{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0001","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-12, 5)、Q=(-12, 5)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"0","explanation_zh":"平方距離=0。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064001,"parameters":{"P":[-12,5],"Q":[-12,5],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"0\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0002","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-11, 4)、Q=(-12, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"5","explanation_zh":"平方距離=5。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064002,"parameters":{"P":[-11,4],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"1\",\"2\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0003","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-10, 3)、Q=(-8, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"13","explanation_zh":"平方距離=13。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064003,"parameters":{"P":[-10,3],"Q":[-8,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"2\",\"3\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0004","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-9, 2)、Q=(-12, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"25","explanation_zh":"平方距離=25。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064004,"parameters":{"P":[-9,2],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"3\",\"4\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0005","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-8, 1)、Q=(-8, 5)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"16","explanation_zh":"平方距離=16。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064005,"parameters":{"P":[-8,1],"Q":[-8,5],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"4\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0006","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-7, 0)、Q=(-12, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"61","explanation_zh":"平方距離=61。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064006,"parameters":{"P":[-7,0],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"5\",\"6\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0007","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-6, -1)、Q=(0, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"85","explanation_zh":"平方距離=85。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064007,"parameters":{"P":[-6,-1],"Q":[0,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"6\",\"7\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0008","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-5, -2)、Q=(-12, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"113","explanation_zh":"平方距離=113。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064008,"parameters":{"P":[-5,-2],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"7\",\"8\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0009","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-4, -3)、Q=(-4, 5)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"64","explanation_zh":"平方距離=64。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064009,"parameters":{"P":[-4,-3],"Q":[-4,5],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"8\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0010","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-3, -4)、Q=(-12, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"181","explanation_zh":"平方距離=181。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064010,"parameters":{"P":[-3,-4],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"9\",\"10\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0011","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-2, -5)、Q=(8, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"221","explanation_zh":"平方距離=221。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064011,"parameters":{"P":[-2,-5],"Q":[8,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"10\",\"11\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0012","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-1, -6)、Q=(-12, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"265","explanation_zh":"平方距離=265。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064012,"parameters":{"P":[-1,-6],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"11\",\"12\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0013","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(0, -7)、Q=(0, 5)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"144","explanation_zh":"平方距離=144。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064013,"parameters":{"P":[0,-7],"Q":[0,5],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"12\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0014","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(1, -8)、Q=(-12, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"365","explanation_zh":"平方距離=365。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064014,"parameters":{"P":[1,-8],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"13\",\"14\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0015","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(2, -9)、Q=(16, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"421","explanation_zh":"平方距離=421。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064015,"parameters":{"P":[2,-9],"Q":[16,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"14\",\"15\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0016","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(3, -10)、Q=(-12, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"481","explanation_zh":"平方距離=481。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064016,"parameters":{"P":[3,-10],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"15\",\"16\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0017","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(4, -11)、Q=(4, 5)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"256","explanation_zh":"平方距離=256。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064017,"parameters":{"P":[4,-11],"Q":[4,5],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"16\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0018","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(5, -12)、Q=(-12, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"613","explanation_zh":"平方距離=613。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064018,"parameters":{"P":[5,-12],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"17\",\"18\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0019","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(6, -13)、Q=(24, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"685","explanation_zh":"平方距離=685。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064019,"parameters":{"P":[6,-13],"Q":[24,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"18\",\"19\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0020","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(7, -14)、Q=(-12, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"761","explanation_zh":"平方距離=761。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064020,"parameters":{"P":[7,-14],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"19\",\"20\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0021","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(8, -15)、Q=(8, 5)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"400","explanation_zh":"平方距離=400。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064021,"parameters":{"P":[8,-15],"Q":[8,5],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"20\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0022","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(9, -16)、Q=(-12, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"925","explanation_zh":"平方距離=925。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064022,"parameters":{"P":[9,-16],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"21\",\"22\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0023","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(10, -17)、Q=(32, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"1013","explanation_zh":"平方距離=1013。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064023,"parameters":{"P":[10,-17],"Q":[32,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"22\",\"23\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0024","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(11, -18)、Q=(-12, 6)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"1105","explanation_zh":"平方距離=1105。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064024,"parameters":{"P":[11,-18],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"23\",\"24\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0025","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"歐氏距離的平方","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(12, -19)、Q=(12, 5)。求精確平方距離 d(P,Q)²，不對平方和開浮點根。","answer":{"kind":"short_text","value":"576","explanation_zh":"平方距離=576。在指定歐氏 L2 框架，兩個座標差各自平方後相加；距離平方不是距離本身。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["歐氏距離的平方"]},"variant":{"family_id":"qf-distance-squared","seed":202610064025,"parameters":{"P":[12,-19],"Q":[12,5],"frame":"euclidean_orthonormal_equal_units","operation":"squared_distance","semantic_template":"qf-distance-squared/{\"inputs\":[[\"0\",\"24\"]],\"operation\":\"squared_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0026","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-12, 5)、Q=(-12, 5)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"0","explanation_zh":"距離=0；平方距離=0。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064026,"parameters":{"P":[-12,5],"Q":[-12,5],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"0\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0027","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-11, 4)、Q=(-19, 10)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"10","explanation_zh":"距離=10；平方距離=100。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064027,"parameters":{"P":[-11,4],"Q":[-19,10],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"6\",\"8\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0028","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-10, 3)、Q=(-10, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"3","explanation_zh":"距離=3；平方距離=9。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064028,"parameters":{"P":[-10,3],"Q":[-10,6],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"3\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0029","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-9, 2)、Q=(-33, 12)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"26","explanation_zh":"距離=26；平方距離=676。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064029,"parameters":{"P":[-9,2],"Q":[-33,12],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"10\",\"24\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0030","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-8, 1)、Q=(-8, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"5","explanation_zh":"距離=5；平方距離=25。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064030,"parameters":{"P":[-8,1],"Q":[-8,6],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"5\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0031","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-7, 0)、Q=(-55, 14)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"50","explanation_zh":"距離=50；平方距離=2500。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064031,"parameters":{"P":[-7,0],"Q":[-55,14],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"14\",\"48\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0032","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-6, -1)、Q=(-6, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"7","explanation_zh":"距離=7；平方距離=49。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064032,"parameters":{"P":[-6,-1],"Q":[-6,6],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"7\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0033","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-5, -2)、Q=(-85, 16)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"82","explanation_zh":"距離=82；平方距離=6724。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064033,"parameters":{"P":[-5,-2],"Q":[-85,16],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"18\",\"80\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0034","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-4, -3)、Q=(-4, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"9","explanation_zh":"距離=9；平方距離=81。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064034,"parameters":{"P":[-4,-3],"Q":[-4,6],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"9\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0035","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-3, -4)、Q=(-123, 18)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"122","explanation_zh":"距離=122；平方距離=14884。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064035,"parameters":{"P":[-3,-4],"Q":[-123,18],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"22\",\"120\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0036","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-2, -5)、Q=(-2, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"11","explanation_zh":"距離=11；平方距離=121。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064036,"parameters":{"P":[-2,-5],"Q":[-2,6],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"11\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0037","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-1, -6)、Q=(-169, 20)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"170","explanation_zh":"距離=170；平方距離=28900。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064037,"parameters":{"P":[-1,-6],"Q":[-169,20],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"26\",\"168\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0038","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(0, -7)、Q=(0, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"13","explanation_zh":"距離=13；平方距離=169。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064038,"parameters":{"P":[0,-7],"Q":[0,6],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"13\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0039","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(1, -8)、Q=(-223, 22)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"226","explanation_zh":"距離=226；平方距離=51076。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064039,"parameters":{"P":[1,-8],"Q":[-223,22],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"30\",\"224\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0040","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(2, -9)、Q=(2, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"15","explanation_zh":"距離=15；平方距離=225。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064040,"parameters":{"P":[2,-9],"Q":[2,6],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"15\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0041","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(3, -10)、Q=(-285, 24)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"290","explanation_zh":"距離=290；平方距離=84100。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064041,"parameters":{"P":[3,-10],"Q":[-285,24],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"34\",\"288\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0042","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(4, -11)、Q=(4, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"17","explanation_zh":"距離=17；平方距離=289。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064042,"parameters":{"P":[4,-11],"Q":[4,6],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"17\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0043","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(5, -12)、Q=(-355, 26)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"362","explanation_zh":"距離=362；平方距離=131044。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064043,"parameters":{"P":[5,-12],"Q":[-355,26],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"38\",\"360\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0044","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(6, -13)、Q=(6, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"19","explanation_zh":"距離=19；平方距離=361。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064044,"parameters":{"P":[6,-13],"Q":[6,6],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"19\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0045","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(7, -14)、Q=(-433, 28)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"442","explanation_zh":"距離=442；平方距離=195364。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064045,"parameters":{"P":[7,-14],"Q":[-433,28],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"42\",\"440\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0046","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(8, -15)、Q=(8, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"21","explanation_zh":"距離=21；平方距離=441。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064046,"parameters":{"P":[8,-15],"Q":[8,6],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"21\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0047","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(9, -16)、Q=(-519, 30)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"530","explanation_zh":"距離=530；平方距離=280900。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064047,"parameters":{"P":[9,-16],"Q":[-519,30],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"46\",\"528\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0048","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(10, -17)、Q=(10, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"23","explanation_zh":"距離=23；平方距離=529。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064048,"parameters":{"P":[10,-17],"Q":[10,6],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"23\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0049","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(11, -18)、Q=(-613, 32)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"626","explanation_zh":"距離=626；平方距離=391876。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064049,"parameters":{"P":[11,-18],"Q":[-613,32],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"50\",\"624\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0050","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"完全平方切片的非負距離","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(12, -19)、Q=(12, 6)。已明示座標差平方和為完全平方，求非負整數距離 d(P,Q)。","answer":{"kind":"short_text","value":"25","explanation_zh":"距離=25；平方距離=625。產生端以整數二分搜尋確認根的平方等於平方和，採非負主根；重合點距離零合法。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["完全平方切片的非負距離"]},"variant":{"family_id":"qf-distance-perfect-square","seed":202610064050,"parameters":{"P":[12,-19],"Q":[12,6],"frame":"euclidean_orthonormal_equal_units","operation":"perfect_square_distance","semantic_template":"qf-distance-perfect-square/{\"inputs\":[[\"0\",\"25\"]],\"operation\":\"perfect_square_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0051","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-12, 5)、Q=(-12, 5)。整數候選 d=0 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=0；平方距離=0。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064051,"parameters":{"P":[-12,5],"Q":[-12,5],"frame":"euclidean_orthonormal_equal_units","d":0,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"0\"],\"0\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0052","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-11, 4)、Q=(-19, 10)。整數候選 d=-10 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"否","explanation_zh":"候選距離=-10；平方距離=100。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064052,"parameters":{"P":[-11,4],"Q":[-19,10],"frame":"euclidean_orthonormal_equal_units","d":-10,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"6\",\"8\"],\"-10\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0053","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-10, 3)、Q=(-10, 6)。整數候選 d=4 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"否","explanation_zh":"候選距離=4；平方距離=9。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064053,"parameters":{"P":[-10,3],"Q":[-10,6],"frame":"euclidean_orthonormal_equal_units","d":4,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"3\"],\"4\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0054","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-9, 2)、Q=(-33, 12)。整數候選 d=26 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=26；平方距離=676。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064054,"parameters":{"P":[-9,2],"Q":[-33,12],"frame":"euclidean_orthonormal_equal_units","d":26,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"10\",\"24\"],\"26\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0055","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-8, 1)、Q=(-8, 6)。整數候選 d=5 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=5；平方距離=25。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064055,"parameters":{"P":[-8,1],"Q":[-8,6],"frame":"euclidean_orthonormal_equal_units","d":5,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"5\"],\"5\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0056","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-7, 0)、Q=(-55, 14)。整數候選 d=-50 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"否","explanation_zh":"候選距離=-50；平方距離=2500。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064056,"parameters":{"P":[-7,0],"Q":[-55,14],"frame":"euclidean_orthonormal_equal_units","d":-50,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"14\",\"48\"],\"-50\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0057","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-6, -1)、Q=(-6, 6)。整數候選 d=8 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"否","explanation_zh":"候選距離=8；平方距離=49。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064057,"parameters":{"P":[-6,-1],"Q":[-6,6],"frame":"euclidean_orthonormal_equal_units","d":8,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"7\"],\"8\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0058","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-5, -2)、Q=(-85, 16)。整數候選 d=82 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=82；平方距離=6724。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064058,"parameters":{"P":[-5,-2],"Q":[-85,16],"frame":"euclidean_orthonormal_equal_units","d":82,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"18\",\"80\"],\"82\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0059","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-4, -3)、Q=(-4, 6)。整數候選 d=9 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=9；平方距離=81。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064059,"parameters":{"P":[-4,-3],"Q":[-4,6],"frame":"euclidean_orthonormal_equal_units","d":9,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"9\"],\"9\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0060","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-3, -4)、Q=(-123, 18)。整數候選 d=-122 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"否","explanation_zh":"候選距離=-122；平方距離=14884。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064060,"parameters":{"P":[-3,-4],"Q":[-123,18],"frame":"euclidean_orthonormal_equal_units","d":-122,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"22\",\"120\"],\"-122\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0061","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-2, -5)、Q=(-2, 6)。整數候選 d=12 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"否","explanation_zh":"候選距離=12；平方距離=121。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064061,"parameters":{"P":[-2,-5],"Q":[-2,6],"frame":"euclidean_orthonormal_equal_units","d":12,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"11\"],\"12\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0062","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-1, -6)、Q=(-169, 20)。整數候選 d=170 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=170；平方距離=28900。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064062,"parameters":{"P":[-1,-6],"Q":[-169,20],"frame":"euclidean_orthonormal_equal_units","d":170,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"26\",\"168\"],\"170\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0063","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(0, -7)、Q=(0, 6)。整數候選 d=13 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=13；平方距離=169。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064063,"parameters":{"P":[0,-7],"Q":[0,6],"frame":"euclidean_orthonormal_equal_units","d":13,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"13\"],\"13\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0064","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(1, -8)、Q=(-223, 22)。整數候選 d=-226 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"否","explanation_zh":"候選距離=-226；平方距離=51076。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064064,"parameters":{"P":[1,-8],"Q":[-223,22],"frame":"euclidean_orthonormal_equal_units","d":-226,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"30\",\"224\"],\"-226\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0065","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(2, -9)、Q=(2, 6)。整數候選 d=16 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"否","explanation_zh":"候選距離=16；平方距離=225。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064065,"parameters":{"P":[2,-9],"Q":[2,6],"frame":"euclidean_orthonormal_equal_units","d":16,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"15\"],\"16\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0066","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(3, -10)、Q=(-285, 24)。整數候選 d=290 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=290；平方距離=84100。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064066,"parameters":{"P":[3,-10],"Q":[-285,24],"frame":"euclidean_orthonormal_equal_units","d":290,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"34\",\"288\"],\"290\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0067","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(4, -11)、Q=(4, 6)。整數候選 d=17 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=17；平方距離=289。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064067,"parameters":{"P":[4,-11],"Q":[4,6],"frame":"euclidean_orthonormal_equal_units","d":17,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"17\"],\"17\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0068","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(5, -12)、Q=(-355, 26)。整數候選 d=-362 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"否","explanation_zh":"候選距離=-362；平方距離=131044。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064068,"parameters":{"P":[5,-12],"Q":[-355,26],"frame":"euclidean_orthonormal_equal_units","d":-362,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"38\",\"360\"],\"-362\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0069","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(6, -13)、Q=(6, 6)。整數候選 d=20 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"否","explanation_zh":"候選距離=20；平方距離=361。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064069,"parameters":{"P":[6,-13],"Q":[6,6],"frame":"euclidean_orthonormal_equal_units","d":20,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"19\"],\"20\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0070","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(7, -14)、Q=(-433, 28)。整數候選 d=442 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=442；平方距離=195364。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064070,"parameters":{"P":[7,-14],"Q":[-433,28],"frame":"euclidean_orthonormal_equal_units","d":442,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"42\",\"440\"],\"442\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0071","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(8, -15)、Q=(8, 6)。整數候選 d=21 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=21；平方距離=441。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064071,"parameters":{"P":[8,-15],"Q":[8,6],"frame":"euclidean_orthonormal_equal_units","d":21,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"21\"],\"21\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0072","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(9, -16)、Q=(-519, 30)。整數候選 d=-530 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"否","explanation_zh":"候選距離=-530；平方距離=280900。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064072,"parameters":{"P":[9,-16],"Q":[-519,30],"frame":"euclidean_orthonormal_equal_units","d":-530,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"46\",\"528\"],\"-530\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0073","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(10, -17)、Q=(10, 6)。整數候選 d=24 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"否","explanation_zh":"候選距離=24；平方距離=529。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064073,"parameters":{"P":[10,-17],"Q":[10,6],"frame":"euclidean_orthonormal_equal_units","d":24,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"23\"],\"24\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0074","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(11, -18)、Q=(-613, 32)。整數候選 d=626 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=626；平方距離=391876。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064074,"parameters":{"P":[11,-18],"Q":[-613,32],"frame":"euclidean_orthonormal_equal_units","d":626,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"50\",\"624\"],\"626\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0075","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"候選距離的符號與平方資格","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(12, -19)、Q=(12, 6)。整數候選 d=25 是否等於歐氏距離？候選可以錯誤，須同時滿足非負與平方條件。","answer":{"kind":"short_text","value":"是","explanation_zh":"候選距離=25；平方距離=625。即使候選平方相等，負候選仍不是距離；非負且平方相等才接受，不從另一邊長模型推論座標證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選距離的符號與平方資格"]},"variant":{"family_id":"qf-distance-candidate","seed":202610064075,"parameters":{"P":[12,-19],"Q":[12,6],"frame":"euclidean_orthonormal_equal_units","d":25,"operation":"candidate_distance","semantic_template":"qf-distance-candidate/{\"inputs\":[[\"0\",\"25\"],\"25\"],\"operation\":\"candidate_distance\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0076","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-12, 5)、Q=(-12, 5)，同一框架另有 U=(5, -3)、V=(5, -3)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"=","explanation_zh":"平方距離=(0, 0)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064076,"parameters":{"P":[-12,5],"Q":[-12,5],"frame":"euclidean_orthonormal_equal_units","U":[5,-3],"V":[5,-3],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"0\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0077","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-11, 4)、Q=(-12, 6)，同一框架另有 U=(4, -4)、V=(5, 0)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"<","explanation_zh":"平方距離=(5, 17)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064077,"parameters":{"P":[-11,4],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","U":[4,-4],"V":[5,0],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"1\",\"2\"],[\"1\",\"4\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0078","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-10, 3)、Q=(-8, 6)，同一框架另有 U=(3, -5)、V=(3, -5)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":">","explanation_zh":"平方距離=(13, 0)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064078,"parameters":{"P":[-10,3],"Q":[-8,6],"frame":"euclidean_orthonormal_equal_units","U":[3,-5],"V":[3,-5],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"2\",\"3\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0079","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-9, 2)、Q=(-12, 6)，同一框架另有 U=(2, -6)、V=(5, -2)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"=","explanation_zh":"平方距離=(25, 25)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064079,"parameters":{"P":[-9,2],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","U":[2,-6],"V":[5,-2],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"3\",\"4\"],[\"3\",\"4\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0080","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-8, 1)、Q=(-8, 5)，同一框架另有 U=(1, -7)、V=(1, -1)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"<","explanation_zh":"平方距離=(16, 36)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064080,"parameters":{"P":[-8,1],"Q":[-8,5],"frame":"euclidean_orthonormal_equal_units","U":[1,-7],"V":[1,-1],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"4\"],[\"0\",\"6\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0081","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-7, 0)、Q=(-12, 6)，同一框架另有 U=(0, -8)、V=(0, -8)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":">","explanation_zh":"平方距離=(61, 0)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064081,"parameters":{"P":[-7,0],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","U":[0,-8],"V":[0,-8],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"5\",\"6\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0082","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-6, -1)、Q=(0, 6)，同一框架另有 U=(-1, -9)、V=(5, -2)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"=","explanation_zh":"平方距離=(85, 85)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064082,"parameters":{"P":[-6,-1],"Q":[0,6],"frame":"euclidean_orthonormal_equal_units","U":[-1,-9],"V":[5,-2],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"6\",\"7\"],[\"6\",\"7\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0083","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-5, -2)、Q=(-12, 6)，同一框架另有 U=(-2, -10)、V=(5, 0)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"<","explanation_zh":"平方距離=(113, 149)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064083,"parameters":{"P":[-5,-2],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","U":[-2,-10],"V":[5,0],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"7\",\"8\"],[\"7\",\"10\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0084","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-4, -3)、Q=(-4, 5)，同一框架另有 U=(-3, -11)、V=(-3, -11)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":">","explanation_zh":"平方距離=(64, 0)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064084,"parameters":{"P":[-4,-3],"Q":[-4,5],"frame":"euclidean_orthonormal_equal_units","U":[-3,-11],"V":[-3,-11],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"8\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0085","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-3, -4)、Q=(-12, 6)，同一框架另有 U=(-4, -12)、V=(5, -2)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"=","explanation_zh":"平方距離=(181, 181)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064085,"parameters":{"P":[-3,-4],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","U":[-4,-12],"V":[5,-2],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"9\",\"10\"],[\"9\",\"10\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0086","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-2, -5)、Q=(8, 6)，同一框架另有 U=(-5, -13)、V=(5, 0)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"<","explanation_zh":"平方距離=(221, 269)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064086,"parameters":{"P":[-2,-5],"Q":[8,6],"frame":"euclidean_orthonormal_equal_units","U":[-5,-13],"V":[5,0],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"10\",\"11\"],[\"10\",\"13\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0087","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(-1, -6)、Q=(-12, 6)，同一框架另有 U=(-6, -14)、V=(-6, -14)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":">","explanation_zh":"平方距離=(265, 0)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064087,"parameters":{"P":[-1,-6],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","U":[-6,-14],"V":[-6,-14],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"11\",\"12\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0088","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(0, -7)、Q=(0, 5)，同一框架另有 U=(-7, -15)、V=(-7, -3)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"=","explanation_zh":"平方距離=(144, 144)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064088,"parameters":{"P":[0,-7],"Q":[0,5],"frame":"euclidean_orthonormal_equal_units","U":[-7,-15],"V":[-7,-3],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"12\"],[\"0\",\"12\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0089","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(1, -8)、Q=(-12, 6)，同一框架另有 U=(-8, -16)、V=(5, 0)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"<","explanation_zh":"平方距離=(365, 425)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064089,"parameters":{"P":[1,-8],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","U":[-8,-16],"V":[5,0],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"13\",\"14\"],[\"13\",\"16\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0090","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(2, -9)、Q=(16, 6)，同一框架另有 U=(-9, -17)、V=(-9, -17)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":">","explanation_zh":"平方距離=(421, 0)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064090,"parameters":{"P":[2,-9],"Q":[16,6],"frame":"euclidean_orthonormal_equal_units","U":[-9,-17],"V":[-9,-17],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"14\",\"15\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0091","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(3, -10)、Q=(-12, 6)，同一框架另有 U=(-10, -18)、V=(5, -2)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"=","explanation_zh":"平方距離=(481, 481)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064091,"parameters":{"P":[3,-10],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","U":[-10,-18],"V":[5,-2],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"15\",\"16\"],[\"15\",\"16\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0092","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(4, -11)、Q=(4, 5)，同一框架另有 U=(-11, -19)、V=(-11, -1)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"<","explanation_zh":"平方距離=(256, 324)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064092,"parameters":{"P":[4,-11],"Q":[4,5],"frame":"euclidean_orthonormal_equal_units","U":[-11,-19],"V":[-11,-1],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"16\"],[\"0\",\"18\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0093","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(5, -12)、Q=(-12, 6)，同一框架另有 U=(-12, -20)、V=(-12, -20)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":">","explanation_zh":"平方距離=(613, 0)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064093,"parameters":{"P":[5,-12],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","U":[-12,-20],"V":[-12,-20],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"17\",\"18\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0094","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(6, -13)、Q=(24, 6)，同一框架另有 U=(-13, -21)、V=(5, -2)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"=","explanation_zh":"平方距離=(685, 685)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064094,"parameters":{"P":[6,-13],"Q":[24,6],"frame":"euclidean_orthonormal_equal_units","U":[-13,-21],"V":[5,-2],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"18\",\"19\"],[\"18\",\"19\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0095","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(7, -14)、Q=(-12, 6)，同一框架另有 U=(-14, -22)、V=(5, 0)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"<","explanation_zh":"平方距離=(761, 845)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064095,"parameters":{"P":[7,-14],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","U":[-14,-22],"V":[5,0],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"19\",\"20\"],[\"19\",\"22\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0096","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(8, -15)、Q=(8, 5)，同一框架另有 U=(-15, -23)、V=(-15, -23)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":">","explanation_zh":"平方距離=(400, 0)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064096,"parameters":{"P":[8,-15],"Q":[8,5],"frame":"euclidean_orthonormal_equal_units","U":[-15,-23],"V":[-15,-23],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"20\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0097","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(9, -16)、Q=(-12, 6)，同一框架另有 U=(-16, -24)、V=(5, -2)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"=","explanation_zh":"平方距離=(925, 925)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064097,"parameters":{"P":[9,-16],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","U":[-16,-24],"V":[5,-2],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"21\",\"22\"],[\"21\",\"22\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0098","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(10, -17)、Q=(32, 6)，同一框架另有 U=(-17, -25)、V=(5, 0)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"<","explanation_zh":"平方距離=(1013, 1109)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064098,"parameters":{"P":[10,-17],"Q":[32,6],"frame":"euclidean_orthonormal_equal_units","U":[-17,-25],"V":[5,0],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"22\",\"23\"],[\"22\",\"25\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0099","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(11, -18)、Q=(-12, 6)，同一框架另有 U=(-18, -26)、V=(-18, -26)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":">","explanation_zh":"平方距離=(1105, 0)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064099,"parameters":{"P":[11,-18],"Q":[-12,6],"frame":"euclidean_orthonormal_equal_units","U":[-18,-26],"V":[-18,-26],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"23\",\"24\"],[\"0\",\"0\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261005-advance-20261006-r04-distance-formula-0100","batch_id":"batch-20261005-advance-20261006-r04-distance-formula-foundations","language":"zh-Hant","source_mko_ids":["mko-distance-formula"],"title_zh":"兩段歐氏距離的精確比較","stem_zh":"在等單位正交座標系中，兩軸使用同一長度單位，P=(12, -19)、Q=(12, 5)，同一框架另有 U=(-19, -27)、V=(-19, -3)。比較 d(P,Q) 與 d(U,V)，回答 <、= 或 >。","answer":{"kind":"short_text","value":"=","explanation_zh":"平方距離=(576, 576)。兩個距離都非負，故比較它們的平方即可保留順序；不以圖形外觀或不同軸尺度猜距離。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩段歐氏距離的精確比較"]},"variant":{"family_id":"qf-distance-comparison","seed":202610064100,"parameters":{"P":[12,-19],"Q":[12,5],"frame":"euclidean_orthonormal_equal_units","U":[-19,-27],"V":[-19,-3],"operation":"distance_comparison","semantic_template":"qf-distance-comparison/{\"inputs\":[[\"0\",\"24\"],[\"0\",\"24\"]],\"operation\":\"distance_comparison\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-05","contract_version":"ocme-question-factory-v0.1"}}