{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0001","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[0, 0], [3, 4]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(0, 25)","explanation_zh":"兩個L2平方=(0, 25)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103001,"parameters":{"vectors":[[0,0],[3,4]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"0\",\"0\"],[\"3\",\"4\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0002","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[1, -1], [2, 3]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(2, 13)","explanation_zh":"兩個L2平方=(2, 13)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103002,"parameters":{"vectors":[[1,-1],[2,3]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"1\"],[\"2\",\"3\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0003","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[2, -1], [3, 4]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(5, 25)","explanation_zh":"兩個L2平方=(5, 25)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103003,"parameters":{"vectors":[[2,-1],[3,4]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"2\"],[\"3\",\"4\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0004","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[3, -1], [4, 2]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(10, 20)","explanation_zh":"兩個L2平方=(10, 20)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103004,"parameters":{"vectors":[[3,-1],[4,2]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"3\"],[\"2\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0005","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[4, -1], [5, 3]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(17, 34)","explanation_zh":"兩個L2平方=(17, 34)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103005,"parameters":{"vectors":[[4,-1],[5,3]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"4\"],[\"3\",\"5\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0006","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[5, -1], [6, 4]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(26, 52)","explanation_zh":"兩個L2平方=(26, 52)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103006,"parameters":{"vectors":[[5,-1],[6,4]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"5\"],[\"4\",\"6\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0007","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[6, -1], [7, 2]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(37, 53)","explanation_zh":"兩個L2平方=(37, 53)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103007,"parameters":{"vectors":[[6,-1],[7,2]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"6\"],[\"2\",\"7\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0008","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[7, -1], [8, 3]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(50, 73)","explanation_zh":"兩個L2平方=(50, 73)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103008,"parameters":{"vectors":[[7,-1],[8,3]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"7\"],[\"3\",\"8\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0009","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[8, -1], [9, 4]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(65, 97)","explanation_zh":"兩個L2平方=(65, 97)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103009,"parameters":{"vectors":[[8,-1],[9,4]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"8\"],[\"4\",\"9\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0010","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[9, -1], [10, 2]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(82, 104)","explanation_zh":"兩個L2平方=(82, 104)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103010,"parameters":{"vectors":[[9,-1],[10,2]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"9\"],[\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0011","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[10, -1], [11, 3]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(101, 130)","explanation_zh":"兩個L2平方=(101, 130)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103011,"parameters":{"vectors":[[10,-1],[11,3]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"10\"],[\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0012","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[11, -1], [12, 4]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(122, 160)","explanation_zh":"兩個L2平方=(122, 160)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103012,"parameters":{"vectors":[[11,-1],[12,4]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"11\"],[\"12\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0013","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[12, -1], [13, 2]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(145, 173)","explanation_zh":"兩個L2平方=(145, 173)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103013,"parameters":{"vectors":[[12,-1],[13,2]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"12\"],[\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0014","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[13, -1], [14, 3]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(170, 205)","explanation_zh":"兩個L2平方=(170, 205)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103014,"parameters":{"vectors":[[13,-1],[14,3]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"13\"],[\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0015","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[14, -1], [15, 4]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(197, 241)","explanation_zh":"兩個L2平方=(197, 241)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103015,"parameters":{"vectors":[[14,-1],[15,4]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"14\"],[\"15\",\"4\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0016","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[15, -1], [16, 2]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(226, 260)","explanation_zh":"兩個L2平方=(226, 260)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103016,"parameters":{"vectors":[[15,-1],[16,2]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"15\"],[\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0017","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[16, -1], [17, 3]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(257, 298)","explanation_zh":"兩個L2平方=(257, 298)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103017,"parameters":{"vectors":[[16,-1],[17,3]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"16\"],[\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0018","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[17, -1], [18, 4]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(290, 340)","explanation_zh":"兩個L2平方=(290, 340)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103018,"parameters":{"vectors":[[17,-1],[18,4]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"17\"],[\"18\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0019","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[18, -1], [19, 2]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(325, 365)","explanation_zh":"兩個L2平方=(325, 365)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103019,"parameters":{"vectors":[[18,-1],[19,2]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"18\"],[\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0020","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[19, -1], [20, 3]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(362, 409)","explanation_zh":"兩個L2平方=(362, 409)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103020,"parameters":{"vectors":[[19,-1],[20,3]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"19\"],[\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0021","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[20, -1], [21, 4]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(401, 457)","explanation_zh":"兩個L2平方=(401, 457)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103021,"parameters":{"vectors":[[20,-1],[21,4]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"20\"],[\"21\",\"4\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0022","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[21, -1], [22, 2]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(442, 488)","explanation_zh":"兩個L2平方=(442, 488)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103022,"parameters":{"vectors":[[21,-1],[22,2]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"21\"],[\"2\",\"22\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0023","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[22, -1], [23, 3]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(485, 538)","explanation_zh":"兩個L2平方=(485, 538)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103023,"parameters":{"vectors":[[22,-1],[23,3]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"22\"],[\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0024","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[23, -1], [24, 4]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(530, 592)","explanation_zh":"兩個L2平方=(530, 592)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103024,"parameters":{"vectors":[[23,-1],[24,4]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"23\"],[\"24\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0025","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩個L2距離的有序平方","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[24, -1], [25, 2]]。求兩個L2平方，保持向量次序。","answer":{"kind":"expression","value":"(577, 629)","explanation_zh":"兩個L2平方=(577, 629)。各向量兩座標平方相加；重合原點允許零，只給平方值並保留精確整數運算。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩個L2距離的有序平方"]},"variant":{"family_id":"qf-density10-distance-squares-pair","seed":202610103025,"parameters":{"vectors":[[24,-1],[25,2]],"frame":"orthogonal_equal_unit","operation":"density10_distance_squares_pair","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-squares-pair/[[\"1\",\"24\"],[\"2\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0026","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[3, 4]、整數候選c=-5。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(否, 0)","explanation_zh":"候選非負與平方缺陷=(否, 0)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103026,"parameters":{"vector":[3,4],"candidate":-5,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"3\",\"4\"],-5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0027","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[3, 4]、整數候選c=5。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, 0)","explanation_zh":"候選非負與平方缺陷=(是, 0)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103027,"parameters":{"vector":[3,4],"candidate":5,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"3\",\"4\"],5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0028","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[3, 4]、整數候選c=4。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, -9)","explanation_zh":"候選非負與平方缺陷=(是, -9)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103028,"parameters":{"vector":[3,4],"candidate":4,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"3\",\"4\"],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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0029","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[0, 0]、整數候選c=0。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, 0)","explanation_zh":"候選非負與平方缺陷=(是, 0)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103029,"parameters":{"vector":[0,0],"candidate":0,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"0\",\"0\"],0]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0030","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[4, 5]、整數候選c=7。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, 8)","explanation_zh":"候選非負與平方缺陷=(是, 8)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103030,"parameters":{"vector":[4,5],"candidate":7,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"4\",\"5\"],7]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0031","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[5, 6]、整數候選c=-8。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(否, 3)","explanation_zh":"候選非負與平方缺陷=(否, 3)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103031,"parameters":{"vector":[5,6],"candidate":-8,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"5\",\"6\"],-8]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0032","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[6, 7]、整數候選c=9。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, -4)","explanation_zh":"候選非負與平方缺陷=(是, -4)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103032,"parameters":{"vector":[6,7],"candidate":9,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"6\",\"7\"],9]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0033","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[7, 8]、整數候選c=-10。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(否, -13)","explanation_zh":"候選非負與平方缺陷=(否, -13)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103033,"parameters":{"vector":[7,8],"candidate":-10,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"7\",\"8\"],-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0034","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[8, 9]、整數候選c=11。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, -24)","explanation_zh":"候選非負與平方缺陷=(是, -24)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103034,"parameters":{"vector":[8,9],"candidate":11,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"8\",\"9\"],11]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0035","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[9, 10]、整數候選c=-12。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(否, -37)","explanation_zh":"候選非負與平方缺陷=(否, -37)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103035,"parameters":{"vector":[9,10],"candidate":-12,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"10\",\"9\"],-12]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0036","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[10, 11]、整數候選c=13。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, -52)","explanation_zh":"候選非負與平方缺陷=(是, -52)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103036,"parameters":{"vector":[10,11],"candidate":13,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"10\",\"11\"],13]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0037","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[11, 12]、整數候選c=-14。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(否, -69)","explanation_zh":"候選非負與平方缺陷=(否, -69)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103037,"parameters":{"vector":[11,12],"candidate":-14,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"11\",\"12\"],-14]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0038","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[12, 13]、整數候選c=15。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, -88)","explanation_zh":"候選非負與平方缺陷=(是, -88)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103038,"parameters":{"vector":[12,13],"candidate":15,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"12\",\"13\"],15]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0039","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[13, 14]、整數候選c=-16。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(否, -109)","explanation_zh":"候選非負與平方缺陷=(否, -109)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103039,"parameters":{"vector":[13,14],"candidate":-16,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"13\",\"14\"],-16]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0040","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[14, 15]、整數候選c=17。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, -132)","explanation_zh":"候選非負與平方缺陷=(是, -132)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103040,"parameters":{"vector":[14,15],"candidate":17,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"14\",\"15\"],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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0041","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[15, 16]、整數候選c=-18。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(否, -157)","explanation_zh":"候選非負與平方缺陷=(否, -157)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103041,"parameters":{"vector":[15,16],"candidate":-18,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"15\",\"16\"],-18]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0042","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[16, 17]、整數候選c=19。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, -184)","explanation_zh":"候選非負與平方缺陷=(是, -184)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103042,"parameters":{"vector":[16,17],"candidate":19,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"16\",\"17\"],19]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0043","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[17, 18]、整數候選c=-20。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(否, -213)","explanation_zh":"候選非負與平方缺陷=(否, -213)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103043,"parameters":{"vector":[17,18],"candidate":-20,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"17\",\"18\"],-20]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0044","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[18, 19]、整數候選c=21。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, -244)","explanation_zh":"候選非負與平方缺陷=(是, -244)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103044,"parameters":{"vector":[18,19],"candidate":21,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"18\",\"19\"],21]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0045","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[19, 20]、整數候選c=-22。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(否, -277)","explanation_zh":"候選非負與平方缺陷=(否, -277)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103045,"parameters":{"vector":[19,20],"candidate":-22,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"19\",\"20\"],-22]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0046","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[20, 21]、整數候選c=23。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, -312)","explanation_zh":"候選非負與平方缺陷=(是, -312)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103046,"parameters":{"vector":[20,21],"candidate":23,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"20\",\"21\"],23]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0047","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[21, 22]、整數候選c=-24。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(否, -349)","explanation_zh":"候選非負與平方缺陷=(否, -349)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103047,"parameters":{"vector":[21,22],"candidate":-24,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"21\",\"22\"],-24]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0048","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[22, 23]、整數候選c=25。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, -388)","explanation_zh":"候選非負與平方缺陷=(是, -388)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103048,"parameters":{"vector":[22,23],"candidate":25,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"22\",\"23\"],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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0049","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[23, 24]、整數候選c=-26。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(否, -429)","explanation_zh":"候選非負與平方缺陷=(否, -429)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103049,"parameters":{"vector":[23,24],"candidate":-26,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"23\",\"24\"],-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0050","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選符號與有號平方缺陷","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量=[24, 25]、整數候選c=27。依序答 (c是否非負, c²−L2²)；第一項只判非負，不宣稱完整距離資格。","answer":{"kind":"expression","value":"(是, -472)","explanation_zh":"候選非負與平方缺陷=(是, -472)。符號與平方缺陷分開檢查；負候選即使缺陷零仍不是距離，缺陷保留正負號。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選符號與有號平方缺陷"]},"variant":{"family_id":"qf-density10-distance-candidate-defect","seed":202610103050,"parameters":{"vector":[24,25],"candidate":27,"frame":"orthogonal_equal_unit","operation":"density10_distance_candidate_defect","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-candidate-defect/[[\"24\",\"25\"],27]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0051","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[0, 0], [0, 0], [0, 0]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[1, 2, 3]]","explanation_zh":"三距離同距分組=[[1, 2, 3]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103051,"parameters":{"vectors":[[0,0],[0,0],[0,0]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"0\"],[\"0\",\"0\"],[\"0\",\"0\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0052","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[1, 0], [0, -1], [2, 0]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[1, 2], [3]]","explanation_zh":"三距離同距分組=[[1, 2], [3]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103052,"parameters":{"vectors":[[1,0],[0,-1],[2,0]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"1\"],[\"0\",\"1\"],[\"0\",\"2\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0053","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[2, 0], [1, 0], [1, 1]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[2], [3], [1]]","explanation_zh":"三距離同距分組=[[2], [3], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103053,"parameters":{"vectors":[[2,0],[1,0],[1,1]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"2\"],[\"0\",\"1\"],[\"1\",\"1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0054","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[1, 1], [1, 0], [-1, 1]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[2], [1, 3]]","explanation_zh":"三距離同距分組=[[2], [1, 3]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103054,"parameters":{"vectors":[[1,1],[1,0],[-1,1]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"1\",\"1\"],[\"0\",\"1\"],[\"1\",\"1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0055","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[2, 1], [1, 2], [0, 0]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[3], [1, 2]]","explanation_zh":"三距離同距分組=[[3], [1, 2]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103055,"parameters":{"vectors":[[2,1],[1,2],[0,0]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"1\",\"2\"],[\"1\",\"2\"],[\"0\",\"0\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0056","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[6, 0], [5, 1], [5, 0]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation_zh":"三距離同距分組=[[3], [2], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103056,"parameters":{"vectors":[[6,0],[5,1],[5,0]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"6\"],[\"1\",\"5\"],[\"0\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0057","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[7, 1], [6, 0], [0, -6]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation_zh":"三距離同距分組=[[2, 3], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103057,"parameters":{"vectors":[[7,1],[6,0],[0,-6]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"1\",\"7\"],[\"0\",\"6\"],[\"0\",\"6\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0058","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[7, 0], [8, 0], [9, 1]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[1], [2], [3]]","explanation_zh":"三距離同距分組=[[1], [2], [3]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103058,"parameters":{"vectors":[[7,0],[8,0],[9,1]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"7\"],[\"0\",\"8\"],[\"1\",\"9\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0059","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[9, 0], [8, 1], [8, 0]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation_zh":"三距離同距分組=[[3], [2], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103059,"parameters":{"vectors":[[9,0],[8,1],[8,0]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"9\"],[\"1\",\"8\"],[\"0\",\"8\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0060","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[10, 1], [9, 0], [0, -9]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation_zh":"三距離同距分組=[[2, 3], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103060,"parameters":{"vectors":[[10,1],[9,0],[0,-9]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"1\",\"10\"],[\"0\",\"9\"],[\"0\",\"9\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0061","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[10, 0], [11, 0], [12, 1]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[1], [2], [3]]","explanation_zh":"三距離同距分組=[[1], [2], [3]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103061,"parameters":{"vectors":[[10,0],[11,0],[12,1]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"10\"],[\"0\",\"11\"],[\"1\",\"12\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0062","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[12, 0], [11, 1], [11, 0]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation_zh":"三距離同距分組=[[3], [2], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103062,"parameters":{"vectors":[[12,0],[11,1],[11,0]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"12\"],[\"1\",\"11\"],[\"0\",\"11\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0063","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[13, 1], [12, 0], [0, -12]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation_zh":"三距離同距分組=[[2, 3], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103063,"parameters":{"vectors":[[13,1],[12,0],[0,-12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"1\",\"13\"],[\"0\",\"12\"],[\"0\",\"12\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0064","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[13, 0], [14, 0], [15, 1]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[1], [2], [3]]","explanation_zh":"三距離同距分組=[[1], [2], [3]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103064,"parameters":{"vectors":[[13,0],[14,0],[15,1]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"13\"],[\"0\",\"14\"],[\"1\",\"15\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0065","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[15, 0], [14, 1], [14, 0]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation_zh":"三距離同距分組=[[3], [2], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103065,"parameters":{"vectors":[[15,0],[14,1],[14,0]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"15\"],[\"1\",\"14\"],[\"0\",\"14\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0066","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[16, 1], [15, 0], [0, -15]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation_zh":"三距離同距分組=[[2, 3], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103066,"parameters":{"vectors":[[16,1],[15,0],[0,-15]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"1\",\"16\"],[\"0\",\"15\"],[\"0\",\"15\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0067","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[16, 0], [17, 0], [18, 1]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[1], [2], [3]]","explanation_zh":"三距離同距分組=[[1], [2], [3]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103067,"parameters":{"vectors":[[16,0],[17,0],[18,1]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"16\"],[\"0\",\"17\"],[\"1\",\"18\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0068","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[18, 0], [17, 1], [17, 0]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation_zh":"三距離同距分組=[[3], [2], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103068,"parameters":{"vectors":[[18,0],[17,1],[17,0]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"18\"],[\"1\",\"17\"],[\"0\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0069","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[19, 1], [18, 0], [0, -18]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation_zh":"三距離同距分組=[[2, 3], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103069,"parameters":{"vectors":[[19,1],[18,0],[0,-18]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"1\",\"19\"],[\"0\",\"18\"],[\"0\",\"18\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0070","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[19, 0], [20, 0], [21, 1]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[1], [2], [3]]","explanation_zh":"三距離同距分組=[[1], [2], [3]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103070,"parameters":{"vectors":[[19,0],[20,0],[21,1]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"19\"],[\"0\",\"20\"],[\"1\",\"21\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0071","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[21, 0], [20, 1], [20, 0]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation_zh":"三距離同距分組=[[3], [2], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103071,"parameters":{"vectors":[[21,0],[20,1],[20,0]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"21\"],[\"1\",\"20\"],[\"0\",\"20\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0072","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[22, 1], [21, 0], [0, -21]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation_zh":"三距離同距分組=[[2, 3], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103072,"parameters":{"vectors":[[22,1],[21,0],[0,-21]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"1\",\"22\"],[\"0\",\"21\"],[\"0\",\"21\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0073","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[22, 0], [23, 0], [24, 1]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[1], [2], [3]]","explanation_zh":"三距離同距分組=[[1], [2], [3]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103073,"parameters":{"vectors":[[22,0],[23,0],[24,1]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"22\"],[\"0\",\"23\"],[\"1\",\"24\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0074","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[24, 0], [23, 1], [23, 0]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[3], [2], [1]]","explanation_zh":"三距離同距分組=[[3], [2], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103074,"parameters":{"vectors":[[24,0],[23,1],[23,0]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"0\",\"24\"],[\"1\",\"23\"],[\"0\",\"23\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0075","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"三個帶索引距離的含同距排序","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；索引1、2、3依序對應 [[25, 1], [24, 0], [0, -24]]。由小至大列索引同距分組，同組索引遞增。","answer":{"kind":"expression","value":"[[2, 3], [1]]","explanation_zh":"三距離同距分組=[[2, 3], [1]]。比較非負平方值即可排序；相等平方分同組，保留原索引而不以座標字典序代替距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個帶索引距離的含同距排序"]},"variant":{"family_id":"qf-density10-distance-three-order","seed":202610103075,"parameters":{"vectors":[[25,1],[24,0],[0,-24]],"frame":"orthogonal_equal_unit","operation":"density10_distance_three_order","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-three-order/[[\"1\",\"25\"],[\"0\",\"24\"],[\"0\",\"24\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0076","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[0, 0], [0, 0]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"0","explanation_zh":"兩段整數距離和=0。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103076,"parameters":{"vectors":[[0,0],[0,0]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"0\",\"0\"],[\"0\",\"0\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0077","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[3, 4], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"18","explanation_zh":"兩段整數距離和=18。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103077,"parameters":{"vectors":[[3,4],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"3\",\"4\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0078","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[0, 0], [3, 4]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"5","explanation_zh":"兩段整數距離和=5。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103078,"parameters":{"vectors":[[0,0],[3,4]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"0\",\"0\"],[\"3\",\"4\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0079","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[9, 12], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"28","explanation_zh":"兩段整數距離和=28。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103079,"parameters":{"vectors":[[9,12],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"12\",\"9\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0080","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[12, 16], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"33","explanation_zh":"兩段整數距離和=33。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103080,"parameters":{"vectors":[[12,16],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"16\"],[\"12\",\"5\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0081","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[15, 20], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"38","explanation_zh":"兩段整數距離和=38。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103081,"parameters":{"vectors":[[15,20],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"15\",\"20\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0082","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[18, 24], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"43","explanation_zh":"兩段整數距離和=43。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103082,"parameters":{"vectors":[[18,24],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"18\",\"24\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0083","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[21, 28], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"48","explanation_zh":"兩段整數距離和=48。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103083,"parameters":{"vectors":[[21,28],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"21\",\"28\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0084","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[24, 32], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"53","explanation_zh":"兩段整數距離和=53。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103084,"parameters":{"vectors":[[24,32],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"24\",\"32\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0085","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[27, 36], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"58","explanation_zh":"兩段整數距離和=58。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103085,"parameters":{"vectors":[[27,36],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"27\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0086","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[30, 40], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"63","explanation_zh":"兩段整數距離和=63。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103086,"parameters":{"vectors":[[30,40],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"30\",\"40\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0087","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[33, 44], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"68","explanation_zh":"兩段整數距離和=68。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103087,"parameters":{"vectors":[[33,44],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"33\",\"44\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0088","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[36, 48], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"73","explanation_zh":"兩段整數距離和=73。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103088,"parameters":{"vectors":[[36,48],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"36\",\"48\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0089","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[39, 52], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"78","explanation_zh":"兩段整數距離和=78。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103089,"parameters":{"vectors":[[39,52],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"39\",\"52\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0090","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[42, 56], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"83","explanation_zh":"兩段整數距離和=83。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103090,"parameters":{"vectors":[[42,56],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"42\",\"56\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0091","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[45, 60], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"88","explanation_zh":"兩段整數距離和=88。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103091,"parameters":{"vectors":[[45,60],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"45\",\"60\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0092","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[48, 64], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"93","explanation_zh":"兩段整數距離和=93。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103092,"parameters":{"vectors":[[48,64],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"48\",\"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0093","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[51, 68], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"98","explanation_zh":"兩段整數距離和=98。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103093,"parameters":{"vectors":[[51,68],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"51\",\"68\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0094","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[54, 72], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"103","explanation_zh":"兩段整數距離和=103。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103094,"parameters":{"vectors":[[54,72],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"54\",\"72\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0095","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[57, 76], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"108","explanation_zh":"兩段整數距離和=108。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103095,"parameters":{"vectors":[[57,76],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"57\",\"76\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0096","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[60, 80], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"113","explanation_zh":"兩段整數距離和=113。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103096,"parameters":{"vectors":[[60,80],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"60\",\"80\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0097","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[63, 84], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"118","explanation_zh":"兩段整數距離和=118。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103097,"parameters":{"vectors":[[63,84],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"63\",\"84\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0098","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[66, 88], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"123","explanation_zh":"兩段整數距離和=123。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103098,"parameters":{"vectors":[[66,88],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"66\",\"88\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0099","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[69, 92], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"128","explanation_zh":"兩段整數距離和=128。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103099,"parameters":{"vectors":[[69,92],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"69\",\"92\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r03-distance-formula-0100","batch_id":"batch-20261008-advance-20261010-r03-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段完全平方距離的整數和","stem_zh":"同一等單位正交實平面；L2為從共同原點到每個座標向量的歐氏距離；向量依序為 [[72, 96], [5, 12]]，兩平方和均為完全平方。求兩個非負L2距離的整數和。","answer":{"kind":"expression","value":"133","explanation_zh":"兩段整數距離和=133。以精確整數主根分別求距離再相加；兩段和不當作兩端直線距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段完全平方距離的整數和"]},"variant":{"family_id":"qf-density10-distance-integer-sum","seed":202610103100,"parameters":{"vectors":[[72,96],[5,12]],"frame":"orthogonal_equal_unit","operation":"density10_distance_integer_sum","domain":"integer_euclidean_plane","semantic_template":"qf-density10-distance-integer-sum/[[\"12\",\"5\"],[\"72\",\"96\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}