{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0001","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1000,\"b\":97}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1000, 97, 10, 30], [97, 30, 3, 7], [30, 7, 4, 2], [7, 2, 3, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1000, 97, 10, 30], [97, 30, 3, 7], [30, 7, 4, 2], [7, 2, 3, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053001,"parameters":{"a":1000,"b":97,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1000,\"b\":97}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0002","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1037,\"b\":98}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1037, 98, 10, 57], [98, 57, 1, 41], [57, 41, 1, 16], [41, 16, 2, 9], [16, 9, 1, 7], [9, 7, 1, 2], [7, 2, 3, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1037, 98, 10, 57], [98, 57, 1, 41], [57, 41, 1, 16], [41, 16, 2, 9], [16, 9, 1, 7], [9, 7, 1, 2], [7, 2, 3, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053002,"parameters":{"a":1037,"b":98,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1037,\"b\":98}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0003","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1074,\"b\":99}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1074, 99, 10, 84], [99, 84, 1, 15], [84, 15, 5, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1074, 99, 10, 84], [99, 84, 1, 15], [84, 15, 5, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053003,"parameters":{"a":1074,"b":99,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1074,\"b\":99}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0004","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1111,\"b\":100}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1111, 100, 11, 11], [100, 11, 9, 1], [11, 1, 11, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1111, 100, 11, 11], [100, 11, 9, 1], [11, 1, 11, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053004,"parameters":{"a":1111,"b":100,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1111,\"b\":100}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0005","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1148,\"b\":101}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1148, 101, 11, 37], [101, 37, 2, 27], [37, 27, 1, 10], [27, 10, 2, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1148, 101, 11, 37], [101, 37, 2, 27], [37, 27, 1, 10], [27, 10, 2, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053005,"parameters":{"a":1148,"b":101,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1148,\"b\":101}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0006","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1185,\"b\":102}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1185, 102, 11, 63], [102, 63, 1, 39], [63, 39, 1, 24], [39, 24, 1, 15], [24, 15, 1, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1185, 102, 11, 63], [102, 63, 1, 39], [63, 39, 1, 24], [39, 24, 1, 15], [24, 15, 1, 9], [15, 9, 1, 6], [9, 6, 1, 3], [6, 3, 2, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053006,"parameters":{"a":1185,"b":102,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1185,\"b\":102}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0007","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1222,\"b\":103}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1222, 103, 11, 89], [103, 89, 1, 14], [89, 14, 6, 5], [14, 5, 2, 4], [5, 4, 1, 1], [4, 1, 4, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1222, 103, 11, 89], [103, 89, 1, 14], [89, 14, 6, 5], [14, 5, 2, 4], [5, 4, 1, 1], [4, 1, 4, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053007,"parameters":{"a":1222,"b":103,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1222,\"b\":103}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0008","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1259,\"b\":104}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1259, 104, 12, 11], [104, 11, 9, 5], [11, 5, 2, 1], [5, 1, 5, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1259, 104, 12, 11], [104, 11, 9, 5], [11, 5, 2, 1], [5, 1, 5, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053008,"parameters":{"a":1259,"b":104,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1259,\"b\":104}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0009","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1296,\"b\":105}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1296, 105, 12, 36], [105, 36, 2, 33], [36, 33, 1, 3], [33, 3, 11, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1296, 105, 12, 36], [105, 36, 2, 33], [36, 33, 1, 3], [33, 3, 11, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053009,"parameters":{"a":1296,"b":105,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1296,\"b\":105}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0010","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1333,\"b\":106}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1333, 106, 12, 61], [106, 61, 1, 45], [61, 45, 1, 16], [45, 16, 2, 13], [16, 13, 1, 3], [13, 3, 4, 1], [3, 1, 3, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1333, 106, 12, 61], [106, 61, 1, 45], [61, 45, 1, 16], [45, 16, 2, 13], [16, 13, 1, 3], [13, 3, 4, 1], [3, 1, 3, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053010,"parameters":{"a":1333,"b":106,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1333,\"b\":106}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0011","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1370,\"b\":107}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1370, 107, 12, 86], [107, 86, 1, 21], [86, 21, 4, 2], [21, 2, 10, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1370, 107, 12, 86], [107, 86, 1, 21], [86, 21, 4, 2], [21, 2, 10, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053011,"parameters":{"a":1370,"b":107,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1370,\"b\":107}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0012","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1407,\"b\":108}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1407, 108, 13, 3], [108, 3, 36, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1407, 108, 13, 3], [108, 3, 36, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053012,"parameters":{"a":1407,"b":108,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1407,\"b\":108}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0013","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1444,\"b\":109}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1444, 109, 13, 27], [109, 27, 4, 1], [27, 1, 27, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1444, 109, 13, 27], [109, 27, 4, 1], [27, 1, 27, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053013,"parameters":{"a":1444,"b":109,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1444,\"b\":109}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0014","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1481,\"b\":110}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1481, 110, 13, 51], [110, 51, 2, 8], [51, 8, 6, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1481, 110, 13, 51], [110, 51, 2, 8], [51, 8, 6, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053014,"parameters":{"a":1481,"b":110,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1481,\"b\":110}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0015","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1518,\"b\":111}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1518, 111, 13, 75], [111, 75, 1, 36], [75, 36, 2, 3], [36, 3, 12, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1518, 111, 13, 75], [111, 75, 1, 36], [75, 36, 2, 3], [36, 3, 12, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053015,"parameters":{"a":1518,"b":111,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1518,\"b\":111}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0016","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1555,\"b\":112}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1555, 112, 13, 99], [112, 99, 1, 13], [99, 13, 7, 8], [13, 8, 1, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1555, 112, 13, 99], [112, 99, 1, 13], [99, 13, 7, 8], [13, 8, 1, 5], [8, 5, 1, 3], [5, 3, 1, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053016,"parameters":{"a":1555,"b":112,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1555,\"b\":112}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0017","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1592,\"b\":113}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1592, 113, 14, 10], [113, 10, 11, 3], [10, 3, 3, 1], [3, 1, 3, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1592, 113, 14, 10], [113, 10, 11, 3], [10, 3, 3, 1], [3, 1, 3, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053017,"parameters":{"a":1592,"b":113,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1592,\"b\":113}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0018","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1629,\"b\":114}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1629, 114, 14, 33], [114, 33, 3, 15], [33, 15, 2, 3], [15, 3, 5, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1629, 114, 14, 33], [114, 33, 3, 15], [33, 15, 2, 3], [15, 3, 5, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053018,"parameters":{"a":1629,"b":114,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1629,\"b\":114}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0019","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1666,\"b\":115}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1666, 115, 14, 56], [115, 56, 2, 3], [56, 3, 18, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1666, 115, 14, 56], [115, 56, 2, 3], [56, 3, 18, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053019,"parameters":{"a":1666,"b":115,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1666,\"b\":115}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0020","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1703,\"b\":116}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1703, 116, 14, 79], [116, 79, 1, 37], [79, 37, 2, 5], [37, 5, 7, 2], [5, 2, 2, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1703, 116, 14, 79], [116, 79, 1, 37], [79, 37, 2, 5], [37, 5, 7, 2], [5, 2, 2, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053020,"parameters":{"a":1703,"b":116,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1703,\"b\":116}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0021","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1740,\"b\":117}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1740, 117, 14, 102], [117, 102, 1, 15], [102, 15, 6, 12], [15, 12, 1, 3], [12, 3, 4, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1740, 117, 14, 102], [117, 102, 1, 15], [102, 15, 6, 12], [15, 12, 1, 3], [12, 3, 4, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053021,"parameters":{"a":1740,"b":117,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1740,\"b\":117}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0022","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1777,\"b\":118}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1777, 118, 15, 7], [118, 7, 16, 6], [7, 6, 1, 1], [6, 1, 6, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1777, 118, 15, 7], [118, 7, 16, 6], [7, 6, 1, 1], [6, 1, 6, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053022,"parameters":{"a":1777,"b":118,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1777,\"b\":118}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0023","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1814,\"b\":119}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1814, 119, 15, 29], [119, 29, 4, 3], [29, 3, 9, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1814, 119, 15, 29], [119, 29, 4, 3], [29, 3, 9, 2], [3, 2, 1, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053023,"parameters":{"a":1814,"b":119,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1814,\"b\":119}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0024","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1851,\"b\":120}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1851, 120, 15, 51], [120, 51, 2, 18], [51, 18, 2, 15], [18, 15, 1, 3], [15, 3, 5, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1851, 120, 15, 51], [120, 51, 2, 18], [51, 18, 2, 15], [18, 15, 1, 3], [15, 3, 5, 0]], 3, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053024,"parameters":{"a":1851,"b":120,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1851,\"b\":120}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0025","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1888,\"b\":121}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1888, 121, 15, 73], [121, 73, 1, 48], [73, 48, 1, 25], [48, 25, 1, 23], [25, 23, 1, 2], [23, 2, 11, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)","explanation_zh":"Wave5第五層核對=([[1888, 121, 15, 73], [121, 73, 1, 48], [73, 48, 1, 25], [48, 25, 1, 23], [25, 23, 1, 2], [23, 2, 11, 1], [2, 1, 2, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-division-chain","seed":202611053025,"parameters":{"a":1888,"b":121,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-division-chain","semantic_template":"qf-mod36-euclid-division-chain/{\"a\":1888,\"b\":121}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0026","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1000,\"b\":97}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053026,"parameters":{"a":1000,"b":97,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1000,\"b\":97}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0027","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1037,\"b\":98}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053027,"parameters":{"a":1037,"b":98,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1037,\"b\":98}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0028","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1074,\"b\":99}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053028,"parameters":{"a":1074,"b":99,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1074,\"b\":99}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0029","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1111,\"b\":100}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053029,"parameters":{"a":1111,"b":100,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1111,\"b\":100}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0030","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1148,\"b\":101}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053030,"parameters":{"a":1148,"b":101,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1148,\"b\":101}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0031","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1185,\"b\":102}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053031,"parameters":{"a":1185,"b":102,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1185,\"b\":102}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0032","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1222,\"b\":103}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053032,"parameters":{"a":1222,"b":103,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1222,\"b\":103}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0033","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1259,\"b\":104}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053033,"parameters":{"a":1259,"b":104,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1259,\"b\":104}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0034","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1296,\"b\":105}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053034,"parameters":{"a":1296,"b":105,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1296,\"b\":105}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0035","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1333,\"b\":106}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053035,"parameters":{"a":1333,"b":106,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1333,\"b\":106}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0036","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1370,\"b\":107}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053036,"parameters":{"a":1370,"b":107,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1370,\"b\":107}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0037","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1407,\"b\":108}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053037,"parameters":{"a":1407,"b":108,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1407,\"b\":108}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0038","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1444,\"b\":109}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053038,"parameters":{"a":1444,"b":109,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1444,\"b\":109}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0039","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1481,\"b\":110}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053039,"parameters":{"a":1481,"b":110,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1481,\"b\":110}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0040","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1518,\"b\":111}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053040,"parameters":{"a":1518,"b":111,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1518,\"b\":111}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0041","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1555,\"b\":112}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053041,"parameters":{"a":1555,"b":112,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1555,\"b\":112}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0043","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1629,\"b\":114}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053043,"parameters":{"a":1629,"b":114,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1629,\"b\":114}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0044","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1666,\"b\":115}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053044,"parameters":{"a":1666,"b":115,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1666,\"b\":115}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0045","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1703,\"b\":116}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053045,"parameters":{"a":1703,"b":116,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1703,\"b\":116}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0046","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1740,\"b\":117}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053046,"parameters":{"a":1740,"b":117,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1740,\"b\":117}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0047","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1777,\"b\":118}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053047,"parameters":{"a":1777,"b":118,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1777,\"b\":118}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0048","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1814,\"b\":119}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053048,"parameters":{"a":1814,"b":119,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1814,\"b\":119}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0049","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1851,\"b\":120}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[3, 3], [3, 3], [3, 3], [3, 3], [3, 3]], 3, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053049,"parameters":{"a":1851,"b":120,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1851,\"b\":120}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0050","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-gcd-invariant；資料={\"a\":1888,\"b\":121}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)","explanation_zh":"Wave5第五層核對=([[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]], 1, 是, gcd_invariant_separate_from_termination)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-gcd-invariant","seed":202611053050,"parameters":{"a":1888,"b":121,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-gcd-invariant","semantic_template":"qf-mod36-euclid-gcd-invariant/{\"a\":1888,\"b\":121}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0051","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":401,\"b\":89}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 2, -9, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, 2, -9, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053051,"parameters":{"a":401,"b":89,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":401,\"b\":89}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0052","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":-424,\"b\":91}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, -44, -205, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, -44, -205, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053052,"parameters":{"a":-424,"b":91,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-424,\"b\":91}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0053","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":447,\"b\":93}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, -24, 3, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(3, 5, -24, 3, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053053,"parameters":{"a":447,"b":93,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":447,\"b\":93}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0054","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":-470,\"b\":95}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 1, 5, 5, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(5, 1, 5, 5, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053054,"parameters":{"a":-470,"b":95,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-470,\"b\":95}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0055","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":493,\"b\":97}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, -12, 61, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, -12, 61, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053055,"parameters":{"a":493,"b":97,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":493,\"b\":97}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0056","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":-516,\"b\":99}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 14, 73, 3, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(3, 14, 73, 3, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053056,"parameters":{"a":-516,"b":99,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-516,\"b\":99}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0057","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":539,\"b\":101}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, -16, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, 3, -16, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053057,"parameters":{"a":539,"b":101,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":539,\"b\":101}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0058","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":-562,\"b\":103}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 46, 251, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, 46, 251, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053058,"parameters":{"a":-562,"b":103,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-562,\"b\":103}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0059","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":585,\"b\":105}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 2, -11, 15, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(15, 2, -11, 15, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053059,"parameters":{"a":585,"b":105,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":585,\"b\":105}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0060","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":-608,\"b\":107}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, -22, -125, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, -22, -125, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053060,"parameters":{"a":-608,"b":107,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-608,\"b\":107}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0061","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":631,\"b\":109}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, -19, 110, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, -19, 110, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053061,"parameters":{"a":631,"b":109,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":631,\"b\":109}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0062","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":-654,\"b\":111}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, -9, -53, 3, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(3, -9, -53, 3, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053062,"parameters":{"a":-654,"b":111,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-654,\"b\":111}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0063","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":677,\"b\":113}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, -1, 6, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, -1, 6, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053063,"parameters":{"a":677,"b":113,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":677,\"b\":113}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0064","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":-700,\"b\":115}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 11, 67, 5, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(5, 11, 67, 5, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053064,"parameters":{"a":-700,"b":115,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-700,\"b\":115}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0065","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":723,\"b\":117}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, -11, 68, 3, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(3, -11, 68, 3, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053065,"parameters":{"a":723,"b":117,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":723,\"b\":117}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0066","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":-746,\"b\":119}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 26, 163, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, 26, 163, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053066,"parameters":{"a":-746,"b":119,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-746,\"b\":119}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0067","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":769,\"b\":121}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, -45, 286, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, -45, 286, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053067,"parameters":{"a":769,"b":121,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":769,\"b\":121}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0068","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":-792,\"b\":123}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, -16, -103, 3, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(3, -16, -103, 3, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053068,"parameters":{"a":-792,"b":123,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-792,\"b\":123}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0069","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":815,\"b\":125}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 2, -13, 5, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(5, 2, -13, 5, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053069,"parameters":{"a":815,"b":125,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":815,\"b\":125}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0070","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":-838,\"b\":127}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 5, 33, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, 5, 33, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053070,"parameters":{"a":-838,"b":127,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-838,\"b\":127}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0071","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":861,\"b\":129}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, -20, 3, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(3, 3, -20, 3, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053071,"parameters":{"a":861,"b":129,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":861,\"b\":129}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0072","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":-884,\"b\":131}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 4, 27, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, 4, 27, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053072,"parameters":{"a":-884,"b":131,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-884,\"b\":131}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0073","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":907,\"b\":133}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, -61, 416, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, -61, 416, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053073,"parameters":{"a":907,"b":133,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":907,\"b\":133}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0074","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":-930,\"b\":135}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 1, 7, 15, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(15, 1, 7, 15, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053074,"parameters":{"a":-930,"b":135,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":-930,\"b\":135}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0075","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-bezout-certificate；資料={\"a\":953,\"b\":137}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, -23, 160, 1, 是, extended_euclidean_back_substitution)","explanation_zh":"Wave5第五層核對=(1, -23, 160, 1, 是, extended_euclidean_back_substitution)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-bezout-certificate","seed":202611053075,"parameters":{"a":953,"b":137,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-bezout-certificate","semantic_template":"qf-mod36-euclid-bezout-certificate/{\"a\":953,\"b\":137}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0076","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 1, 0, 0, gcd_0_0_equals_0)","explanation_zh":"Wave5第五層核對=(0, 0, 0, 1, 0, 0, gcd_0_0_equals_0)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053076,"parameters":{"a":0,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0077","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 2, 0, 1, 2, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(0, 2, 2, 0, 1, 2, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053077,"parameters":{"a":0,"b":2,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0078","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-3,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 0, 3, -1, 0, 3, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(-3, 0, 3, -1, 0, 3, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053078,"parameters":{"a":-3,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-3,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0079","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 4, 4, 0, 1, 4, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(0, 4, 4, 0, 1, 4, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053079,"parameters":{"a":0,"b":4,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0080","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-5,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 0, 5, -1, 0, 5, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(-5, 0, 5, -1, 0, 5, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053080,"parameters":{"a":-5,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-5,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0081","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 6, 6, 0, 1, 6, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(0, 6, 6, 0, 1, 6, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053081,"parameters":{"a":0,"b":6,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0082","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-7,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 0, 7, -1, 0, 7, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(-7, 0, 7, -1, 0, 7, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053082,"parameters":{"a":-7,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-7,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0083","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 8, 8, 0, 1, 8, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(0, 8, 8, 0, 1, 8, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053083,"parameters":{"a":0,"b":8,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0084","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-9,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 0, 9, -1, 0, 9, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(-9, 0, 9, -1, 0, 9, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053084,"parameters":{"a":-9,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-9,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0085","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 10, 10, 0, 1, 10, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(0, 10, 10, 0, 1, 10, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053085,"parameters":{"a":0,"b":10,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0086","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-11,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 0, 11, -1, 0, 11, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(-11, 0, 11, -1, 0, 11, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053086,"parameters":{"a":-11,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-11,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0087","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 12, 12, 0, 1, 12, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(0, 12, 12, 0, 1, 12, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053087,"parameters":{"a":0,"b":12,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0088","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-13,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, 0, 13, -1, 0, 13, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(-13, 0, 13, -1, 0, 13, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053088,"parameters":{"a":-13,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-13,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0089","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 14, 14, 0, 1, 14, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(0, 14, 14, 0, 1, 14, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053089,"parameters":{"a":0,"b":14,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0090","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-15,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-15, 0, 15, -1, 0, 15, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(-15, 0, 15, -1, 0, 15, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053090,"parameters":{"a":-15,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-15,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0091","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 16, 16, 0, 1, 16, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(0, 16, 16, 0, 1, 16, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053091,"parameters":{"a":0,"b":16,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0092","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-17,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-17, 0, 17, -1, 0, 17, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(-17, 0, 17, -1, 0, 17, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053092,"parameters":{"a":-17,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-17,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0093","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 18, 18, 0, 1, 18, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(0, 18, 18, 0, 1, 18, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053093,"parameters":{"a":0,"b":18,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0094","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-19,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-19, 0, 19, -1, 0, 19, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(-19, 0, 19, -1, 0, 19, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053094,"parameters":{"a":-19,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-19,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0095","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 20, 20, 0, 1, 20, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(0, 20, 20, 0, 1, 20, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053095,"parameters":{"a":0,"b":20,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0096","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-21,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-21, 0, 21, -1, 0, 21, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(-21, 0, 21, -1, 0, 21, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053096,"parameters":{"a":-21,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-21,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0097","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 22, 22, 0, 1, 22, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(0, 22, 22, 0, 1, 22, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053097,"parameters":{"a":0,"b":22,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0098","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-23,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-23, 0, 23, -1, 0, 23, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(-23, 0, 23, -1, 0, 23, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053098,"parameters":{"a":-23,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-23,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0099","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":0,\"b\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 24, 24, 0, 1, 24, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(0, 24, 24, 0, 1, 24, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053099,"parameters":{"a":0,"b":24,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":0,\"b\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261105-r03-euclidean-algorithm-0100","batch_id":"batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations","language":"zh-Hant","source_mko_ids":["mko-euclidean-algorithm"],"title_zh":"Euclidean演算法","stem_zh":"exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-zero-input-boundary；資料={\"a\":-25,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-25, 0, 25, -1, 0, 25, no_division_by_zero_executed)","explanation_zh":"Wave5第五層核對=(-25, 0, 25, -1, 0, 25, no_division_by_zero_executed)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euclidean演算法"]},"variant":{"family_id":"qf-mod36-euclid-zero-input-boundary","seed":202611053100,"parameters":{"a":-25,"b":0,"domain":"Exact_integer_Euclidean_chain","operation":"euclid-zero-input-boundary","semantic_template":"qf-mod36-euclid-zero-input-boundary/{\"a\":-25,\"b\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}