{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r06-legendre-symbol-0001","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[0, 0], [1, 1], [2, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(3, [[0, 0], [1, 1], [2, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126001,"parameters":{"prime":3,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":3}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0002","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(5, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126002,"parameters":{"prime":5,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":5}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0003","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, -1], [6, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(7, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, -1], [6, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126003,"parameters":{"prime":7,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":7}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0004","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, -1], [8, -1], [9, 1], [10, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(11, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, -1], [8, -1], [9, 1], [10, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126004,"parameters":{"prime":11,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":11}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0005","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, -1], [8, -1], [9, 1], [10, 1], [11, -1], [12, 1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(13, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, -1], [8, -1], [9, 1], [10, 1], [11, -1], [12, 1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126005,"parameters":{"prime":13,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":13}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0006","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, -1], [6, -1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, -1], [15, 1], [16, 1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(17, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, -1], [6, -1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, -1], [15, 1], [16, 1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126006,"parameters":{"prime":17,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":17}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0007","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(19, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126007,"parameters":{"prime":19,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":19}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0008","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, -1], [21, -1], [22, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(23, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, -1], [21, -1], [22, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126008,"parameters":{"prime":23,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":23}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0009","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":29}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, -1], [19, -1], [20, 1], [21, -1], [22, 1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, 1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(29, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, -1], [19, -1], [20, 1], [21, -1], [22, 1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, 1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126009,"parameters":{"prime":29,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":29}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0010","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":31}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, 1], [8, 1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, 1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, -1], [22, -1], [23, -1], [24, -1], [25, 1], [26, -1], [27, -1], [28, 1], [29, -1], [30, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(31, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, 1], [8, 1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, 1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, -1], [22, -1], [23, -1], [24, -1], [25, 1], [26, -1], [27, -1], [28, 1], [29, -1], [30, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126010,"parameters":{"prime":31,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":31}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0011","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":37}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(37, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, -1], [30, 1], [31, -1], [32, -1], [33, 1], [34, 1], [35, -1], [36, 1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(37, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, -1], [30, 1], [31, -1], [32, -1], [33, 1], [34, 1], [35, -1], [36, 1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126011,"parameters":{"prime":37,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":37}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0012","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":41}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(41, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, 1], [21, 1], [22, -1], [23, 1], [24, -1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, -1], [39, 1], [40, 1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(41, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, 1], [21, 1], [22, -1], [23, 1], [24, -1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, -1], [39, 1], [40, 1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126012,"parameters":{"prime":41,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":41}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0013","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":43}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(43, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, -1], [8, -1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, 1], [42, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(43, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, -1], [8, -1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, 1], [42, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126013,"parameters":{"prime":43,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":43}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0014","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":47}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, 1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, 1], [15, -1], [16, 1], [17, 1], [18, 1], [19, -1], [20, -1], [21, 1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, 1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, 1], [35, -1], [36, 1], [37, 1], [38, -1], [39, -1], [40, -1], [41, -1], [42, 1], [43, -1], [44, -1], [45, -1], [46, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(47, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, 1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, 1], [15, -1], [16, 1], [17, 1], [18, 1], [19, -1], [20, -1], [21, 1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, 1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, 1], [35, -1], [36, 1], [37, 1], [38, -1], [39, -1], [40, -1], [41, -1], [42, 1], [43, -1], [44, -1], [45, -1], [46, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126014,"parameters":{"prime":47,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":47}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0015","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":53}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(53, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, -1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, -1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, 1], [44, 1], [45, -1], [46, 1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, 1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(53, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, -1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, -1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, 1], [44, 1], [45, -1], [46, 1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, 1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126015,"parameters":{"prime":53,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":53}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0016","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":59}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(59, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, 1], [46, 1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, 1], [54, -1], [55, -1], [56, -1], [57, 1], [58, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(59, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, 1], [46, 1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, 1], [54, -1], [55, -1], [56, -1], [57, 1], [58, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126016,"parameters":{"prime":59,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":59}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0017","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":61}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(61, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, -1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, -1], [18, -1], [19, 1], [20, 1], [21, -1], [22, 1], [23, -1], [24, -1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, -1], [32, -1], [33, -1], [34, 1], [35, -1], [36, 1], [37, -1], [38, -1], [39, 1], [40, -1], [41, 1], [42, 1], [43, -1], [44, -1], [45, 1], [46, 1], [47, 1], [48, 1], [49, 1], [50, -1], [51, -1], [52, 1], [53, -1], [54, -1], [55, -1], [56, 1], [57, 1], [58, 1], [59, -1], [60, 1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(61, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, -1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, -1], [18, -1], [19, 1], [20, 1], [21, -1], [22, 1], [23, -1], [24, -1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, -1], [32, -1], [33, -1], [34, 1], [35, -1], [36, 1], [37, -1], [38, -1], [39, 1], [40, -1], [41, 1], [42, 1], [43, -1], [44, -1], [45, 1], [46, 1], [47, 1], [48, 1], [49, 1], [50, -1], [51, -1], [52, 1], [53, -1], [54, -1], [55, -1], [56, 1], [57, 1], [58, 1], [59, -1], [60, 1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126017,"parameters":{"prime":61,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":61}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0018","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":67}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(67, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, -1], [8, -1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, 1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, -1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, 1], [27, -1], [28, -1], [29, 1], [30, -1], [31, -1], [32, -1], [33, 1], [34, -1], [35, 1], [36, 1], [37, 1], [38, -1], [39, 1], [40, 1], [41, -1], [42, -1], [43, -1], [44, -1], [45, -1], [46, -1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, -1], [53, -1], [54, 1], [55, 1], [56, 1], [57, -1], [58, -1], [59, 1], [60, 1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(67, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, -1], [8, -1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, 1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, -1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, 1], [27, -1], [28, -1], [29, 1], [30, -1], [31, -1], [32, -1], [33, 1], [34, -1], [35, 1], [36, 1], [37, 1], [38, -1], [39, 1], [40, 1], [41, -1], [42, -1], [43, -1], [44, -1], [45, -1], [46, -1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, -1], [53, -1], [54, 1], [55, 1], [56, 1], [57, -1], [58, -1], [59, 1], [60, 1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126018,"parameters":{"prime":67,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":67}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0019","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":71}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(71, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, -1], [8, 1], [9, 1], [10, 1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, -1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, 1], [30, 1], [31, -1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, -1], [42, -1], [43, 1], [44, -1], [45, 1], [46, -1], [47, -1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, -1], [54, 1], [55, -1], [56, -1], [57, 1], [58, 1], [59, -1], [60, 1], [61, -1], [62, -1], [63, -1], [64, 1], [65, -1], [66, -1], [67, -1], [68, -1], [69, -1], [70, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(71, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, -1], [8, 1], [9, 1], [10, 1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, -1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, 1], [30, 1], [31, -1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, -1], [42, -1], [43, 1], [44, -1], [45, 1], [46, -1], [47, -1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, -1], [54, 1], [55, -1], [56, -1], [57, 1], [58, 1], [59, -1], [60, 1], [61, -1], [62, -1], [63, -1], [64, 1], [65, -1], [66, -1], [67, -1], [68, -1], [69, -1], [70, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126019,"parameters":{"prime":71,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":71}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0020","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":73}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(73, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, -1], [21, -1], [22, -1], [23, 1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, -1], [35, 1], [36, 1], [37, 1], [38, 1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, -1], [46, 1], [47, -1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, -1], [54, 1], [55, 1], [56, -1], [57, 1], [58, -1], [59, -1], [60, -1], [61, 1], [62, -1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, 1], [70, 1], [71, 1], [72, 1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(73, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, -1], [21, -1], [22, -1], [23, 1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, -1], [35, 1], [36, 1], [37, 1], [38, 1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, -1], [46, 1], [47, -1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, -1], [54, 1], [55, 1], [56, -1], [57, 1], [58, -1], [59, -1], [60, -1], [61, 1], [62, -1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, 1], [70, 1], [71, 1], [72, 1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126020,"parameters":{"prime":73,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":73}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0021","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":79}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(79, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, -1], [25, 1], [26, 1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, 1], [47, -1], [48, -1], [49, 1], [50, 1], [51, 1], [52, 1], [53, -1], [54, -1], [55, 1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, -1], [70, -1], [71, -1], [72, 1], [73, 1], [74, -1], [75, -1], [76, 1], [77, -1], [78, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(79, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, -1], [25, 1], [26, 1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, 1], [47, -1], [48, -1], [49, 1], [50, 1], [51, 1], [52, 1], [53, -1], [54, -1], [55, 1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, -1], [70, -1], [71, -1], [72, 1], [73, 1], [74, -1], [75, -1], [76, 1], [77, -1], [78, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126021,"parameters":{"prime":79,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":79}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0022","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":83}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(83, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, 1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, 1], [31, 1], [32, -1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, 1], [42, -1], [43, -1], [44, 1], [45, -1], [46, -1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, -1], [54, -1], [55, -1], [56, -1], [57, -1], [58, -1], [59, 1], [60, -1], [61, 1], [62, -1], [63, 1], [64, 1], [65, 1], [66, -1], [67, -1], [68, 1], [69, 1], [70, 1], [71, -1], [72, -1], [73, -1], [74, -1], [75, 1], [76, -1], [77, 1], [78, 1], [79, -1], [80, -1], [81, 1], [82, -1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(83, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, 1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, 1], [31, 1], [32, -1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, 1], [42, -1], [43, -1], [44, 1], [45, -1], [46, -1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, -1], [54, -1], [55, -1], [56, -1], [57, -1], [58, -1], [59, 1], [60, -1], [61, 1], [62, -1], [63, 1], [64, 1], [65, 1], [66, -1], [67, -1], [68, 1], [69, 1], [70, 1], [71, -1], [72, -1], [73, -1], [74, -1], [75, 1], [76, -1], [77, 1], [78, 1], [79, -1], [80, -1], [81, 1], [82, -1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126022,"parameters":{"prime":83,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":83}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0023","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":89}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(89, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, 1], [19, -1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, 1], [35, -1], [36, 1], [37, -1], [38, -1], [39, 1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, -1], [47, 1], [48, -1], [49, 1], [50, 1], [51, -1], [52, -1], [53, 1], [54, -1], [55, 1], [56, -1], [57, 1], [58, -1], [59, -1], [60, -1], [61, -1], [62, -1], [63, -1], [64, 1], [65, -1], [66, -1], [67, 1], [68, 1], [69, 1], [70, -1], [71, 1], [72, 1], [73, 1], [74, -1], [75, -1], [76, -1], [77, -1], [78, 1], [79, 1], [80, 1], [81, 1], [82, -1], [83, -1], [84, 1], [85, 1], [86, -1], [87, 1], [88, 1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(89, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, 1], [19, -1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, 1], [35, -1], [36, 1], [37, -1], [38, -1], [39, 1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, -1], [47, 1], [48, -1], [49, 1], [50, 1], [51, -1], [52, -1], [53, 1], [54, -1], [55, 1], [56, -1], [57, 1], [58, -1], [59, -1], [60, -1], [61, -1], [62, -1], [63, -1], [64, 1], [65, -1], [66, -1], [67, 1], [68, 1], [69, 1], [70, -1], [71, 1], [72, 1], [73, 1], [74, -1], [75, -1], [76, -1], [77, -1], [78, 1], [79, 1], [80, 1], [81, 1], [82, -1], [83, -1], [84, 1], [85, 1], [86, -1], [87, 1], [88, 1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126023,"parameters":{"prime":89,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0024","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":97}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(97, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, -1], [21, -1], [22, 1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, 1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, -1], [42, -1], [43, 1], [44, 1], [45, -1], [46, -1], [47, 1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, 1], [54, 1], [55, -1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, 1], [62, 1], [63, -1], [64, 1], [65, 1], [66, 1], [67, -1], [68, -1], [69, -1], [70, 1], [71, -1], [72, 1], [73, 1], [74, -1], [75, 1], [76, -1], [77, -1], [78, -1], [79, 1], [80, -1], [81, 1], [82, -1], [83, -1], [84, -1], [85, 1], [86, 1], [87, -1], [88, 1], [89, 1], [90, -1], [91, 1], [92, -1], [93, 1], [94, 1], [95, 1], [96, 1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(97, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, -1], [21, -1], [22, 1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, 1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, -1], [42, -1], [43, 1], [44, 1], [45, -1], [46, -1], [47, 1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, 1], [54, 1], [55, -1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, 1], [62, 1], [63, -1], [64, 1], [65, 1], [66, 1], [67, -1], [68, -1], [69, -1], [70, 1], [71, -1], [72, 1], [73, 1], [74, -1], [75, 1], [76, -1], [77, -1], [78, -1], [79, 1], [80, -1], [81, 1], [82, -1], [83, -1], [84, -1], [85, 1], [86, 1], [87, -1], [88, 1], [89, 1], [90, -1], [91, 1], [92, -1], [93, 1], [94, 1], [95, 1], [96, 1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126024,"parameters":{"prime":97,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0025","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={\"prime\":101}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(101, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, -1], [8, -1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, 1], [15, -1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, 1], [31, 1], [32, -1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, -1], [39, -1], [40, -1], [41, -1], [42, -1], [43, 1], [44, -1], [45, 1], [46, -1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, 1], [53, -1], [54, 1], [55, -1], [56, 1], [57, -1], [58, 1], [59, -1], [60, -1], [61, -1], [62, -1], [63, -1], [64, 1], [65, 1], [66, -1], [67, -1], [68, 1], [69, -1], [70, 1], [71, 1], [72, -1], [73, -1], [74, -1], [75, -1], [76, 1], [77, 1], [78, 1], [79, 1], [80, 1], [81, 1], [82, 1], [83, -1], [84, 1], [85, 1], [86, -1], [87, 1], [88, 1], [89, -1], [90, -1], [91, -1], [92, 1], [93, -1], [94, -1], [95, 1], [96, 1], [97, 1], [98, -1], [99, -1], [100, 1]], 是, zero_residue_nonresidue_partition_is_complete)","explanation_zh":"Wave6第六層核對=(101, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, -1], [8, -1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, 1], [15, -1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, 1], [31, 1], [32, -1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, -1], [39, -1], [40, -1], [41, -1], [42, -1], [43, 1], [44, -1], [45, 1], [46, -1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, 1], [53, -1], [54, 1], [55, -1], [56, 1], [57, -1], [58, 1], [59, -1], [60, -1], [61, -1], [62, -1], [63, -1], [64, 1], [65, 1], [66, -1], [67, -1], [68, 1], [69, -1], [70, 1], [71, 1], [72, -1], [73, -1], [74, -1], [75, -1], [76, 1], [77, 1], [78, 1], [79, 1], [80, 1], [81, 1], [82, 1], [83, -1], [84, 1], [85, 1], [86, -1], [87, 1], [88, 1], [89, -1], [90, -1], [91, -1], [92, 1], [93, -1], [94, -1], [95, 1], [96, 1], [97, 1], [98, -1], [99, -1], [100, 1]], 是, zero_residue_nonresidue_partition_is_complete)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-classification-table","seed":202611126025,"parameters":{"prime":101,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-classification-table","semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0026","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":3,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 1, [1, 2], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(3, 1, 1, [1, 2], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126026,"parameters":{"prime":3,"a":1,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":1,\"prime\":3}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0027","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":5,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 2, [], -1, complete_square_table_has_no_root)","explanation_zh":"Wave6第六層核對=(5, 2, 2, [], -1, complete_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126027,"parameters":{"prime":5,"a":2,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":2,\"prime\":5}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0028","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":7,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 3, 3, [], -1, complete_square_table_has_no_root)","explanation_zh":"Wave6第六層核對=(7, 3, 3, [], -1, complete_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126028,"parameters":{"prime":7,"a":3,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":3,\"prime\":7}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0029","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":11,\"a\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 4, 4, [2, 9], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(11, 4, 4, [2, 9], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126029,"parameters":{"prime":11,"a":4,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":4,\"prime\":11}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0030","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":13,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 5, 5, [], -1, complete_square_table_has_no_root)","explanation_zh":"Wave6第六層核對=(13, 5, 5, [], -1, complete_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126030,"parameters":{"prime":13,"a":5,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":5,\"prime\":13}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0031","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":17,\"a\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 6, 6, [], -1, complete_square_table_has_no_root)","explanation_zh":"Wave6第六層核對=(17, 6, 6, [], -1, complete_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126031,"parameters":{"prime":17,"a":6,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":6,\"prime\":17}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0032","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":19,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 7, 7, [8, 11], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(19, 7, 7, [8, 11], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126032,"parameters":{"prime":19,"a":7,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":7,\"prime\":19}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0033","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":23,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 8, 8, [10, 13], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(23, 8, 8, [10, 13], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126033,"parameters":{"prime":23,"a":8,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":8,\"prime\":23}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0034","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":29,\"a\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 9, 9, [3, 26], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(29, 9, 9, [3, 26], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126034,"parameters":{"prime":29,"a":9,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":9,\"prime\":29}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0035","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":31,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, 10, 10, [14, 17], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(31, 10, 10, [14, 17], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126035,"parameters":{"prime":31,"a":10,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":10,\"prime\":31}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0036","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":37,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(37, 11, 11, [14, 23], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(37, 11, 11, [14, 23], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126036,"parameters":{"prime":37,"a":11,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":11,\"prime\":37}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0037","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":41,\"a\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(41, 12, 12, [], -1, complete_square_table_has_no_root)","explanation_zh":"Wave6第六層核對=(41, 12, 12, [], -1, complete_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126037,"parameters":{"prime":41,"a":12,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":12,\"prime\":41}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0038","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":43,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(43, 13, 13, [20, 23], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(43, 13, 13, [20, 23], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126038,"parameters":{"prime":43,"a":13,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":13,\"prime\":43}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0039","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":47,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, 14, 14, [22, 25], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(47, 14, 14, [22, 25], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126039,"parameters":{"prime":47,"a":14,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":14,\"prime\":47}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0040","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":53,\"a\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(53, 15, 15, [11, 42], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(53, 15, 15, [11, 42], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126040,"parameters":{"prime":53,"a":15,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":15,\"prime\":53}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0041","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":59,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(59, 16, 16, [4, 55], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(59, 16, 16, [4, 55], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126041,"parameters":{"prime":59,"a":16,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":16,\"prime\":59}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0042","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":61,\"a\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(61, 17, 17, [], -1, complete_square_table_has_no_root)","explanation_zh":"Wave6第六層核對=(61, 17, 17, [], -1, complete_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126042,"parameters":{"prime":61,"a":17,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":17,\"prime\":61}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0043","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":67,\"a\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(67, 18, 18, [], -1, complete_square_table_has_no_root)","explanation_zh":"Wave6第六層核對=(67, 18, 18, [], -1, complete_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126043,"parameters":{"prime":67,"a":18,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":18,\"prime\":67}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0044","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":71,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(71, 19, 19, [27, 44], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(71, 19, 19, [27, 44], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126044,"parameters":{"prime":71,"a":19,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":19,\"prime\":71}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0045","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":73,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(73, 20, 20, [], -1, complete_square_table_has_no_root)","explanation_zh":"Wave6第六層核對=(73, 20, 20, [], -1, complete_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126045,"parameters":{"prime":73,"a":20,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":20,\"prime\":73}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0046","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":79,\"a\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(79, 21, 21, [10, 69], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(79, 21, 21, [10, 69], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126046,"parameters":{"prime":79,"a":21,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":21,\"prime\":79}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0047","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":83,\"a\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(83, 22, 22, [], -1, complete_square_table_has_no_root)","explanation_zh":"Wave6第六層核對=(83, 22, 22, [], -1, complete_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126047,"parameters":{"prime":83,"a":22,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":22,\"prime\":83}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0048","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":89,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(89, 23, 23, [], -1, complete_square_table_has_no_root)","explanation_zh":"Wave6第六層核對=(89, 23, 23, [], -1, complete_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126048,"parameters":{"prime":89,"a":23,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":23,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0049","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":97,\"a\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(97, 24, 24, [11, 86], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(97, 24, 24, [11, 86], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126049,"parameters":{"prime":97,"a":24,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":24,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0050","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-residue-nonresidue-witness；資料={\"prime\":101,\"a\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(101, 25, 25, [5, 96], 1, complete_nonzero_square_root_witness)","explanation_zh":"Wave6第六層核對=(101, 25, 25, [5, 96], 1, complete_nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126050,"parameters":{"prime":101,"a":25,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-residue-nonresidue-witness","semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":25,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0051","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":3,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(3, 3, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126051,"parameters":{"prime":3,"a":3,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":3,\"prime\":3}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0052","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":5,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 10, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(5, 10, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126052,"parameters":{"prime":5,"a":10,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":10,\"prime\":5}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0053","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":7,\"a\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 21, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(7, 21, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126053,"parameters":{"prime":7,"a":21,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":21,\"prime\":7}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0054","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":11,\"a\":44}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 44, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(11, 44, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126054,"parameters":{"prime":11,"a":44,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":44,\"prime\":11}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0055","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":13,\"a\":65}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 65, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(13, 65, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126055,"parameters":{"prime":13,"a":65,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":65,\"prime\":13}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0056","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":17,\"a\":102}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 102, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(17, 102, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126056,"parameters":{"prime":17,"a":102,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":102,\"prime\":17}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0057","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":19,\"a\":133}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 133, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(19, 133, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126057,"parameters":{"prime":19,"a":133,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":133,\"prime\":19}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0058","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":23,\"a\":184}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 184, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(23, 184, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126058,"parameters":{"prime":23,"a":184,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":184,\"prime\":23}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0059","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":29,\"a\":261}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 261, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(29, 261, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126059,"parameters":{"prime":29,"a":261,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":261,\"prime\":29}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0060","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":31,\"a\":310}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, 310, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(31, 310, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126060,"parameters":{"prime":31,"a":310,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":310,\"prime\":31}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0061","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":37,\"a\":407}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(37, 407, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(37, 407, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126061,"parameters":{"prime":37,"a":407,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":407,\"prime\":37}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0062","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":41,\"a\":492}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(41, 492, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(41, 492, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126062,"parameters":{"prime":41,"a":492,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":492,\"prime\":41}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0063","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":43,\"a\":559}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(43, 559, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(43, 559, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126063,"parameters":{"prime":43,"a":559,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":559,\"prime\":43}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0064","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":47,\"a\":658}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, 658, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(47, 658, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126064,"parameters":{"prime":47,"a":658,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":658,\"prime\":47}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0065","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":53,\"a\":795}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(53, 795, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(53, 795, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126065,"parameters":{"prime":53,"a":795,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":795,\"prime\":53}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0066","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":59,\"a\":944}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(59, 944, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(59, 944, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126066,"parameters":{"prime":59,"a":944,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":944,\"prime\":59}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0067","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":61,\"a\":1037}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(61, 1037, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(61, 1037, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126067,"parameters":{"prime":61,"a":1037,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":1037,\"prime\":61}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0068","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":67,\"a\":1206}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(67, 1206, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(67, 1206, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126068,"parameters":{"prime":67,"a":1206,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":1206,\"prime\":67}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0069","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":71,\"a\":1349}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(71, 1349, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(71, 1349, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126069,"parameters":{"prime":71,"a":1349,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":1349,\"prime\":71}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0070","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":73,\"a\":1460}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(73, 1460, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(73, 1460, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126070,"parameters":{"prime":73,"a":1460,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":1460,\"prime\":73}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0071","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":79,\"a\":1659}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(79, 1659, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(79, 1659, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126071,"parameters":{"prime":79,"a":1659,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":1659,\"prime\":79}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0072","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":83,\"a\":1826}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(83, 1826, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(83, 1826, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126072,"parameters":{"prime":83,"a":1826,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":1826,\"prime\":83}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0073","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":89,\"a\":2047}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(89, 2047, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(89, 2047, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126073,"parameters":{"prime":89,"a":2047,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":2047,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0074","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":97,\"a\":2328}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(97, 2328, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(97, 2328, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126074,"parameters":{"prime":97,"a":2328,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":2328,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0075","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-zero-class；資料={\"prime\":101,\"a\":2525}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(101, 2525, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation_zh":"Wave6第六層核對=(101, 2525, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-zero-class","seed":202611126075,"parameters":{"prime":101,"a":2525,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-zero-class","semantic_template":"qf-mod43-legendre-zero-class/{\"a\":2525,\"prime\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0076","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(even_prime, 2, 1, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126076,"parameters":{"mode":"even_prime","modulus":2,"a":1,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"even_prime\",\"modulus\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0077","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":9,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 9, 2, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(composite_modulus, 9, 2, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126077,"parameters":{"mode":"composite_modulus","modulus":9,"a":2,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":2,\"mode\":\"composite_modulus\",\"modulus\":9}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0078","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"premature_euler\",\"modulus\":7,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(premature_euler, 7, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation_zh":"Wave6第六層核對=(premature_euler, 7, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126078,"parameters":{"mode":"premature_euler","modulus":7,"a":1,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":7}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0079","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 0, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(even_prime, 2, 0, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126079,"parameters":{"mode":"even_prime","modulus":2,"a":4,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":4,\"mode\":\"even_prime\",\"modulus\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0080","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":15,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 15, 5, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(composite_modulus, 15, 5, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126080,"parameters":{"mode":"composite_modulus","modulus":15,"a":5,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":5,\"mode\":\"composite_modulus\",\"modulus\":15}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0081","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"premature_euler\",\"modulus\":17,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(premature_euler, 17, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation_zh":"Wave6第六層核對=(premature_euler, 17, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126081,"parameters":{"mode":"premature_euler","modulus":17,"a":1,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":17}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0082","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(even_prime, 2, 1, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126082,"parameters":{"mode":"even_prime","modulus":2,"a":7,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":7,\"mode\":\"even_prime\",\"modulus\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0083","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":21,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 21, 8, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(composite_modulus, 21, 8, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126083,"parameters":{"mode":"composite_modulus","modulus":21,"a":8,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":8,\"mode\":\"composite_modulus\",\"modulus\":21}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0084","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"premature_euler\",\"modulus\":29,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(premature_euler, 29, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation_zh":"Wave6第六層核對=(premature_euler, 29, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126084,"parameters":{"mode":"premature_euler","modulus":29,"a":1,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":29}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0085","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 0, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(even_prime, 2, 0, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126085,"parameters":{"mode":"even_prime","modulus":2,"a":10,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":10,\"mode\":\"even_prime\",\"modulus\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0086","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":27,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 27, 11, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(composite_modulus, 27, 11, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126086,"parameters":{"mode":"composite_modulus","modulus":27,"a":11,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":11,\"mode\":\"composite_modulus\",\"modulus\":27}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0087","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"premature_euler\",\"modulus\":41,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(premature_euler, 41, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation_zh":"Wave6第六層核對=(premature_euler, 41, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126087,"parameters":{"mode":"premature_euler","modulus":41,"a":1,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":41}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0088","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(even_prime, 2, 1, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126088,"parameters":{"mode":"even_prime","modulus":2,"a":13,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":13,\"mode\":\"even_prime\",\"modulus\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0089","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":33,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 33, 14, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(composite_modulus, 33, 14, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126089,"parameters":{"mode":"composite_modulus","modulus":33,"a":14,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":14,\"mode\":\"composite_modulus\",\"modulus\":33}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0090","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"premature_euler\",\"modulus\":53,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(premature_euler, 53, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation_zh":"Wave6第六層核對=(premature_euler, 53, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126090,"parameters":{"mode":"premature_euler","modulus":53,"a":1,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":53}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0091","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 0, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(even_prime, 2, 0, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126091,"parameters":{"mode":"even_prime","modulus":2,"a":16,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":16,\"mode\":\"even_prime\",\"modulus\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0092","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":39,\"a\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 39, 17, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(composite_modulus, 39, 17, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126092,"parameters":{"mode":"composite_modulus","modulus":39,"a":17,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":17,\"mode\":\"composite_modulus\",\"modulus\":39}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0093","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"premature_euler\",\"modulus\":67,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(premature_euler, 67, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation_zh":"Wave6第六層核對=(premature_euler, 67, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126093,"parameters":{"mode":"premature_euler","modulus":67,"a":1,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":67}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0094","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(even_prime, 2, 1, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126094,"parameters":{"mode":"even_prime","modulus":2,"a":19,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":19,\"mode\":\"even_prime\",\"modulus\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0095","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":45,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 45, 20, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(composite_modulus, 45, 20, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126095,"parameters":{"mode":"composite_modulus","modulus":45,"a":20,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":20,\"mode\":\"composite_modulus\",\"modulus\":45}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0096","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"premature_euler\",\"modulus\":79,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(premature_euler, 79, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation_zh":"Wave6第六層核對=(premature_euler, 79, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126096,"parameters":{"mode":"premature_euler","modulus":79,"a":1,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":79}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0097","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 0, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(even_prime, 2, 0, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126097,"parameters":{"mode":"even_prime","modulus":2,"a":22,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":22,\"mode\":\"even_prime\",\"modulus\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0098","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":51,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 51, 23, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(composite_modulus, 51, 23, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126098,"parameters":{"mode":"composite_modulus","modulus":51,"a":23,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":23,\"mode\":\"composite_modulus\",\"modulus\":51}"}},"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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0099","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"premature_euler\",\"modulus\":97,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(premature_euler, 97, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation_zh":"Wave6第六層核對=(premature_euler, 97, 1, 否, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126099,"parameters":{"mode":"premature_euler","modulus":97,"a":1,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、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-20261112-r06-legendre-symbol-0100","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","language":"zh-Hant","source_mko_ids":["mko-legendre-symbol"],"title_zh":"Legendre符號","stem_zh":"odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-domain-method-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation_zh":"Wave6第六層核對=(even_prime, 2, 1, 否, Legendre_symbol_domain_requires_an_odd_prime_modulus)；依完整明示資料獨立重算；a先normalize modulo p且zero class輸出0；nonzero classification由完整square table取得；p=2與composite modulus不在domain，且不得提前使用Euler criterion。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Legendre符號"]},"variant":{"family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126100,"parameters":{"mode":"even_prime","modulus":2,"a":25,"domain":"Odd_prime_Legendre_symbol","operation":"legendre-domain-method-boundary","semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":25,\"mode\":\"even_prime\",\"modulus\":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、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}