{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261111-r06-quadratic-residue-0001","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, 1], [2, 1]], [1], 1, 1, 是)","explanation_zh":"Wave6第五層核對=(3, [[1, 1], [2, 1]], [1], 1, 1, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116001,"parameters":{"p":3,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0002","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[1, 1], [2, 4], [3, 4], [4, 1]], [1, 4], 2, 2, 是)","explanation_zh":"Wave6第五層核對=(5, [[1, 1], [2, 4], [3, 4], [4, 1]], [1, 4], 2, 2, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116002,"parameters":{"p":5,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0003","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, [[1, 1], [2, 4], [3, 2], [4, 2], [5, 4], [6, 1]], [1, 2, 4], 3, 3, 是)","explanation_zh":"Wave6第五層核對=(7, [[1, 1], [2, 4], [3, 2], [4, 2], [5, 4], [6, 1]], [1, 2, 4], 3, 3, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116003,"parameters":{"p":7,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0004","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, [[1, 1], [2, 4], [3, 9], [4, 5], [5, 3], [6, 3], [7, 5], [8, 9], [9, 4], [10, 1]], [1, 3, 4, 5, 9], 5, 5, 是)","explanation_zh":"Wave6第五層核對=(11, [[1, 1], [2, 4], [3, 9], [4, 5], [5, 3], [6, 3], [7, 5], [8, 9], [9, 4], [10, 1]], [1, 3, 4, 5, 9], 5, 5, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116004,"parameters":{"p":11,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0005","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, [[1, 1], [2, 4], [3, 9], [4, 3], [5, 12], [6, 10], [7, 10], [8, 12], [9, 3], [10, 9], [11, 4], [12, 1]], [1, 3, 4, 9, 10, 12], 6, 6, 是)","explanation_zh":"Wave6第五層核對=(13, [[1, 1], [2, 4], [3, 9], [4, 3], [5, 12], [6, 10], [7, 10], [8, 12], [9, 3], [10, 9], [11, 4], [12, 1]], [1, 3, 4, 9, 10, 12], 6, 6, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116005,"parameters":{"p":13,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0006","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 8], [6, 2], [7, 15], [8, 13], [9, 13], [10, 15], [11, 2], [12, 8], [13, 16], [14, 9], [15, 4], [16, 1]], [1, 2, 4, 8, 9, 13, 15, 16], 8, 8, 是)","explanation_zh":"Wave6第五層核對=(17, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 8], [6, 2], [7, 15], [8, 13], [9, 13], [10, 15], [11, 2], [12, 8], [13, 16], [14, 9], [15, 4], [16, 1]], [1, 2, 4, 8, 9, 13, 15, 16], 8, 8, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116006,"parameters":{"p":17,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0007","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 6], [6, 17], [7, 11], [8, 7], [9, 5], [10, 5], [11, 7], [12, 11], [13, 17], [14, 6], [15, 16], [16, 9], [17, 4], [18, 1]], [1, 4, 5, 6, 7, 9, 11, 16, 17], 9, 9, 是)","explanation_zh":"Wave6第五層核對=(19, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 6], [6, 17], [7, 11], [8, 7], [9, 5], [10, 5], [11, 7], [12, 11], [13, 17], [14, 6], [15, 16], [16, 9], [17, 4], [18, 1]], [1, 4, 5, 6, 7, 9, 11, 16, 17], 9, 9, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116007,"parameters":{"p":19,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0008","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 2], [6, 13], [7, 3], [8, 18], [9, 12], [10, 8], [11, 6], [12, 6], [13, 8], [14, 12], [15, 18], [16, 3], [17, 13], [18, 2], [19, 16], [20, 9], [21, 4], [22, 1]], [1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18], 11, 11, 是)","explanation_zh":"Wave6第五層核對=(23, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 2], [6, 13], [7, 3], [8, 18], [9, 12], [10, 8], [11, 6], [12, 6], [13, 8], [14, 12], [15, 18], [16, 3], [17, 13], [18, 2], [19, 16], [20, 9], [21, 4], [22, 1]], [1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18], 11, 11, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116008,"parameters":{"p":23,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0009","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":29}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 7], [7, 20], [8, 6], [9, 23], [10, 13], [11, 5], [12, 28], [13, 24], [14, 22], [15, 22], [16, 24], [17, 28], [18, 5], [19, 13], [20, 23], [21, 6], [22, 20], [23, 7], [24, 25], [25, 16], [26, 9], [27, 4], [28, 1]], [1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28], 14, 14, 是)","explanation_zh":"Wave6第五層核對=(29, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 7], [7, 20], [8, 6], [9, 23], [10, 13], [11, 5], [12, 28], [13, 24], [14, 22], [15, 22], [16, 24], [17, 28], [18, 5], [19, 13], [20, 23], [21, 6], [22, 20], [23, 7], [24, 25], [25, 16], [26, 9], [27, 4], [28, 1]], [1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28], 14, 14, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116009,"parameters":{"p":29,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0010","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":31}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 5], [7, 18], [8, 2], [9, 19], [10, 7], [11, 28], [12, 20], [13, 14], [14, 10], [15, 8], [16, 8], [17, 10], [18, 14], [19, 20], [20, 28], [21, 7], [22, 19], [23, 2], [24, 18], [25, 5], [26, 25], [27, 16], [28, 9], [29, 4], [30, 1]], [1, 2, 4, 5, 7, 8, 9, 10, 14, 16, 18, 19, 20, 25, 28], 15, 15, 是)","explanation_zh":"Wave6第五層核對=(31, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 5], [7, 18], [8, 2], [9, 19], [10, 7], [11, 28], [12, 20], [13, 14], [14, 10], [15, 8], [16, 8], [17, 10], [18, 14], [19, 20], [20, 28], [21, 7], [22, 19], [23, 2], [24, 18], [25, 5], [26, 25], [27, 16], [28, 9], [29, 4], [30, 1]], [1, 2, 4, 5, 7, 8, 9, 10, 14, 16, 18, 19, 20, 25, 28], 15, 15, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116010,"parameters":{"p":31,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0011","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":37}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(37, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 12], [8, 27], [9, 7], [10, 26], [11, 10], [12, 33], [13, 21], [14, 11], [15, 3], [16, 34], [17, 30], [18, 28], [19, 28], [20, 30], [21, 34], [22, 3], [23, 11], [24, 21], [25, 33], [26, 10], [27, 26], [28, 7], [29, 27], [30, 12], [31, 36], [32, 25], [33, 16], [34, 9], [35, 4], [36, 1]], [1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26, 27, 28, 30, 33, 34, 36], 18, 18, 是)","explanation_zh":"Wave6第五層核對=(37, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 12], [8, 27], [9, 7], [10, 26], [11, 10], [12, 33], [13, 21], [14, 11], [15, 3], [16, 34], [17, 30], [18, 28], [19, 28], [20, 30], [21, 34], [22, 3], [23, 11], [24, 21], [25, 33], [26, 10], [27, 26], [28, 7], [29, 27], [30, 12], [31, 36], [32, 25], [33, 16], [34, 9], [35, 4], [36, 1]], [1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26, 27, 28, 30, 33, 34, 36], 18, 18, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116011,"parameters":{"p":37,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0012","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":41}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(41, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 8], [8, 23], [9, 40], [10, 18], [11, 39], [12, 21], [13, 5], [14, 32], [15, 20], [16, 10], [17, 2], [18, 37], [19, 33], [20, 31], [21, 31], [22, 33], [23, 37], [24, 2], [25, 10], [26, 20], [27, 32], [28, 5], [29, 21], [30, 39], [31, 18], [32, 40], [33, 23], [34, 8], [35, 36], [36, 25], [37, 16], [38, 9], [39, 4], [40, 1]], [1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40], 20, 20, 是)","explanation_zh":"Wave6第五層核對=(41, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 8], [8, 23], [9, 40], [10, 18], [11, 39], [12, 21], [13, 5], [14, 32], [15, 20], [16, 10], [17, 2], [18, 37], [19, 33], [20, 31], [21, 31], [22, 33], [23, 37], [24, 2], [25, 10], [26, 20], [27, 32], [28, 5], [29, 21], [30, 39], [31, 18], [32, 40], [33, 23], [34, 8], [35, 36], [36, 25], [37, 16], [38, 9], [39, 4], [40, 1]], [1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40], 20, 20, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116012,"parameters":{"p":41,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0013","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":43}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(43, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 6], [8, 21], [9, 38], [10, 14], [11, 35], [12, 15], [13, 40], [14, 24], [15, 10], [16, 41], [17, 31], [18, 23], [19, 17], [20, 13], [21, 11], [22, 11], [23, 13], [24, 17], [25, 23], [26, 31], [27, 41], [28, 10], [29, 24], [30, 40], [31, 15], [32, 35], [33, 14], [34, 38], [35, 21], [36, 6], [37, 36], [38, 25], [39, 16], [40, 9], [41, 4], [42, 1]], [1, 4, 6, 9, 10, 11, 13, 14, 15, 16, 17, 21, 23, 24, 25, 31, 35, 36, 38, 40, 41], 21, 21, 是)","explanation_zh":"Wave6第五層核對=(43, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 6], [8, 21], [9, 38], [10, 14], [11, 35], [12, 15], [13, 40], [14, 24], [15, 10], [16, 41], [17, 31], [18, 23], [19, 17], [20, 13], [21, 11], [22, 11], [23, 13], [24, 17], [25, 23], [26, 31], [27, 41], [28, 10], [29, 24], [30, 40], [31, 15], [32, 35], [33, 14], [34, 38], [35, 21], [36, 6], [37, 36], [38, 25], [39, 16], [40, 9], [41, 4], [42, 1]], [1, 4, 6, 9, 10, 11, 13, 14, 15, 16, 17, 21, 23, 24, 25, 31, 35, 36, 38, 40, 41], 21, 21, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116013,"parameters":{"p":43,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0014","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":47}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 2], [8, 17], [9, 34], [10, 6], [11, 27], [12, 3], [13, 28], [14, 8], [15, 37], [16, 21], [17, 7], [18, 42], [19, 32], [20, 24], [21, 18], [22, 14], [23, 12], [24, 12], [25, 14], [26, 18], [27, 24], [28, 32], [29, 42], [30, 7], [31, 21], [32, 37], [33, 8], [34, 28], [35, 3], [36, 27], [37, 6], [38, 34], [39, 17], [40, 2], [41, 36], [42, 25], [43, 16], [44, 9], [45, 4], [46, 1]], [1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 17, 18, 21, 24, 25, 27, 28, 32, 34, 36, 37, 42], 23, 23, 是)","explanation_zh":"Wave6第五層核對=(47, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 2], [8, 17], [9, 34], [10, 6], [11, 27], [12, 3], [13, 28], [14, 8], [15, 37], [16, 21], [17, 7], [18, 42], [19, 32], [20, 24], [21, 18], [22, 14], [23, 12], [24, 12], [25, 14], [26, 18], [27, 24], [28, 32], [29, 42], [30, 7], [31, 21], [32, 37], [33, 8], [34, 28], [35, 3], [36, 27], [37, 6], [38, 34], [39, 17], [40, 2], [41, 36], [42, 25], [43, 16], [44, 9], [45, 4], [46, 1]], [1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 17, 18, 21, 24, 25, 27, 28, 32, 34, 36, 37, 42], 23, 23, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116014,"parameters":{"p":47,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0015","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":53}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(53, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 11], [9, 28], [10, 47], [11, 15], [12, 38], [13, 10], [14, 37], [15, 13], [16, 44], [17, 24], [18, 6], [19, 43], [20, 29], [21, 17], [22, 7], [23, 52], [24, 46], [25, 42], [26, 40], [27, 40], [28, 42], [29, 46], [30, 52], [31, 7], [32, 17], [33, 29], [34, 43], [35, 6], [36, 24], [37, 44], [38, 13], [39, 37], [40, 10], [41, 38], [42, 15], [43, 47], [44, 28], [45, 11], [46, 49], [47, 36], [48, 25], [49, 16], [50, 9], [51, 4], [52, 1]], [1, 4, 6, 7, 9, 10, 11, 13, 15, 16, 17, 24, 25, 28, 29, 36, 37, 38, 40, 42, 43, 44, 46, 47, 49, 52], 26, 26, 是)","explanation_zh":"Wave6第五層核對=(53, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 11], [9, 28], [10, 47], [11, 15], [12, 38], [13, 10], [14, 37], [15, 13], [16, 44], [17, 24], [18, 6], [19, 43], [20, 29], [21, 17], [22, 7], [23, 52], [24, 46], [25, 42], [26, 40], [27, 40], [28, 42], [29, 46], [30, 52], [31, 7], [32, 17], [33, 29], [34, 43], [35, 6], [36, 24], [37, 44], [38, 13], [39, 37], [40, 10], [41, 38], [42, 15], [43, 47], [44, 28], [45, 11], [46, 49], [47, 36], [48, 25], [49, 16], [50, 9], [51, 4], [52, 1]], [1, 4, 6, 7, 9, 10, 11, 13, 15, 16, 17, 24, 25, 28, 29, 36, 37, 38, 40, 42, 43, 44, 46, 47, 49, 52], 26, 26, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116015,"parameters":{"p":53,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0016","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":59}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(59, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 5], [9, 22], [10, 41], [11, 3], [12, 26], [13, 51], [14, 19], [15, 48], [16, 20], [17, 53], [18, 29], [19, 7], [20, 46], [21, 28], [22, 12], [23, 57], [24, 45], [25, 35], [26, 27], [27, 21], [28, 17], [29, 15], [30, 15], [31, 17], [32, 21], [33, 27], [34, 35], [35, 45], [36, 57], [37, 12], [38, 28], [39, 46], [40, 7], [41, 29], [42, 53], [43, 20], [44, 48], [45, 19], [46, 51], [47, 26], [48, 3], [49, 41], [50, 22], [51, 5], [52, 49], [53, 36], [54, 25], [55, 16], [56, 9], [57, 4], [58, 1]], [1, 3, 4, 5, 7, 9, 12, 15, 16, 17, 19, 20, 21, 22, 25, 26, 27, 28, 29, 35, 36, 41, 45, 46, 48, 49, 51, 53, 57], 29, 29, 是)","explanation_zh":"Wave6第五層核對=(59, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 5], [9, 22], [10, 41], [11, 3], [12, 26], [13, 51], [14, 19], [15, 48], [16, 20], [17, 53], [18, 29], [19, 7], [20, 46], [21, 28], [22, 12], [23, 57], [24, 45], [25, 35], [26, 27], [27, 21], [28, 17], [29, 15], [30, 15], [31, 17], [32, 21], [33, 27], [34, 35], [35, 45], [36, 57], [37, 12], [38, 28], [39, 46], [40, 7], [41, 29], [42, 53], [43, 20], [44, 48], [45, 19], [46, 51], [47, 26], [48, 3], [49, 41], [50, 22], [51, 5], [52, 49], [53, 36], [54, 25], [55, 16], [56, 9], [57, 4], [58, 1]], [1, 3, 4, 5, 7, 9, 12, 15, 16, 17, 19, 20, 21, 22, 25, 26, 27, 28, 29, 35, 36, 41, 45, 46, 48, 49, 51, 53, 57], 29, 29, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116016,"parameters":{"p":59,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0017","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":61}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(61, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 3], [9, 20], [10, 39], [11, 60], [12, 22], [13, 47], [14, 13], [15, 42], [16, 12], [17, 45], [18, 19], [19, 56], [20, 34], [21, 14], [22, 57], [23, 41], [24, 27], [25, 15], [26, 5], [27, 58], [28, 52], [29, 48], [30, 46], [31, 46], [32, 48], [33, 52], [34, 58], [35, 5], [36, 15], [37, 27], [38, 41], [39, 57], [40, 14], [41, 34], [42, 56], [43, 19], [44, 45], [45, 12], [46, 42], [47, 13], [48, 47], [49, 22], [50, 60], [51, 39], [52, 20], [53, 3], [54, 49], [55, 36], [56, 25], [57, 16], [58, 9], [59, 4], [60, 1]], [1, 3, 4, 5, 9, 12, 13, 14, 15, 16, 19, 20, 22, 25, 27, 34, 36, 39, 41, 42, 45, 46, 47, 48, 49, 52, 56, 57, 58, 60], 30, 30, 是)","explanation_zh":"Wave6第五層核對=(61, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 3], [9, 20], [10, 39], [11, 60], [12, 22], [13, 47], [14, 13], [15, 42], [16, 12], [17, 45], [18, 19], [19, 56], [20, 34], [21, 14], [22, 57], [23, 41], [24, 27], [25, 15], [26, 5], [27, 58], [28, 52], [29, 48], [30, 46], [31, 46], [32, 48], [33, 52], [34, 58], [35, 5], [36, 15], [37, 27], [38, 41], [39, 57], [40, 14], [41, 34], [42, 56], [43, 19], [44, 45], [45, 12], [46, 42], [47, 13], [48, 47], [49, 22], [50, 60], [51, 39], [52, 20], [53, 3], [54, 49], [55, 36], [56, 25], [57, 16], [58, 9], [59, 4], [60, 1]], [1, 3, 4, 5, 9, 12, 13, 14, 15, 16, 19, 20, 22, 25, 27, 34, 36, 39, 41, 42, 45, 46, 47, 48, 49, 52, 56, 57, 58, 60], 30, 30, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116017,"parameters":{"p":61,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0018","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":67}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(67, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 14], [10, 33], [11, 54], [12, 10], [13, 35], [14, 62], [15, 24], [16, 55], [17, 21], [18, 56], [19, 26], [20, 65], [21, 39], [22, 15], [23, 60], [24, 40], [25, 22], [26, 6], [27, 59], [28, 47], [29, 37], [30, 29], [31, 23], [32, 19], [33, 17], [34, 17], [35, 19], [36, 23], [37, 29], [38, 37], [39, 47], [40, 59], [41, 6], [42, 22], [43, 40], [44, 60], [45, 15], [46, 39], [47, 65], [48, 26], [49, 56], [50, 21], [51, 55], [52, 24], [53, 62], [54, 35], [55, 10], [56, 54], [57, 33], [58, 14], [59, 64], [60, 49], [61, 36], [62, 25], [63, 16], [64, 9], [65, 4], [66, 1]], [1, 4, 6, 9, 10, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 29, 33, 35, 36, 37, 39, 40, 47, 49, 54, 55, 56, 59, 60, 62, 64, 65], 33, 33, 是)","explanation_zh":"Wave6第五層核對=(67, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 14], [10, 33], [11, 54], [12, 10], [13, 35], [14, 62], [15, 24], [16, 55], [17, 21], [18, 56], [19, 26], [20, 65], [21, 39], [22, 15], [23, 60], [24, 40], [25, 22], [26, 6], [27, 59], [28, 47], [29, 37], [30, 29], [31, 23], [32, 19], [33, 17], [34, 17], [35, 19], [36, 23], [37, 29], [38, 37], [39, 47], [40, 59], [41, 6], [42, 22], [43, 40], [44, 60], [45, 15], [46, 39], [47, 65], [48, 26], [49, 56], [50, 21], [51, 55], [52, 24], [53, 62], [54, 35], [55, 10], [56, 54], [57, 33], [58, 14], [59, 64], [60, 49], [61, 36], [62, 25], [63, 16], [64, 9], [65, 4], [66, 1]], [1, 4, 6, 9, 10, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 29, 33, 35, 36, 37, 39, 40, 47, 49, 54, 55, 56, 59, 60, 62, 64, 65], 33, 33, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116018,"parameters":{"p":67,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0019","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":71}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(71, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 10], [10, 29], [11, 50], [12, 2], [13, 27], [14, 54], [15, 12], [16, 43], [17, 5], [18, 40], [19, 6], [20, 45], [21, 15], [22, 58], [23, 32], [24, 8], [25, 57], [26, 37], [27, 19], [28, 3], [29, 60], [30, 48], [31, 38], [32, 30], [33, 24], [34, 20], [35, 18], [36, 18], [37, 20], [38, 24], [39, 30], [40, 38], [41, 48], [42, 60], [43, 3], [44, 19], [45, 37], [46, 57], [47, 8], [48, 32], [49, 58], [50, 15], [51, 45], [52, 6], [53, 40], [54, 5], [55, 43], [56, 12], [57, 54], [58, 27], [59, 2], [60, 50], [61, 29], [62, 10], [63, 64], [64, 49], [65, 36], [66, 25], [67, 16], [68, 9], [69, 4], [70, 1]], [1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 19, 20, 24, 25, 27, 29, 30, 32, 36, 37, 38, 40, 43, 45, 48, 49, 50, 54, 57, 58, 60, 64], 35, 35, 是)","explanation_zh":"Wave6第五層核對=(71, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 10], [10, 29], [11, 50], [12, 2], [13, 27], [14, 54], [15, 12], [16, 43], [17, 5], [18, 40], [19, 6], [20, 45], [21, 15], [22, 58], [23, 32], [24, 8], [25, 57], [26, 37], [27, 19], [28, 3], [29, 60], [30, 48], [31, 38], [32, 30], [33, 24], [34, 20], [35, 18], [36, 18], [37, 20], [38, 24], [39, 30], [40, 38], [41, 48], [42, 60], [43, 3], [44, 19], [45, 37], [46, 57], [47, 8], [48, 32], [49, 58], [50, 15], [51, 45], [52, 6], [53, 40], [54, 5], [55, 43], [56, 12], [57, 54], [58, 27], [59, 2], [60, 50], [61, 29], [62, 10], [63, 64], [64, 49], [65, 36], [66, 25], [67, 16], [68, 9], [69, 4], [70, 1]], [1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 19, 20, 24, 25, 27, 29, 30, 32, 36, 37, 38, 40, 43, 45, 48, 49, 50, 54, 57, 58, 60, 64], 35, 35, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116019,"parameters":{"p":71,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0020","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":73}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(73, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 8], [10, 27], [11, 48], [12, 71], [13, 23], [14, 50], [15, 6], [16, 37], [17, 70], [18, 32], [19, 69], [20, 35], [21, 3], [22, 46], [23, 18], [24, 65], [25, 41], [26, 19], [27, 72], [28, 54], [29, 38], [30, 24], [31, 12], [32, 2], [33, 67], [34, 61], [35, 57], [36, 55], [37, 55], [38, 57], [39, 61], [40, 67], [41, 2], [42, 12], [43, 24], [44, 38], [45, 54], [46, 72], [47, 19], [48, 41], [49, 65], [50, 18], [51, 46], [52, 3], [53, 35], [54, 69], [55, 32], [56, 70], [57, 37], [58, 6], [59, 50], [60, 23], [61, 71], [62, 48], [63, 27], [64, 8], [65, 64], [66, 49], [67, 36], [68, 25], [69, 16], [70, 9], [71, 4], [72, 1]], [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 19, 23, 24, 25, 27, 32, 35, 36, 37, 38, 41, 46, 48, 49, 50, 54, 55, 57, 61, 64, 65, 67, 69, 70, 71, 72], 36, 36, 是)","explanation_zh":"Wave6第五層核對=(73, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 8], [10, 27], [11, 48], [12, 71], [13, 23], [14, 50], [15, 6], [16, 37], [17, 70], [18, 32], [19, 69], [20, 35], [21, 3], [22, 46], [23, 18], [24, 65], [25, 41], [26, 19], [27, 72], [28, 54], [29, 38], [30, 24], [31, 12], [32, 2], [33, 67], [34, 61], [35, 57], [36, 55], [37, 55], [38, 57], [39, 61], [40, 67], [41, 2], [42, 12], [43, 24], [44, 38], [45, 54], [46, 72], [47, 19], [48, 41], [49, 65], [50, 18], [51, 46], [52, 3], [53, 35], [54, 69], [55, 32], [56, 70], [57, 37], [58, 6], [59, 50], [60, 23], [61, 71], [62, 48], [63, 27], [64, 8], [65, 64], [66, 49], [67, 36], [68, 25], [69, 16], [70, 9], [71, 4], [72, 1]], [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 19, 23, 24, 25, 27, 32, 35, 36, 37, 38, 41, 46, 48, 49, 50, 54, 55, 57, 61, 64, 65, 67, 69, 70, 71, 72], 36, 36, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116020,"parameters":{"p":73,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0021","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":79}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(79, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 2], [10, 21], [11, 42], [12, 65], [13, 11], [14, 38], [15, 67], [16, 19], [17, 52], [18, 8], [19, 45], [20, 5], [21, 46], [22, 10], [23, 55], [24, 23], [25, 72], [26, 44], [27, 18], [28, 73], [29, 51], [30, 31], [31, 13], [32, 76], [33, 62], [34, 50], [35, 40], [36, 32], [37, 26], [38, 22], [39, 20], [40, 20], [41, 22], [42, 26], [43, 32], [44, 40], [45, 50], [46, 62], [47, 76], [48, 13], [49, 31], [50, 51], [51, 73], [52, 18], [53, 44], [54, 72], [55, 23], [56, 55], [57, 10], [58, 46], [59, 5], [60, 45], [61, 8], [62, 52], [63, 19], [64, 67], [65, 38], [66, 11], [67, 65], [68, 42], [69, 21], [70, 2], [71, 64], [72, 49], [73, 36], [74, 25], [75, 16], [76, 9], [77, 4], [78, 1]], [1, 2, 4, 5, 8, 9, 10, 11, 13, 16, 18, 19, 20, 21, 22, 23, 25, 26, 31, 32, 36, 38, 40, 42, 44, 45, 46, 49, 50, 51, 52, 55, 62, 64, 65, 67, 72, 73, 76], 39, 39, 是)","explanation_zh":"Wave6第五層核對=(79, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 2], [10, 21], [11, 42], [12, 65], [13, 11], [14, 38], [15, 67], [16, 19], [17, 52], [18, 8], [19, 45], [20, 5], [21, 46], [22, 10], [23, 55], [24, 23], [25, 72], [26, 44], [27, 18], [28, 73], [29, 51], [30, 31], [31, 13], [32, 76], [33, 62], [34, 50], [35, 40], [36, 32], [37, 26], [38, 22], [39, 20], [40, 20], [41, 22], [42, 26], [43, 32], [44, 40], [45, 50], [46, 62], [47, 76], [48, 13], [49, 31], [50, 51], [51, 73], [52, 18], [53, 44], [54, 72], [55, 23], [56, 55], [57, 10], [58, 46], [59, 5], [60, 45], [61, 8], [62, 52], [63, 19], [64, 67], [65, 38], [66, 11], [67, 65], [68, 42], [69, 21], [70, 2], [71, 64], [72, 49], [73, 36], [74, 25], [75, 16], [76, 9], [77, 4], [78, 1]], [1, 2, 4, 5, 8, 9, 10, 11, 13, 16, 18, 19, 20, 21, 22, 23, 25, 26, 31, 32, 36, 38, 40, 42, 44, 45, 46, 49, 50, 51, 52, 55, 62, 64, 65, 67, 72, 73, 76], 39, 39, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116021,"parameters":{"p":79,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0022","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":83}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(83, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 17], [11, 38], [12, 61], [13, 3], [14, 30], [15, 59], [16, 7], [17, 40], [18, 75], [19, 29], [20, 68], [21, 26], [22, 69], [23, 31], [24, 78], [25, 44], [26, 12], [27, 65], [28, 37], [29, 11], [30, 70], [31, 48], [32, 28], [33, 10], [34, 77], [35, 63], [36, 51], [37, 41], [38, 33], [39, 27], [40, 23], [41, 21], [42, 21], [43, 23], [44, 27], [45, 33], [46, 41], [47, 51], [48, 63], [49, 77], [50, 10], [51, 28], [52, 48], [53, 70], [54, 11], [55, 37], [56, 65], [57, 12], [58, 44], [59, 78], [60, 31], [61, 69], [62, 26], [63, 68], [64, 29], [65, 75], [66, 40], [67, 7], [68, 59], [69, 30], [70, 3], [71, 61], [72, 38], [73, 17], [74, 81], [75, 64], [76, 49], [77, 36], [78, 25], [79, 16], [80, 9], [81, 4], [82, 1]], [1, 3, 4, 7, 9, 10, 11, 12, 16, 17, 21, 23, 25, 26, 27, 28, 29, 30, 31, 33, 36, 37, 38, 40, 41, 44, 48, 49, 51, 59, 61, 63, 64, 65, 68, 69, 70, 75, 77, 78, 81], 41, 41, 是)","explanation_zh":"Wave6第五層核對=(83, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 17], [11, 38], [12, 61], [13, 3], [14, 30], [15, 59], [16, 7], [17, 40], [18, 75], [19, 29], [20, 68], [21, 26], [22, 69], [23, 31], [24, 78], [25, 44], [26, 12], [27, 65], [28, 37], [29, 11], [30, 70], [31, 48], [32, 28], [33, 10], [34, 77], [35, 63], [36, 51], [37, 41], [38, 33], [39, 27], [40, 23], [41, 21], [42, 21], [43, 23], [44, 27], [45, 33], [46, 41], [47, 51], [48, 63], [49, 77], [50, 10], [51, 28], [52, 48], [53, 70], [54, 11], [55, 37], [56, 65], [57, 12], [58, 44], [59, 78], [60, 31], [61, 69], [62, 26], [63, 68], [64, 29], [65, 75], [66, 40], [67, 7], [68, 59], [69, 30], [70, 3], [71, 61], [72, 38], [73, 17], [74, 81], [75, 64], [76, 49], [77, 36], [78, 25], [79, 16], [80, 9], [81, 4], [82, 1]], [1, 3, 4, 7, 9, 10, 11, 12, 16, 17, 21, 23, 25, 26, 27, 28, 29, 30, 31, 33, 36, 37, 38, 40, 41, 44, 48, 49, 51, 59, 61, 63, 64, 65, 68, 69, 70, 75, 77, 78, 81], 41, 41, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116022,"parameters":{"p":83,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0023","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":89}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(89, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 11], [11, 32], [12, 55], [13, 80], [14, 18], [15, 47], [16, 78], [17, 22], [18, 57], [19, 5], [20, 44], [21, 85], [22, 39], [23, 84], [24, 42], [25, 2], [26, 53], [27, 17], [28, 72], [29, 40], [30, 10], [31, 71], [32, 45], [33, 21], [34, 88], [35, 68], [36, 50], [37, 34], [38, 20], [39, 8], [40, 87], [41, 79], [42, 73], [43, 69], [44, 67], [45, 67], [46, 69], [47, 73], [48, 79], [49, 87], [50, 8], [51, 20], [52, 34], [53, 50], [54, 68], [55, 88], [56, 21], [57, 45], [58, 71], [59, 10], [60, 40], [61, 72], [62, 17], [63, 53], [64, 2], [65, 42], [66, 84], [67, 39], [68, 85], [69, 44], [70, 5], [71, 57], [72, 22], [73, 78], [74, 47], [75, 18], [76, 80], [77, 55], [78, 32], [79, 11], [80, 81], [81, 64], [82, 49], [83, 36], [84, 25], [85, 16], [86, 9], [87, 4], [88, 1]], [1, 2, 4, 5, 8, 9, 10, 11, 16, 17, 18, 20, 21, 22, 25, 32, 34, 36, 39, 40, 42, 44, 45, 47, 49, 50, 53, 55, 57, 64, 67, 68, 69, 71, 72, 73, 78, 79, 80, 81, 84, 85, 87, 88], 44, 44, 是)","explanation_zh":"Wave6第五層核對=(89, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 11], [11, 32], [12, 55], [13, 80], [14, 18], [15, 47], [16, 78], [17, 22], [18, 57], [19, 5], [20, 44], [21, 85], [22, 39], [23, 84], [24, 42], [25, 2], [26, 53], [27, 17], [28, 72], [29, 40], [30, 10], [31, 71], [32, 45], [33, 21], [34, 88], [35, 68], [36, 50], [37, 34], [38, 20], [39, 8], [40, 87], [41, 79], [42, 73], [43, 69], [44, 67], [45, 67], [46, 69], [47, 73], [48, 79], [49, 87], [50, 8], [51, 20], [52, 34], [53, 50], [54, 68], [55, 88], [56, 21], [57, 45], [58, 71], [59, 10], [60, 40], [61, 72], [62, 17], [63, 53], [64, 2], [65, 42], [66, 84], [67, 39], [68, 85], [69, 44], [70, 5], [71, 57], [72, 22], [73, 78], [74, 47], [75, 18], [76, 80], [77, 55], [78, 32], [79, 11], [80, 81], [81, 64], [82, 49], [83, 36], [84, 25], [85, 16], [86, 9], [87, 4], [88, 1]], [1, 2, 4, 5, 8, 9, 10, 11, 16, 17, 18, 20, 21, 22, 25, 32, 34, 36, 39, 40, 42, 44, 45, 47, 49, 50, 53, 55, 57, 64, 67, 68, 69, 71, 72, 73, 78, 79, 80, 81, 84, 85, 87, 88], 44, 44, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116023,"parameters":{"p":89,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0024","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":97}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(97, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 3], [11, 24], [12, 47], [13, 72], [14, 2], [15, 31], [16, 62], [17, 95], [18, 33], [19, 70], [20, 12], [21, 53], [22, 96], [23, 44], [24, 91], [25, 43], [26, 94], [27, 50], [28, 8], [29, 65], [30, 27], [31, 88], [32, 54], [33, 22], [34, 89], [35, 61], [36, 35], [37, 11], [38, 86], [39, 66], [40, 48], [41, 32], [42, 18], [43, 6], [44, 93], [45, 85], [46, 79], [47, 75], [48, 73], [49, 73], [50, 75], [51, 79], [52, 85], [53, 93], [54, 6], [55, 18], [56, 32], [57, 48], [58, 66], [59, 86], [60, 11], [61, 35], [62, 61], [63, 89], [64, 22], [65, 54], [66, 88], [67, 27], [68, 65], [69, 8], [70, 50], [71, 94], [72, 43], [73, 91], [74, 44], [75, 96], [76, 53], [77, 12], [78, 70], [79, 33], [80, 95], [81, 62], [82, 31], [83, 2], [84, 72], [85, 47], [86, 24], [87, 3], [88, 81], [89, 64], [90, 49], [91, 36], [92, 25], [93, 16], [94, 9], [95, 4], [96, 1]], [1, 2, 3, 4, 6, 8, 9, 11, 12, 16, 18, 22, 24, 25, 27, 31, 32, 33, 35, 36, 43, 44, 47, 48, 49, 50, 53, 54, 61, 62, 64, 65, 66, 70, 72, 73, 75, 79, 81, 85, 86, 88, 89, 91, 93, 94, 95, 96], 48, 48, 是)","explanation_zh":"Wave6第五層核對=(97, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 3], [11, 24], [12, 47], [13, 72], [14, 2], [15, 31], [16, 62], [17, 95], [18, 33], [19, 70], [20, 12], [21, 53], [22, 96], [23, 44], [24, 91], [25, 43], [26, 94], [27, 50], [28, 8], [29, 65], [30, 27], [31, 88], [32, 54], [33, 22], [34, 89], [35, 61], [36, 35], [37, 11], [38, 86], [39, 66], [40, 48], [41, 32], [42, 18], [43, 6], [44, 93], [45, 85], [46, 79], [47, 75], [48, 73], [49, 73], [50, 75], [51, 79], [52, 85], [53, 93], [54, 6], [55, 18], [56, 32], [57, 48], [58, 66], [59, 86], [60, 11], [61, 35], [62, 61], [63, 89], [64, 22], [65, 54], [66, 88], [67, 27], [68, 65], [69, 8], [70, 50], [71, 94], [72, 43], [73, 91], [74, 44], [75, 96], [76, 53], [77, 12], [78, 70], [79, 33], [80, 95], [81, 62], [82, 31], [83, 2], [84, 72], [85, 47], [86, 24], [87, 3], [88, 81], [89, 64], [90, 49], [91, 36], [92, 25], [93, 16], [94, 9], [95, 4], [96, 1]], [1, 2, 3, 4, 6, 8, 9, 11, 12, 16, 18, 22, 24, 25, 27, 31, 32, 33, 35, 36, 43, 44, 47, 48, 49, 50, 53, 54, 61, 62, 64, 65, 66, 70, 72, 73, 75, 79, 81, 85, 86, 88, 89, 91, 93, 94, 95, 96], 48, 48, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116024,"parameters":{"p":97,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0025","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-square-table；資料={\"p\":101}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(101, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 100], [11, 20], [12, 43], [13, 68], [14, 95], [15, 23], [16, 54], [17, 87], [18, 21], [19, 58], [20, 97], [21, 37], [22, 80], [23, 24], [24, 71], [25, 19], [26, 70], [27, 22], [28, 77], [29, 33], [30, 92], [31, 52], [32, 14], [33, 79], [34, 45], [35, 13], [36, 84], [37, 56], [38, 30], [39, 6], [40, 85], [41, 65], [42, 47], [43, 31], [44, 17], [45, 5], [46, 96], [47, 88], [48, 82], [49, 78], [50, 76], [51, 76], [52, 78], [53, 82], [54, 88], [55, 96], [56, 5], [57, 17], [58, 31], [59, 47], [60, 65], [61, 85], [62, 6], [63, 30], [64, 56], [65, 84], [66, 13], [67, 45], [68, 79], [69, 14], [70, 52], [71, 92], [72, 33], [73, 77], [74, 22], [75, 70], [76, 19], [77, 71], [78, 24], [79, 80], [80, 37], [81, 97], [82, 58], [83, 21], [84, 87], [85, 54], [86, 23], [87, 95], [88, 68], [89, 43], [90, 20], [91, 100], [92, 81], [93, 64], [94, 49], [95, 36], [96, 25], [97, 16], [98, 9], [99, 4], [100, 1]], [1, 4, 5, 6, 9, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 30, 31, 33, 36, 37, 43, 45, 47, 49, 52, 54, 56, 58, 64, 65, 68, 70, 71, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 88, 92, 95, 96, 97, 100], 50, 50, 是)","explanation_zh":"Wave6第五層核對=(101, [[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 100], [11, 20], [12, 43], [13, 68], [14, 95], [15, 23], [16, 54], [17, 87], [18, 21], [19, 58], [20, 97], [21, 37], [22, 80], [23, 24], [24, 71], [25, 19], [26, 70], [27, 22], [28, 77], [29, 33], [30, 92], [31, 52], [32, 14], [33, 79], [34, 45], [35, 13], [36, 84], [37, 56], [38, 30], [39, 6], [40, 85], [41, 65], [42, 47], [43, 31], [44, 17], [45, 5], [46, 96], [47, 88], [48, 82], [49, 78], [50, 76], [51, 76], [52, 78], [53, 82], [54, 88], [55, 96], [56, 5], [57, 17], [58, 31], [59, 47], [60, 65], [61, 85], [62, 6], [63, 30], [64, 56], [65, 84], [66, 13], [67, 45], [68, 79], [69, 14], [70, 52], [71, 92], [72, 33], [73, 77], [74, 22], [75, 70], [76, 19], [77, 71], [78, 24], [79, 80], [80, 37], [81, 97], [82, 58], [83, 21], [84, 87], [85, 54], [86, 23], [87, 95], [88, 68], [89, 43], [90, 20], [91, 100], [92, 81], [93, 64], [94, 49], [95, 36], [96, 25], [97, 16], [98, 9], [99, 4], [100, 1]], [1, 4, 5, 6, 9, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 30, 31, 33, 36, 37, 43, 45, 47, 49, 52, 54, 56, 58, 64, 65, 68, 70, 71, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 88, 92, 95, 96, 97, 100], 50, 50, 是)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-square-table","seed":202611116025,"parameters":{"p":101,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-square-table","semantic_template":"qf-mod42-quadratic-square-table/{\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0026","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":3,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 1, [1, 2], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(3, 1, 1, [1, 2], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116026,"parameters":{"p":3,"a":1,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":1,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0027","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":5,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 2, [], 否, complete_nonzero_square_table_has_no_root)","explanation_zh":"Wave6第五層核對=(5, 2, 2, [], 否, complete_nonzero_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116027,"parameters":{"p":5,"a":2,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":2,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0028","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":7,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 3, 3, [], 否, complete_nonzero_square_table_has_no_root)","explanation_zh":"Wave6第五層核對=(7, 3, 3, [], 否, complete_nonzero_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116028,"parameters":{"p":7,"a":3,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":3,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0029","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":11,\"a\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 4, 4, [2, 9], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(11, 4, 4, [2, 9], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116029,"parameters":{"p":11,"a":4,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":4,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0030","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":13,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 5, 5, [], 否, complete_nonzero_square_table_has_no_root)","explanation_zh":"Wave6第五層核對=(13, 5, 5, [], 否, complete_nonzero_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116030,"parameters":{"p":13,"a":5,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":5,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0031","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":17,\"a\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 6, 6, [], 否, complete_nonzero_square_table_has_no_root)","explanation_zh":"Wave6第五層核對=(17, 6, 6, [], 否, complete_nonzero_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116031,"parameters":{"p":17,"a":6,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":6,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0032","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":19,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 7, 7, [8, 11], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(19, 7, 7, [8, 11], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116032,"parameters":{"p":19,"a":7,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":7,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0033","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":23,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 8, 8, [10, 13], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(23, 8, 8, [10, 13], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116033,"parameters":{"p":23,"a":8,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":8,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0034","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":29,\"a\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 9, 9, [3, 26], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(29, 9, 9, [3, 26], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116034,"parameters":{"p":29,"a":9,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":9,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0035","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":31,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, 10, 10, [14, 17], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(31, 10, 10, [14, 17], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116035,"parameters":{"p":31,"a":10,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":10,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0036","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":37,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(37, 11, 11, [14, 23], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(37, 11, 11, [14, 23], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116036,"parameters":{"p":37,"a":11,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":11,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0037","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":41,\"a\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(41, 12, 12, [], 否, complete_nonzero_square_table_has_no_root)","explanation_zh":"Wave6第五層核對=(41, 12, 12, [], 否, complete_nonzero_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116037,"parameters":{"p":41,"a":12,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":12,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0038","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":43,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(43, 13, 13, [20, 23], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(43, 13, 13, [20, 23], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116038,"parameters":{"p":43,"a":13,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":13,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0039","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":47,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, 14, 14, [22, 25], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(47, 14, 14, [22, 25], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116039,"parameters":{"p":47,"a":14,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":14,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0040","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":53,\"a\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(53, 15, 15, [11, 42], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(53, 15, 15, [11, 42], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116040,"parameters":{"p":53,"a":15,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":15,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0041","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":59,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(59, 16, 16, [4, 55], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(59, 16, 16, [4, 55], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116041,"parameters":{"p":59,"a":16,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":16,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0042","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":61,\"a\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(61, 17, 17, [], 否, complete_nonzero_square_table_has_no_root)","explanation_zh":"Wave6第五層核對=(61, 17, 17, [], 否, complete_nonzero_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116042,"parameters":{"p":61,"a":17,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":17,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0043","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":67,\"a\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(67, 18, 18, [], 否, complete_nonzero_square_table_has_no_root)","explanation_zh":"Wave6第五層核對=(67, 18, 18, [], 否, complete_nonzero_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116043,"parameters":{"p":67,"a":18,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":18,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0044","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":71,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(71, 19, 19, [27, 44], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(71, 19, 19, [27, 44], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116044,"parameters":{"p":71,"a":19,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":19,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0045","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":73,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(73, 20, 20, [], 否, complete_nonzero_square_table_has_no_root)","explanation_zh":"Wave6第五層核對=(73, 20, 20, [], 否, complete_nonzero_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116045,"parameters":{"p":73,"a":20,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":20,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0046","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":79,\"a\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(79, 21, 21, [10, 69], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(79, 21, 21, [10, 69], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116046,"parameters":{"p":79,"a":21,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":21,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0047","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":83,\"a\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(83, 22, 22, [], 否, complete_nonzero_square_table_has_no_root)","explanation_zh":"Wave6第五層核對=(83, 22, 22, [], 否, complete_nonzero_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116047,"parameters":{"p":83,"a":22,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":22,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0048","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":89,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(89, 23, 23, [], 否, complete_nonzero_square_table_has_no_root)","explanation_zh":"Wave6第五層核對=(89, 23, 23, [], 否, complete_nonzero_square_table_has_no_root)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116048,"parameters":{"p":89,"a":23,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":23,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0049","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":97,\"a\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(97, 24, 24, [11, 86], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(97, 24, 24, [11, 86], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116049,"parameters":{"p":97,"a":24,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":24,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0050","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-membership-witness；資料={\"p\":101,\"a\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(101, 25, 25, [5, 96], 是, nonzero_square_root_witness)","explanation_zh":"Wave6第五層核對=(101, 25, 25, [5, 96], 是, nonzero_square_root_witness)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-membership-witness","seed":202611116050,"parameters":{"p":101,"a":25,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-membership-witness","semantic_template":"qf-mod42-quadratic-membership-witness/{\"a\":25,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0051","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":3,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, [1, 2], 2, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(3, 1, [1, 2], 2, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116051,"parameters":{"p":3,"a":1,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":1,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0052","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":5,\"a\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 4, [2, 3], 3, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(5, 4, [2, 3], 3, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116052,"parameters":{"p":5,"a":4,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":4,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0053","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":7,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 2, [3, 4], 4, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(7, 2, [3, 4], 4, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116053,"parameters":{"p":7,"a":2,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":2,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0054","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":11,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 5, [4, 7], 7, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(11, 5, [4, 7], 7, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116054,"parameters":{"p":11,"a":5,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":5,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0055","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":13,\"a\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 12, [5, 8], 8, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(13, 12, [5, 8], 8, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116055,"parameters":{"p":13,"a":12,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":12,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0056","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":17,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 2, [6, 11], 11, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(17, 2, [6, 11], 11, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116056,"parameters":{"p":17,"a":2,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":2,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0057","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":19,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 11, [7, 12], 12, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(19, 11, [7, 12], 12, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116057,"parameters":{"p":19,"a":11,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":11,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0058","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":23,\"a\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 18, [8, 15], 15, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(23, 18, [8, 15], 15, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116058,"parameters":{"p":23,"a":18,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":18,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0059","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":29,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 23, [9, 20], 20, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(29, 23, [9, 20], 20, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116059,"parameters":{"p":29,"a":23,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":23,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0060","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":31,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, 7, [10, 21], 21, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(31, 7, [10, 21], 21, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116060,"parameters":{"p":31,"a":7,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":7,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0061","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":37,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(37, 10, [11, 26], 26, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(37, 10, [11, 26], 26, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116061,"parameters":{"p":37,"a":10,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":10,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0062","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":41,\"a\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(41, 21, [12, 29], 29, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(41, 21, [12, 29], 29, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116062,"parameters":{"p":41,"a":21,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":21,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0063","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":43,\"a\":40}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(43, 40, [13, 30], 30, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(43, 40, [13, 30], 30, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116063,"parameters":{"p":43,"a":40,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":40,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0064","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":47,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, 8, [14, 33], 33, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(47, 8, [14, 33], 33, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116064,"parameters":{"p":47,"a":8,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":8,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0065","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":53,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(53, 13, [15, 38], 38, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(53, 13, [15, 38], 38, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116065,"parameters":{"p":53,"a":13,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":13,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0066","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":59,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(59, 20, [16, 43], 43, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(59, 20, [16, 43], 43, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116066,"parameters":{"p":59,"a":20,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":20,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0067","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":61,\"a\":45}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(61, 45, [17, 44], 44, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(61, 45, [17, 44], 44, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116067,"parameters":{"p":61,"a":45,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":45,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0068","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":67,\"a\":56}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(67, 56, [18, 49], 49, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(67, 56, [18, 49], 49, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116068,"parameters":{"p":67,"a":56,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":56,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0069","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":71,\"a\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(71, 6, [19, 52], 52, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(71, 6, [19, 52], 52, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116069,"parameters":{"p":71,"a":6,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":6,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0070","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":73,\"a\":35}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(73, 35, [20, 53], 53, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(73, 35, [20, 53], 53, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116070,"parameters":{"p":73,"a":35,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":35,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0071","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":79,\"a\":46}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(79, 46, [21, 58], 58, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(79, 46, [21, 58], 58, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116071,"parameters":{"p":79,"a":46,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":46,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0072","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":83,\"a\":69}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(83, 69, [22, 61], 61, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(83, 69, [22, 61], 61, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116072,"parameters":{"p":83,"a":69,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":69,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0073","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":89,\"a\":84}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(89, 84, [23, 66], 66, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(89, 84, [23, 66], 66, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116073,"parameters":{"p":89,"a":84,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":84,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0074","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":97,\"a\":91}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(97, 91, [24, 73], 73, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(97, 91, [24, 73], 73, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116074,"parameters":{"p":97,"a":91,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":91,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0075","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-paired-roots；資料={\"p\":101,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(101, 19, [25, 76], 76, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)","explanation_zh":"Wave6第五層核對=(101, 19, [25, 76], 76, 是, 是, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-paired-roots","seed":202611116075,"parameters":{"p":101,"a":19,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-paired-roots","semantic_template":"qf-mod42-quadratic-paired-roots/{\"a\":19,\"p\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0076","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":3,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 3, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(zero_class, 3, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116076,"parameters":{"mode":"zero_class","modulus":3,"a":0,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":0,\"mode\":\"zero_class\",\"modulus\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0077","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(even_prime, 2, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116077,"parameters":{"mode":"even_prime","modulus":2,"a":2,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":2,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0078","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":9,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 9, 3, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(composite_modulus, 9, 3, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116078,"parameters":{"mode":"composite_modulus","modulus":9,"a":3,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":3,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0079","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":11,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 11, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(zero_class, 11, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116079,"parameters":{"mode":"zero_class","modulus":11,"a":0,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":0,\"mode\":\"zero_class\",\"modulus\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0080","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(even_prime, 2, 1, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116080,"parameters":{"mode":"even_prime","modulus":2,"a":5,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":5,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0081","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":15,\"a\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 15, 6, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(composite_modulus, 15, 6, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116081,"parameters":{"mode":"composite_modulus","modulus":15,"a":6,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":6,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0082","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":19,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 19, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(zero_class, 19, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116082,"parameters":{"mode":"zero_class","modulus":19,"a":0,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":0,\"mode\":\"zero_class\",\"modulus\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0083","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(even_prime, 2, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116083,"parameters":{"mode":"even_prime","modulus":2,"a":8,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":8,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0084","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":21,\"a\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 21, 9, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(composite_modulus, 21, 9, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116084,"parameters":{"mode":"composite_modulus","modulus":21,"a":9,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":9,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0085","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":31,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 31, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(zero_class, 31, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116085,"parameters":{"mode":"zero_class","modulus":31,"a":0,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":0,\"mode\":\"zero_class\",\"modulus\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0086","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(even_prime, 2, 1, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116086,"parameters":{"mode":"even_prime","modulus":2,"a":11,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":11,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0087","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":27,\"a\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 27, 12, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(composite_modulus, 27, 12, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116087,"parameters":{"mode":"composite_modulus","modulus":27,"a":12,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":12,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0088","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":43,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 43, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(zero_class, 43, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116088,"parameters":{"mode":"zero_class","modulus":43,"a":0,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":0,\"mode\":\"zero_class\",\"modulus\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0089","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(even_prime, 2, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116089,"parameters":{"mode":"even_prime","modulus":2,"a":14,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":14,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0090","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":33,\"a\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 33, 15, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(composite_modulus, 33, 15, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116090,"parameters":{"mode":"composite_modulus","modulus":33,"a":15,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":15,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0091","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":59,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 59, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(zero_class, 59, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116091,"parameters":{"mode":"zero_class","modulus":59,"a":0,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":0,\"mode\":\"zero_class\",\"modulus\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0092","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(even_prime, 2, 1, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116092,"parameters":{"mode":"even_prime","modulus":2,"a":17,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":17,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0093","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":39,\"a\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 39, 18, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(composite_modulus, 39, 18, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116093,"parameters":{"mode":"composite_modulus","modulus":39,"a":18,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":18,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0094","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":71,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 71, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(zero_class, 71, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116094,"parameters":{"mode":"zero_class","modulus":71,"a":0,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":0,\"mode\":\"zero_class\",\"modulus\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0095","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(even_prime, 2, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116095,"parameters":{"mode":"even_prime","modulus":2,"a":20,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":20,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0096","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":45,\"a\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 45, 21, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(composite_modulus, 45, 21, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116096,"parameters":{"mode":"composite_modulus","modulus":45,"a":21,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":21,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0097","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":83,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 83, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(zero_class, 83, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116097,"parameters":{"mode":"zero_class","modulus":83,"a":0,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":0,\"mode\":\"zero_class\",\"modulus\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0098","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(even_prime, 2, 1, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116098,"parameters":{"mode":"even_prime","modulus":2,"a":23,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":23,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0099","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":51,\"a\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 51, 24, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(composite_modulus, 51, 24, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116099,"parameters":{"mode":"composite_modulus","modulus":51,"a":24,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":24,\"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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、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-20261111-r06-quadratic-residue-0100","batch_id":"batch-20261014-advance-20261111-r06-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-quadratic-residue"],"title_zh":"二次剩餘","stem_zh":"odd prime p上的nonzero quadratic-residue convention與complete nonzero square table。操作=quadratic-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":101,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 101, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)","explanation_zh":"Wave6第五層核對=(zero_class, 101, 0, 否, nonzero_quadratic_residue_convention_requires_an_odd_prime_and_nonzero_class)；依完整明示資料獨立重算；a先normalize mod p；nonzero residue有兩個distinct opposite roots；zero、p=2與composite modulus分開；不預先使用Euler criterion或Legendre symbol。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二次剩餘"]},"variant":{"family_id":"qf-mod42-quadratic-domain-boundary","seed":202611116100,"parameters":{"mode":"zero_class","modulus":101,"a":0,"domain":"Odd_prime_nonzero_quadratic_residue","operation":"quadratic-domain-boundary","semantic_template":"qf-mod42-quadratic-domain-boundary/{\"a\":0,\"mode\":\"zero_class\",\"modulus\":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、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}