032","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, -6]、r=6、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([6, -6], [-6, -6], 144, 12)","explanation_zh":"軸向對徑端點平方與直徑=([6, -6], [-6, -6], 144, 12)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147032,"parameters":{"O":[0,-6],"r":6,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,-6],6,\"x\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0033","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, -5]、r=7、axis=y。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([1, 2], [1, -12], 196, 14)","explanation_zh":"軸向對徑端點平方與直徑=([1, 2], [1, -12], 196, 14)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147033,"parameters":{"O":[1,-5],"r":7,"axis":"y","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[1,-5],7,\"y\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0034","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-2, -4]、r=8、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([6, -4], [-10, -4], 256, 16)","explanation_zh":"軸向對徑端點平方與直徑=([6, -4], [-10, -4], 256, 16)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147034,"parameters":{"O":[-2,-4],"r":8,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-2,-4],8,\"x\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0035","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, -3]、r=9、axis=y。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([-1, 6], [-1, -12], 324, 18)","explanation_zh":"軸向對徑端點平方與直徑=([-1, 6], [-1, -12], 324, 18)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147035,"parameters":{"O":[-1,-3],"r":9,"axis":"y","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-1,-3],9,\"y\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0036","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, -2]、r=10、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([10, -2], [-10, -2], 400, 20)","explanation_zh":"軸向對徑端點平方與直徑=([10, -2], [-10, -2], 400, 20)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147036,"parameters":{"O":[0,-2],"r":10,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,-2],10,\"x\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0037","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, -1]、r=11、axis=y。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([1, 10], [1, -12], 484, 22)","explanation_zh":"軸向對徑端點平方與直徑=([1, 10], [1, -12], 484, 22)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147037,"parameters":{"O":[1,-1],"r":11,"axis":"y","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[1,-1],11,\"y\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0038","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-2, 0]、r=12、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([10, 0], [-14, 0], 576, 24)","explanation_zh":"軸向對徑端點平方與直徑=([10, 0], [-14, 0], 576, 24)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147038,"parameters":{"O":[-2,0],"r":12,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-2,0],12,\"x\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0039","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, 1]、r=13、axis=y。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([-1, 14], [-1, -12], 676, 26)","explanation_zh":"軸向對徑端點平方與直徑=([-1, 14], [-1, -12], 676, 26)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147039,"parameters":{"O":[-1,1],"r":13,"axis":"y","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-1,1],13,\"y\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0040","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, 2]、r=14、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([14, 2], [-14, 2], 784, 28)","explanation_zh":"軸向對徑端點平方與直徑=([14, 2], [-14, 2], 784, 28)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147040,"parameters":{"O":[0,2],"r":14,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,2],14,\"x\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0041","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, 3]、r=15、axis=y。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([1, 18], [1, -12], 900, 30)","explanation_zh":"軸向對徑端點平方與直徑=([1, 18], [1, -12], 900, 30)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147041,"parameters":{"O":[1,3],"r":15,"axis":"y","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[1,3],15,\"y\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0042","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-2, 4]、r=16、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([14, 4], [-18, 4], 1024, 32)","explanation_zh":"軸向對徑端點平方與直徑=([14, 4], [-18, 4], 1024, 32)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147042,"parameters":{"O":[-2,4],"r":16,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-2,4],16,\"x\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0043","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, 5]、r=17、axis=y。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([-1, 22], [-1, -12], 1156, 34)","explanation_zh":"軸向對徑端點平方與直徑=([-1, 22], [-1, -12], 1156, 34)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147043,"parameters":{"O":[-1,5],"r":17,"axis":"y","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-1,5],17,\"y\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0044","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, 6]、r=18、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([18, 6], [-18, 6], 1296, 36)","explanation_zh":"軸向對徑端點平方與直徑=([18, 6], [-18, 6], 1296, 36)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147044,"parameters":{"O":[0,6],"r":18,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,6],18,\"x\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0045","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, 7]、r=19、axis=y。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([1, 26], [1, -12], 1444, 38)","explanation_zh":"軸向對徑端點平方與直徑=([1, 26], [1, -12], 1444, 38)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147045,"parameters":{"O":[1,7],"r":19,"axis":"y","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[1,7],19,\"y\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0046","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-2, 8]、r=20、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([18, 8], [-22, 8], 1600, 40)","explanation_zh":"軸向對徑端點平方與直徑=([18, 8], [-22, 8], 1600, 40)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147046,"parameters":{"O":[-2,8],"r":20,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-2,8],20,\"x\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0047","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-1, 9]、r=21、axis=y。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([-1, 30], [-1, -12], 1764, 42)","explanation_zh":"軸向對徑端點平方與直徑=([-1, 30], [-1, -12], 1764, 42)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147047,"parameters":{"O":[-1,9],"r":21,"axis":"y","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-1,9],21,\"y\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0048","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[0, 10]、r=22、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([22, 10], [-22, 10], 1936, 44)","explanation_zh":"軸向對徑端點平方與直徑=([22, 10], [-22, 10], 1936, 44)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147048,"parameters":{"O":[0,10],"r":22,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[0,10],22,\"x\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0049","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[1, 11]、r=23、axis=y。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([1, 34], [1, -12], 2116, 46)","explanation_zh":"軸向對徑端點平方與直徑=([1, 34], [1, -12], 2116, 46)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147049,"parameters":{"O":[1,11],"r":23,"axis":"y","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[1,11],23,\"y\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0050","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"軸向對徑端點與直徑","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；O=[-2, 12]、r=24、axis=x。依序求 (正軸端點, 負軸端點, 端點平方距離, 直徑)。零半徑兩端同一點。","answer":{"kind":"expression","value":"([22, 12], [-26, 12], 2304, 48)","explanation_zh":"軸向對徑端點平方與直徑=([22, 12], [-26, 12], 2304, 48)。在指定軸加減半徑並用精確平方值，不把零半徑當四點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["軸向對徑端點與直徑"]},"variant":{"family_id":"qf-density14-circle-axis-antipodal-pair","seed":202610147050,"parameters":{"O":[-2,12],"r":24,"axis":"x","frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_axis_antipodal_pair","semantic_template":"qf-density14-circle-axis-antipodal-pair/[[-2,12],24,\"x\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0051","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=0、k=5。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(0, 0, 0, 0, 否)","explanation_zh":"半徑尺度平方與直徑=(0, 0, 0, 0, 否)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147051,"parameters":{"r":0,"k":5,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[\"zero_radius\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0052","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=1、k=0。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(0, 1, 0, 0, 否)","explanation_zh":"半徑尺度平方與直徑=(0, 1, 0, 0, 否)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147052,"parameters":{"r":1,"k":0,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[1,0]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0053","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=2、k=3。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(6, 4, 36, 12, 是)","explanation_zh":"半徑尺度平方與直徑=(6, 4, 36, 12, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147053,"parameters":{"r":2,"k":3,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0054","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=3、k=1。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(3, 9, 9, 6, 是)","explanation_zh":"半徑尺度平方與直徑=(3, 9, 9, 6, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147054,"parameters":{"r":3,"k":1,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[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-r07-circle-0055","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=4、k=2。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(8, 16, 64, 16, 是)","explanation_zh":"半徑尺度平方與直徑=(8, 16, 64, 16, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147055,"parameters":{"r":4,"k":2,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[4,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0056","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=5、k=3。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(15, 25, 225, 30, 是)","explanation_zh":"半徑尺度平方與直徑=(15, 25, 225, 30, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147056,"parameters":{"r":5,"k":3,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[5,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0057","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=6、k=1。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(6, 36, 36, 12, 是)","explanation_zh":"半徑尺度平方與直徑=(6, 36, 36, 12, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147057,"parameters":{"r":6,"k":1,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[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-r07-circle-0058","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=7、k=2。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(14, 49, 196, 28, 是)","explanation_zh":"半徑尺度平方與直徑=(14, 49, 196, 28, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147058,"parameters":{"r":7,"k":2,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[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-r07-circle-0059","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=8、k=3。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(24, 64, 576, 48, 是)","explanation_zh":"半徑尺度平方與直徑=(24, 64, 576, 48, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147059,"parameters":{"r":8,"k":3,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[8,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0060","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=9、k=1。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(9, 81, 81, 18, 是)","explanation_zh":"半徑尺度平方與直徑=(9, 81, 81, 18, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147060,"parameters":{"r":9,"k":1,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[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-r07-circle-0061","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=10、k=2。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(20, 100, 400, 40, 是)","explanation_zh":"半徑尺度平方與直徑=(20, 100, 400, 40, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147061,"parameters":{"r":10,"k":2,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[10,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0062","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=11、k=3。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(33, 121, 1089, 66, 是)","explanation_zh":"半徑尺度平方與直徑=(33, 121, 1089, 66, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147062,"parameters":{"r":11,"k":3,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[11,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0063","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=12、k=1。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(12, 144, 144, 24, 是)","explanation_zh":"半徑尺度平方與直徑=(12, 144, 144, 24, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147063,"parameters":{"r":12,"k":1,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[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-r07-circle-0064","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=13、k=2。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(26, 169, 676, 52, 是)","explanation_zh":"半徑尺度平方與直徑=(26, 169, 676, 52, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147064,"parameters":{"r":13,"k":2,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[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-r07-circle-0065","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=14、k=3。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(42, 196, 1764, 84, 是)","explanation_zh":"半徑尺度平方與直徑=(42, 196, 1764, 84, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147065,"parameters":{"r":14,"k":3,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[14,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0066","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=15、k=1。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(15, 225, 225, 30, 是)","explanation_zh":"半徑尺度平方與直徑=(15, 225, 225, 30, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147066,"parameters":{"r":15,"k":1,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[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-r07-circle-0067","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=16、k=2。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(32, 256, 1024, 64, 是)","explanation_zh":"半徑尺度平方與直徑=(32, 256, 1024, 64, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147067,"parameters":{"r":16,"k":2,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[16,2]"}},"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-r07-circle-0068","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=17、k=3。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(51, 289, 2601, 102, 是)","explanation_zh":"半徑尺度平方與直徑=(51, 289, 2601, 102, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147068,"parameters":{"r":17,"k":3,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[17,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0069","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=18、k=1。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(18, 324, 324, 36, 是)","explanation_zh":"半徑尺度平方與直徑=(18, 324, 324, 36, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147069,"parameters":{"r":18,"k":1,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[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-r07-circle-0070","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=19、k=2。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(38, 361, 1444, 76, 是)","explanation_zh":"半徑尺度平方與直徑=(38, 361, 1444, 76, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147070,"parameters":{"r":19,"k":2,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[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-r07-circle-0071","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=20、k=3。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(60, 400, 3600, 120, 是)","explanation_zh":"半徑尺度平方與直徑=(60, 400, 3600, 120, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147071,"parameters":{"r":20,"k":3,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[20,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0072","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=21、k=1。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(21, 441, 441, 42, 是)","explanation_zh":"半徑尺度平方與直徑=(21, 441, 441, 42, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147072,"parameters":{"r":21,"k":1,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[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-r07-circle-0073","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=22、k=2。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(44, 484, 1936, 88, 是)","explanation_zh":"半徑尺度平方與直徑=(44, 484, 1936, 88, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147073,"parameters":{"r":22,"k":2,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[22,2]"}},"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-r07-circle-0074","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=23、k=3。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(69, 529, 4761, 138, 是)","explanation_zh":"半徑尺度平方與直徑=(69, 529, 4761, 138, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147074,"parameters":{"r":23,"k":3,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[23,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0075","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"半徑參數的尺度見證","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=24、k=1。依序求 (kr, r², (kr)², 2kr, kr>0資格)，只問參數乘算。","answer":{"kind":"expression","value":"(24, 576, 576, 48, 是)","explanation_zh":"半徑尺度平方與直徑=(24, 576, 576, 48, 是)。先乘非負半徑參數，再平方及倍增；參數乘算非相似定理證明。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["半徑參數的尺度見證"]},"variant":{"family_id":"qf-density14-circle-radius-scale-witness","seed":202610147075,"parameters":{"r":24,"k":1,"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_radius_scale_witness","semantic_template":"qf-density14-circle-radius-scale-witness/[24,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0076","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=0、水平位移dx=0，候選點(dx,y)的有序ys=[0, 1, -1]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(0, [0, 1, 1], [是, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(0, [0, 1, 1], [是, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147076,"parameters":{"r":0,"dx":0,"ys":[0,1,-1],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"0\",[\"0\",\"1\",\"1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0077","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=5、水平位移dx=3，候選點(dx,y)的有序ys=[4, -4, 0]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(16, [0, 0, -16], [是, 是, 否])","explanation_zh":"水平截線平方餘量與候選=(16, [0, 0, -16], [是, 是, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147077,"parameters":{"r":5,"dx":3,"ys":[4,-4,0],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"16\",[\"16\",\"16\",\"0\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0078","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=1、水平位移dx=2，候選點(dx,y)的有序ys=[0, 1, 2]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(-3, [3, 4, 7], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(-3, [3, 4, 7], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147078,"parameters":{"r":1,"dx":2,"ys":[0,1,2],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"-3\",[\"0\",\"1\",\"4\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-10","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261010-advance-20261014-r07-circle-0079","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=4、水平位移dx=-1，候選點(dx,y)的有序ys=[0, 4, 5]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(15, [-15, 1, 10], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(15, [-15, 1, 10], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147079,"parameters":{"r":4,"dx":-1,"ys":[0,4,5],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"15\",[\"0\",\"16\",\"25\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0080","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=5、水平位移dx=0，候選點(dx,y)的有序ys=[0, 5, 6]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(25, [-25, 0, 11], [否, 是, 否])","explanation_zh":"水平截線平方餘量與候選=(25, [-25, 0, 11], [否, 是, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147080,"parameters":{"r":5,"dx":0,"ys":[0,5,6],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"25\",[\"0\",\"25\",\"36\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0081","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=6、水平位移dx=1，候選點(dx,y)的有序ys=[0, 6, 7]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(35, [-35, 1, 14], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(35, [-35, 1, 14], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147081,"parameters":{"r":6,"dx":1,"ys":[0,6,7],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"35\",[\"0\",\"36\",\"49\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0082","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=7、水平位移dx=-1，候選點(dx,y)的有序ys=[0, 7, 8]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(48, [-48, 1, 16], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(48, [-48, 1, 16], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147082,"parameters":{"r":7,"dx":-1,"ys":[0,7,8],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"48\",[\"0\",\"49\",\"64\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0083","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=8、水平位移dx=0，候選點(dx,y)的有序ys=[0, 8, 9]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(64, [-64, 0, 17], [否, 是, 否])","explanation_zh":"水平截線平方餘量與候選=(64, [-64, 0, 17], [否, 是, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147083,"parameters":{"r":8,"dx":0,"ys":[0,8,9],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"64\",[\"0\",\"64\",\"81\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0084","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=9、水平位移dx=1，候選點(dx,y)的有序ys=[0, 9, 10]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(80, [-80, 1, 20], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(80, [-80, 1, 20], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147084,"parameters":{"r":9,"dx":1,"ys":[0,9,10],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"80\",[\"0\",\"81\",\"100\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0085","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=10、水平位移dx=-1，候選點(dx,y)的有序ys=[0, 10, 11]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(99, [-99, 1, 22], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(99, [-99, 1, 22], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147085,"parameters":{"r":10,"dx":-1,"ys":[0,10,11],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"99\",[\"0\",\"100\",\"121\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0086","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=11、水平位移dx=0，候選點(dx,y)的有序ys=[0, 11, 12]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(121, [-121, 0, 23], [否, 是, 否])","explanation_zh":"水平截線平方餘量與候選=(121, [-121, 0, 23], [否, 是, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147086,"parameters":{"r":11,"dx":0,"ys":[0,11,12],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"121\",[\"0\",\"121\",\"144\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0087","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=12、水平位移dx=1，候選點(dx,y)的有序ys=[0, 12, 13]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(143, [-143, 1, 26], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(143, [-143, 1, 26], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147087,"parameters":{"r":12,"dx":1,"ys":[0,12,13],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"143\",[\"0\",\"144\",\"169\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0088","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=13、水平位移dx=-1，候選點(dx,y)的有序ys=[0, 13, 14]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(168, [-168, 1, 28], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(168, [-168, 1, 28], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147088,"parameters":{"r":13,"dx":-1,"ys":[0,13,14],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"168\",[\"0\",\"169\",\"196\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0089","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=14、水平位移dx=0，候選點(dx,y)的有序ys=[0, 14, 15]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(196, [-196, 0, 29], [否, 是, 否])","explanation_zh":"水平截線平方餘量與候選=(196, [-196, 0, 29], [否, 是, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147089,"parameters":{"r":14,"dx":0,"ys":[0,14,15],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"196\",[\"0\",\"196\",\"225\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0090","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=15、水平位移dx=1，候選點(dx,y)的有序ys=[0, 15, 16]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(224, [-224, 1, 32], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(224, [-224, 1, 32], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147090,"parameters":{"r":15,"dx":1,"ys":[0,15,16],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"224\",[\"0\",\"225\",\"256\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0091","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=16、水平位移dx=-1，候選點(dx,y)的有序ys=[0, 16, 17]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(255, [-255, 1, 34], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(255, [-255, 1, 34], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147091,"parameters":{"r":16,"dx":-1,"ys":[0,16,17],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"255\",[\"0\",\"256\",\"289\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0092","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=17、水平位移dx=0，候選點(dx,y)的有序ys=[0, 17, 18]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(289, [-289, 0, 35], [否, 是, 否])","explanation_zh":"水平截線平方餘量與候選=(289, [-289, 0, 35], [否, 是, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147092,"parameters":{"r":17,"dx":0,"ys":[0,17,18],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"289\",[\"0\",\"289\",\"324\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0093","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=18、水平位移dx=1，候選點(dx,y)的有序ys=[0, 18, 19]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(323, [-323, 1, 38], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(323, [-323, 1, 38], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147093,"parameters":{"r":18,"dx":1,"ys":[0,18,19],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"323\",[\"0\",\"324\",\"361\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0094","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=19、水平位移dx=-1，候選點(dx,y)的有序ys=[0, 19, 20]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(360, [-360, 1, 40], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(360, [-360, 1, 40], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147094,"parameters":{"r":19,"dx":-1,"ys":[0,19,20],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"360\",[\"0\",\"361\",\"400\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0095","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=20、水平位移dx=0，候選點(dx,y)的有序ys=[0, 20, 21]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(400, [-400, 0, 41], [否, 是, 否])","explanation_zh":"水平截線平方餘量與候選=(400, [-400, 0, 41], [否, 是, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147095,"parameters":{"r":20,"dx":0,"ys":[0,20,21],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"400\",[\"0\",\"400\",\"441\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0096","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=21、水平位移dx=1，候選點(dx,y)的有序ys=[0, 21, 22]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(440, [-440, 1, 44], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(440, [-440, 1, 44], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147096,"parameters":{"r":21,"dx":1,"ys":[0,21,22],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"440\",[\"0\",\"441\",\"484\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0097","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=22、水平位移dx=-1，候選點(dx,y)的有序ys=[0, 22, 23]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(483, [-483, 1, 46], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(483, [-483, 1, 46], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147097,"parameters":{"r":22,"dx":-1,"ys":[0,22,23],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"483\",[\"0\",\"484\",\"529\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0098","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=23、水平位移dx=0，候選點(dx,y)的有序ys=[0, 23, 24]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(529, [-529, 0, 47], [否, 是, 否])","explanation_zh":"水平截線平方餘量與候選=(529, [-529, 0, 47], [否, 是, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147098,"parameters":{"r":23,"dx":0,"ys":[0,23,24],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"529\",[\"0\",\"529\",\"576\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0099","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=24、水平位移dx=1，候選點(dx,y)的有序ys=[0, 24, 25]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(575, [-575, 1, 50], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(575, [-575, 1, 50], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147099,"parameters":{"r":24,"dx":1,"ys":[0,24,25],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"575\",[\"0\",\"576\",\"625\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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-r07-circle-0100","batch_id":"batch-20261010-advance-20261014-r07-circle-foundations","language":"zh-Hant","source_mko_ids":["mko-circle"],"title_zh":"固定水平位移的截線候選","stem_zh":"圓邊界不是圓盤；非負整數半徑；歐氏等單位正交L2；r=25、水平位移dx=-1，候選點(dx,y)的有序ys=[0, 25, 26]。依序求 (required=r²−dx², y²−required三殘差, 圓上資格三值)。","answer":{"kind":"expression","value":"(624, [-624, 1, 52], [否, 否, 否])","explanation_zh":"水平截線平方餘量與候選=(624, [-624, 1, 52], [否, 否, 否])。只用平方餘量核候選；負平方餘量不取根，不能補造實數截點。r=0只有中心，有限座標核對非一般圓API形式橋。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["固定水平位移的截線候選"]},"variant":{"family_id":"qf-density14-circle-horizontal-section-candidates","seed":202610147100,"parameters":{"r":25,"dx":-1,"ys":[0,25,26],"frame":"orthogonal_equal_unit","domain":"finite_integer_circle_boundary_protocol","operation":"density14_circle_horizontal_section_candidates","semantic_template":"qf-density14-circle-horizontal-section-candidates/[\"624\",[\"0\",\"625\",\"676\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、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"}}