{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0001","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":3,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, 1, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(3, 1, 1, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134001,"parameters":{"prime":3,"a":1,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":1,\"prime\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0002","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":5,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 2, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(5, 2, 2, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134002,"parameters":{"prime":5,"a":2,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":2,\"prime\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0003","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":7,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 3, 3, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(7, 3, 3, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134003,"parameters":{"prime":7,"a":3,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":3,\"prime\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0004","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":11,\"a\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 4, 4, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(11, 4, 4, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134004,"parameters":{"prime":11,"a":4,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":4,\"prime\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0005","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":13,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 5, 5, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(13, 5, 5, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134005,"parameters":{"prime":13,"a":5,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":5,\"prime\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0006","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":17,\"a\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 6, 6, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(17, 6, 6, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134006,"parameters":{"prime":17,"a":6,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":6,\"prime\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0007","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":19,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 7, 7, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(19, 7, 7, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134007,"parameters":{"prime":19,"a":7,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":7,\"prime\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0008","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":23,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 8, 8, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(23, 8, 8, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134008,"parameters":{"prime":23,"a":8,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":8,\"prime\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0009","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":29,\"a\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 9, 9, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(29, 9, 9, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134009,"parameters":{"prime":29,"a":9,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":9,\"prime\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0010","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":31,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, 10, 10, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(31, 10, 10, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134010,"parameters":{"prime":31,"a":10,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":10,\"prime\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0011","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":37,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(37, 11, 11, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(37, 11, 11, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134011,"parameters":{"prime":37,"a":11,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":11,\"prime\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0012","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":41,\"a\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(41, 12, 12, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(41, 12, 12, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134012,"parameters":{"prime":41,"a":12,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":12,\"prime\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0013","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":43,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(43, 13, 13, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(43, 13, 13, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134013,"parameters":{"prime":43,"a":13,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":13,\"prime\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0014","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":47,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, 14, 14, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(47, 14, 14, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134014,"parameters":{"prime":47,"a":14,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":14,\"prime\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0015","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":53,\"a\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(53, 15, 15, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(53, 15, 15, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134015,"parameters":{"prime":53,"a":15,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":15,\"prime\":53}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0016","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":59,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(59, 16, 16, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(59, 16, 16, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134016,"parameters":{"prime":59,"a":16,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":16,\"prime\":59}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0017","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":61,\"a\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(61, 17, 17, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(61, 17, 17, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134017,"parameters":{"prime":61,"a":17,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":17,\"prime\":61}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0018","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":67,\"a\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(67, 18, 18, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(67, 18, 18, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134018,"parameters":{"prime":67,"a":18,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":18,\"prime\":67}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0019","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":71,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(71, 19, 19, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(71, 19, 19, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134019,"parameters":{"prime":71,"a":19,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":19,\"prime\":71}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0020","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":73,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(73, 20, 20, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(73, 20, 20, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134020,"parameters":{"prime":73,"a":20,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":20,\"prime\":73}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0021","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":79,\"a\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(79, 21, 21, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(79, 21, 21, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134021,"parameters":{"prime":79,"a":21,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":21,\"prime\":79}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0022","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":83,\"a\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(83, 22, 22, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(83, 22, 22, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134022,"parameters":{"prime":83,"a":22,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":22,\"prime\":83}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0023","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":89,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(89, 23, 23, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(89, 23, 23, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134023,"parameters":{"prime":89,"a":23,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":23,\"prime\":89}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0024","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":97,\"a\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(97, 24, 24, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(97, 24, 24, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134024,"parameters":{"prime":97,"a":24,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":24,\"prime\":97}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0025","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-qualification；資料={\"prime\":101,\"a\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(101, 25, 25, 是, 是, odd_prime_and_nonzero_modular_class_qualified)","explanation_zh":"Wave6第七層核對=(101, 25, 25, 是, 是, odd_prime_and_nonzero_modular_class_qualified)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-qualification","seed":202611134025,"parameters":{"prime":101,"a":25,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-qualification","semantic_template":"qf-mod44-eulercrit-domain-qualification/{\"a\":25,\"prime\":101}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0026","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":3,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, [1, 2], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(3, 1, [1, 2], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134026,"parameters":{"prime":3,"a":1,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":1,\"prime\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0027","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":5,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 2, [], -1, 4, -1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(5, 2, [], -1, 4, -1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134027,"parameters":{"prime":5,"a":2,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":2,\"prime\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0028","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":7,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 3, [], -1, 6, -1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(7, 3, [], -1, 6, -1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134028,"parameters":{"prime":7,"a":3,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":3,\"prime\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0029","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":11,\"a\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 4, [2, 9], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(11, 4, [2, 9], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134029,"parameters":{"prime":11,"a":4,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":4,\"prime\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0030","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":13,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 5, [], -1, 12, -1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(13, 5, [], -1, 12, -1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134030,"parameters":{"prime":13,"a":5,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":5,\"prime\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0031","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":17,\"a\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 6, [], -1, 16, -1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(17, 6, [], -1, 16, -1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134031,"parameters":{"prime":17,"a":6,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":6,\"prime\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0032","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":19,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 7, [8, 11], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(19, 7, [8, 11], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134032,"parameters":{"prime":19,"a":7,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":7,\"prime\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0033","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":23,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 8, [10, 13], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(23, 8, [10, 13], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134033,"parameters":{"prime":23,"a":8,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":8,\"prime\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0034","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":29,\"a\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 9, [3, 26], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(29, 9, [3, 26], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134034,"parameters":{"prime":29,"a":9,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":9,\"prime\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0035","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":31,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, 10, [14, 17], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(31, 10, [14, 17], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134035,"parameters":{"prime":31,"a":10,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":10,\"prime\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0036","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":37,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(37, 11, [14, 23], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(37, 11, [14, 23], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134036,"parameters":{"prime":37,"a":11,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":11,\"prime\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0037","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":41,\"a\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(41, 12, [], -1, 40, -1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(41, 12, [], -1, 40, -1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134037,"parameters":{"prime":41,"a":12,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":12,\"prime\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0038","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":43,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(43, 13, [20, 23], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(43, 13, [20, 23], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134038,"parameters":{"prime":43,"a":13,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":13,\"prime\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0039","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":47,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, 14, [22, 25], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(47, 14, [22, 25], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134039,"parameters":{"prime":47,"a":14,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":14,\"prime\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0040","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":53,\"a\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(53, 15, [11, 42], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(53, 15, [11, 42], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134040,"parameters":{"prime":53,"a":15,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":15,\"prime\":53}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0041","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":59,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(59, 16, [4, 55], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(59, 16, [4, 55], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134041,"parameters":{"prime":59,"a":16,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":16,\"prime\":59}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0042","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":61,\"a\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(61, 17, [], -1, 60, -1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(61, 17, [], -1, 60, -1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134042,"parameters":{"prime":61,"a":17,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":17,\"prime\":61}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0043","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":67,\"a\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(67, 18, [], -1, 66, -1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(67, 18, [], -1, 66, -1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134043,"parameters":{"prime":67,"a":18,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":18,\"prime\":67}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0044","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":71,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(71, 19, [27, 44], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(71, 19, [27, 44], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134044,"parameters":{"prime":71,"a":19,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":19,\"prime\":71}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0045","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":73,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(73, 20, [], -1, 72, -1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(73, 20, [], -1, 72, -1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134045,"parameters":{"prime":73,"a":20,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":20,\"prime\":73}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0046","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":79,\"a\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(79, 21, [10, 69], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(79, 21, [10, 69], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134046,"parameters":{"prime":79,"a":21,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":21,\"prime\":79}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0047","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":83,\"a\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(83, 22, [], -1, 82, -1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(83, 22, [], -1, 82, -1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134047,"parameters":{"prime":83,"a":22,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":22,\"prime\":83}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0048","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":89,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(89, 23, [], -1, 88, -1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(89, 23, [], -1, 88, -1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134048,"parameters":{"prime":89,"a":23,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":23,\"prime\":89}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0049","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":97,\"a\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(97, 24, [11, 86], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(97, 24, [11, 86], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134049,"parameters":{"prime":97,"a":24,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":24,\"prime\":97}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0050","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-independent-comparison；資料={\"prime\":101,\"a\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(101, 25, [5, 96], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)","explanation_zh":"Wave6第七層核對=(101, 25, [5, 96], 1, 1, 1, 是, square_table_and_repeated_squaring_are_independent_computations)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-independent-comparison","seed":202611134050,"parameters":{"prime":101,"a":25,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-independent-comparison","semantic_template":"qf-mod44-eulercrit-independent-comparison/{\"a\":25,\"prime\":101}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0051","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":3,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1, quadratic_residue, [1, 2], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(3, 1, quadratic_residue, [1, 2], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134051,"parameters":{"prime":3,"a":1,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":1,\"prime\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0052","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":5,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)","explanation_zh":"Wave6第七層核對=(5, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134052,"parameters":{"prime":5,"a":2,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":2,\"prime\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0053","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":7,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 2, quadratic_residue, [3, 4], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(7, 2, quadratic_residue, [3, 4], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134053,"parameters":{"prime":7,"a":2,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":2,\"prime\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0054","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":11,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)","explanation_zh":"Wave6第七層核對=(11, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134054,"parameters":{"prime":11,"a":2,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":2,\"prime\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0055","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":13,\"a\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 12, quadratic_residue, [5, 8], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(13, 12, quadratic_residue, [5, 8], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134055,"parameters":{"prime":13,"a":12,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":12,\"prime\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0056","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":17,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 3, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)","explanation_zh":"Wave6第七層核對=(17, 3, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134056,"parameters":{"prime":17,"a":3,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":3,\"prime\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0057","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":19,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 11, quadratic_residue, [7, 12], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(19, 11, quadratic_residue, [7, 12], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134057,"parameters":{"prime":19,"a":11,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":11,\"prime\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0058","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":23,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 5, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)","explanation_zh":"Wave6第七層核對=(23, 5, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134058,"parameters":{"prime":23,"a":5,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":5,\"prime\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0059","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":29,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 23, quadratic_residue, [9, 20], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(29, 23, quadratic_residue, [9, 20], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134059,"parameters":{"prime":29,"a":23,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":23,\"prime\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0060","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":31,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, 3, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)","explanation_zh":"Wave6第七層核對=(31, 3, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134060,"parameters":{"prime":31,"a":3,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":3,\"prime\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0061","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":37,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(37, 10, quadratic_residue, [11, 26], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(37, 10, quadratic_residue, [11, 26], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134061,"parameters":{"prime":37,"a":10,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":10,\"prime\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0062","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":41,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(41, 3, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)","explanation_zh":"Wave6第七層核對=(41, 3, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134062,"parameters":{"prime":41,"a":3,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":3,\"prime\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0063","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":43,\"a\":40}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(43, 40, quadratic_residue, [13, 30], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(43, 40, quadratic_residue, [13, 30], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134063,"parameters":{"prime":43,"a":40,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":40,\"prime\":43}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0064","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":47,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, 5, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)","explanation_zh":"Wave6第七層核對=(47, 5, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134064,"parameters":{"prime":47,"a":5,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":5,\"prime\":47}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0065","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":53,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(53, 13, quadratic_residue, [15, 38], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(53, 13, quadratic_residue, [15, 38], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134065,"parameters":{"prime":53,"a":13,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":13,\"prime\":53}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0066","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":59,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(59, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)","explanation_zh":"Wave6第七層核對=(59, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134066,"parameters":{"prime":59,"a":2,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":2,\"prime\":59}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0067","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":61,\"a\":45}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(61, 45, quadratic_residue, [17, 44], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(61, 45, quadratic_residue, [17, 44], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134067,"parameters":{"prime":61,"a":45,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":45,\"prime\":61}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0068","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":67,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(67, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)","explanation_zh":"Wave6第七層核對=(67, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134068,"parameters":{"prime":67,"a":2,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":2,\"prime\":67}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0069","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":71,\"a\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(71, 6, quadratic_residue, [19, 52], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(71, 6, quadratic_residue, [19, 52], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134069,"parameters":{"prime":71,"a":6,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":6,\"prime\":71}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0070","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":73,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(73, 5, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)","explanation_zh":"Wave6第七層核對=(73, 5, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134070,"parameters":{"prime":73,"a":5,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":5,\"prime\":73}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0071","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":79,\"a\":46}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(79, 46, quadratic_residue, [21, 58], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(79, 46, quadratic_residue, [21, 58], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134071,"parameters":{"prime":79,"a":46,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":46,\"prime\":79}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0072","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":83,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(83, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)","explanation_zh":"Wave6第七層核對=(83, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134072,"parameters":{"prime":83,"a":2,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":2,\"prime\":83}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0073","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":89,\"a\":84}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(89, 84, quadratic_residue, [23, 66], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(89, 84, quadratic_residue, [23, 66], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134073,"parameters":{"prime":89,"a":84,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":84,\"prime\":89}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0074","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":97,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(97, 5, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)","explanation_zh":"Wave6第七層核對=(97, 5, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134074,"parameters":{"prime":97,"a":5,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":5,\"prime\":97}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0075","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":101,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(101, 19, quadratic_residue, [25, 76], 1, 1, 是, residue_maps_to_plus_one)","explanation_zh":"Wave6第七層核對=(101, 19, quadratic_residue, [25, 76], 1, 1, 是, residue_maps_to_plus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-residue-nonresidue-cases","seed":202611134075,"parameters":{"prime":101,"a":19,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-residue-nonresidue-cases","semantic_template":"qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":19,\"prime\":101}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0076","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(even_prime, 2, 1, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134076,"parameters":{"mode":"even_prime","modulus":2,"a":1,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":1,\"mode\":\"even_prime\",\"modulus\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0077","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":9,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 9, 2, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(composite_modulus, 9, 2, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134077,"parameters":{"mode":"composite_modulus","modulus":9,"a":2,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":2,\"mode\":\"composite_modulus\",\"modulus\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0078","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":7,\"a\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 7, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(zero_class, 7, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134078,"parameters":{"mode":"zero_class","modulus":7,"a":21,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":21,\"mode\":\"zero_class\",\"modulus\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0079","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(even_prime, 2, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134079,"parameters":{"mode":"even_prime","modulus":2,"a":4,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":4,\"mode\":\"even_prime\",\"modulus\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0080","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":15,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 15, 5, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(composite_modulus, 15, 5, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134080,"parameters":{"mode":"composite_modulus","modulus":15,"a":5,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":5,\"mode\":\"composite_modulus\",\"modulus\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0081","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":17,\"a\":102}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 17, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(zero_class, 17, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134081,"parameters":{"mode":"zero_class","modulus":17,"a":102,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":102,\"mode\":\"zero_class\",\"modulus\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0082","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(even_prime, 2, 1, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134082,"parameters":{"mode":"even_prime","modulus":2,"a":7,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":7,\"mode\":\"even_prime\",\"modulus\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0083","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":21,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 21, 8, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(composite_modulus, 21, 8, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134083,"parameters":{"mode":"composite_modulus","modulus":21,"a":8,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":8,\"mode\":\"composite_modulus\",\"modulus\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0084","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":29,\"a\":261}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 29, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(zero_class, 29, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134084,"parameters":{"mode":"zero_class","modulus":29,"a":261,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":261,\"mode\":\"zero_class\",\"modulus\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0085","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(even_prime, 2, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134085,"parameters":{"mode":"even_prime","modulus":2,"a":10,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":10,\"mode\":\"even_prime\",\"modulus\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0086","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":27,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 27, 11, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(composite_modulus, 27, 11, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134086,"parameters":{"mode":"composite_modulus","modulus":27,"a":11,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":11,\"mode\":\"composite_modulus\",\"modulus\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0087","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":41,\"a\":492}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 41, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(zero_class, 41, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134087,"parameters":{"mode":"zero_class","modulus":41,"a":492,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":492,\"mode\":\"zero_class\",\"modulus\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0088","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(even_prime, 2, 1, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134088,"parameters":{"mode":"even_prime","modulus":2,"a":13,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":13,\"mode\":\"even_prime\",\"modulus\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0089","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":33,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 33, 14, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(composite_modulus, 33, 14, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134089,"parameters":{"mode":"composite_modulus","modulus":33,"a":14,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":14,\"mode\":\"composite_modulus\",\"modulus\":33}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0090","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":53,\"a\":795}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 53, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(zero_class, 53, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134090,"parameters":{"mode":"zero_class","modulus":53,"a":795,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":795,\"mode\":\"zero_class\",\"modulus\":53}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0091","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(even_prime, 2, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134091,"parameters":{"mode":"even_prime","modulus":2,"a":16,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":16,\"mode\":\"even_prime\",\"modulus\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0092","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":39,\"a\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 39, 17, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(composite_modulus, 39, 17, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134092,"parameters":{"mode":"composite_modulus","modulus":39,"a":17,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":17,\"mode\":\"composite_modulus\",\"modulus\":39}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0093","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":67,\"a\":1206}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 67, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(zero_class, 67, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134093,"parameters":{"mode":"zero_class","modulus":67,"a":1206,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":1206,\"mode\":\"zero_class\",\"modulus\":67}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0094","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(even_prime, 2, 1, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134094,"parameters":{"mode":"even_prime","modulus":2,"a":19,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":19,\"mode\":\"even_prime\",\"modulus\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0095","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":45,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 45, 20, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(composite_modulus, 45, 20, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134095,"parameters":{"mode":"composite_modulus","modulus":45,"a":20,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":20,\"mode\":\"composite_modulus\",\"modulus\":45}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0096","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":79,\"a\":1659}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 79, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(zero_class, 79, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134096,"parameters":{"mode":"zero_class","modulus":79,"a":1659,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":1659,\"mode\":\"zero_class\",\"modulus\":79}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0097","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(even_prime, 2, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134097,"parameters":{"mode":"even_prime","modulus":2,"a":22,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":22,\"mode\":\"even_prime\",\"modulus\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0098","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"composite_modulus\",\"modulus\":51,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(composite_modulus, 51, 23, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(composite_modulus, 51, 23, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134098,"parameters":{"mode":"composite_modulus","modulus":51,"a":23,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":23,\"mode\":\"composite_modulus\",\"modulus\":51}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0099","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"zero_class\",\"modulus\":97,\"a\":2328}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_class, 97, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(zero_class, 97, 0, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134099,"parameters":{"mode":"zero_class","modulus":97,"a":2328,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":2328,\"mode\":\"zero_class\",\"modulus\":97}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0100","batch_id":"batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations","language":"zh-Hant","source_mko_ids":["mko-euler-criterion-quadratic-residue"],"title_zh":"Euler二次剩餘判準","stem_zh":"odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(even_prime, 2, 1, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)","explanation_zh":"Wave6第七層核對=(even_prime, 2, 1, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Euler二次剩餘判準"]},"variant":{"family_id":"qf-mod44-eulercrit-domain-boundary","seed":202611134100,"parameters":{"mode":"even_prime","modulus":2,"a":25,"domain":"Odd_prime_Euler_criterion","operation":"eulercrit-domain-boundary","semantic_template":"qf-mod44-eulercrit-domain-boundary/{\"a\":25,\"mode\":\"even_prime\",\"modulus\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}