{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r07-multiplicative-order-mod-n-0001","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":3,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 2, 1, 2, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(3, 2, 2, 1, 2, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167001,"parameters":{"n":3,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0002","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":4,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 3, 3, 1, 2, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(4, 3, 3, 1, 2, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167002,"parameters":{"n":4,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0003","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":5,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 2, 1, 4, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(5, 2, 2, 1, 4, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167003,"parameters":{"n":5,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0004","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":6,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 5, 5, 1, 2, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(6, 5, 5, 1, 2, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167004,"parameters":{"n":6,"a":5,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[5,1],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0005","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":7,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 2, 2, 1, 6, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(7, 2, 2, 1, 6, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167005,"parameters":{"n":7,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0006","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":8,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 3, 3, 1, 4, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(8, 3, 3, 1, 4, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167006,"parameters":{"n":8,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0007","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":9,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 2, 2, 1, 6, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(9, 2, 2, 1, 6, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167007,"parameters":{"n":9,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0008","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":10,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 3, 3, 1, 4, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(10, 3, 3, 1, 4, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167008,"parameters":{"n":10,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0009","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":11,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 2, 2, 1, 10, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(11, 2, 2, 1, 10, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167009,"parameters":{"n":11,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0010","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":12,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 5, 5, 1, 4, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(12, 5, 5, 1, 4, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167010,"parameters":{"n":12,"a":5,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[5,1],\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0011","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":13,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 2, 2, 1, 12, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(13, 2, 2, 1, 12, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167011,"parameters":{"n":13,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0012","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":14,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 3, 3, 1, 6, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(14, 3, 3, 1, 6, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167012,"parameters":{"n":14,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0013","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":15,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 2, 2, 1, 8, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(15, 2, 2, 1, 8, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167013,"parameters":{"n":15,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0014","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":16,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 3, 3, 1, 8, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(16, 3, 3, 1, 8, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167014,"parameters":{"n":16,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0015","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":17,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 2, 2, 1, 16, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(17, 2, 2, 1, 16, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167015,"parameters":{"n":17,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0016","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":18,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 5, 5, 1, 6, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(18, 5, 5, 1, 6, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167016,"parameters":{"n":18,"a":5,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[5,1],\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0017","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":19,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 2, 2, 1, 18, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(19, 2, 2, 1, 18, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167017,"parameters":{"n":19,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0018","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":20,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 3, 3, 1, 8, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(20, 3, 3, 1, 8, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167018,"parameters":{"n":20,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0019","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":21,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 2, 2, 1, 12, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(21, 2, 2, 1, 12, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167019,"parameters":{"n":21,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0020","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":22,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 3, 3, 1, 10, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(22, 3, 3, 1, 10, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167020,"parameters":{"n":22,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0021","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":23,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 2, 2, 1, 22, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(23, 2, 2, 1, 22, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167021,"parameters":{"n":23,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0022","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":24,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 5, 5, 1, 8, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(24, 5, 5, 1, 8, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167022,"parameters":{"n":24,"a":5,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[5,1],\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0023","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":25,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, 2, 2, 1, 20, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(25, 2, 2, 1, 20, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167023,"parameters":{"n":25,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0024","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":26,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(26, 3, 3, 1, 12, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(26, 3, 3, 1, 12, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167024,"parameters":{"n":26,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[3,1],\"n\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0025","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-qualification；資料={\"n\":27,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(27, 2, 2, 1, 18, 是, unit_mod_n_qualified)","explanation_zh":"Wave6第十層核對=(27, 2, 2, 1, 18, 是, unit_mod_n_qualified)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-qualification","seed":202611167025,"parameters":{"n":27,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-qualification","semantic_template":"qf-mod47-morder-qualification/{\"a\":[2,1],\"n\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0026","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":3,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 1]], 2, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 1]], 2, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167026,"parameters":{"n":3,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0027","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":4,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [2, 1]], 2, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 3], [2, 1]], 2, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167027,"parameters":{"n":4,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0028","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":5,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 3], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 4], [3, 3], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167028,"parameters":{"n":5,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0029","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":6,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 5], [2, 1]], 2, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 5], [2, 1]], 2, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167029,"parameters":{"n":6,"a":5,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[5,1],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0030","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":7,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 1]], 3, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 4], [3, 1]], 3, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167030,"parameters":{"n":7,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0031","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":8,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [2, 1]], 2, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 3], [2, 1]], 2, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167031,"parameters":{"n":8,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0032","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":9,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 7], [5, 5], [6, 1]], 6, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 4], [3, 8], [4, 7], [5, 5], [6, 1]], 6, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167032,"parameters":{"n":9,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0033","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":10,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [2, 9], [3, 7], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 3], [2, 9], [3, 7], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167033,"parameters":{"n":10,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0034","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":11,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 5], [5, 10], [6, 9], [7, 7], [8, 3], [9, 6], [10, 1]], 10, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 4], [3, 8], [4, 5], [5, 10], [6, 9], [7, 7], [8, 3], [9, 6], [10, 1]], 10, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167034,"parameters":{"n":11,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0035","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":12,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 5], [2, 1]], 2, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 5], [2, 1]], 2, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167035,"parameters":{"n":12,"a":5,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[5,1],\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0036","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":13,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 3], [5, 6], [6, 12], [7, 11], [8, 9], [9, 5], [10, 10], [11, 7], [12, 1]], 12, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 4], [3, 8], [4, 3], [5, 6], [6, 12], [7, 11], [8, 9], [9, 5], [10, 10], [11, 7], [12, 1]], 12, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167036,"parameters":{"n":13,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0037","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":14,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [2, 9], [3, 13], [4, 11], [5, 5], [6, 1]], 6, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 3], [2, 9], [3, 13], [4, 11], [5, 5], [6, 1]], 6, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167037,"parameters":{"n":14,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0038","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":15,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 4], [3, 8], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167038,"parameters":{"n":15,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0039","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":16,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [2, 9], [3, 11], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 3], [2, 9], [3, 11], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167039,"parameters":{"n":16,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0040","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":17,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], 8, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], 8, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167040,"parameters":{"n":17,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0041","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":18,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], 6, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], 6, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167041,"parameters":{"n":18,"a":5,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[5,1],\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0042","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":19,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 16], [5, 13], [6, 7], [7, 14], [8, 9], [9, 18], [10, 17], [11, 15], [12, 11], [13, 3], [14, 6], [15, 12], [16, 5], [17, 10], [18, 1]], 18, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 4], [3, 8], [4, 16], [5, 13], [6, 7], [7, 14], [8, 9], [9, 18], [10, 17], [11, 15], [12, 11], [13, 3], [14, 6], [15, 12], [16, 5], [17, 10], [18, 1]], 18, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167042,"parameters":{"n":19,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0043","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":20,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [2, 9], [3, 7], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 3], [2, 9], [3, 7], [4, 1]], 4, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167043,"parameters":{"n":20,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0044","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":21,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]], 6, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]], 6, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167044,"parameters":{"n":21,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0045","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":22,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], 5, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], 5, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167045,"parameters":{"n":22,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0046","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":23,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 16], [5, 9], [6, 18], [7, 13], [8, 3], [9, 6], [10, 12], [11, 1]], 11, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 4], [3, 8], [4, 16], [5, 9], [6, 18], [7, 13], [8, 3], [9, 6], [10, 12], [11, 1]], 11, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167046,"parameters":{"n":23,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0047","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":24,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 5], [2, 1]], 2, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 5], [2, 1]], 2, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167047,"parameters":{"n":24,"a":5,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[5,1],\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0048","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":25,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 16], [5, 7], [6, 14], [7, 3], [8, 6], [9, 12], [10, 24], [11, 23], [12, 21], [13, 17], [14, 9], [15, 18], [16, 11], [17, 22], [18, 19], [19, 13], [20, 1]], 20, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 4], [3, 8], [4, 16], [5, 7], [6, 14], [7, 3], [8, 6], [9, 12], [10, 24], [11, 23], [12, 21], [13, 17], [14, 9], [15, 18], [16, 11], [17, 22], [18, 19], [19, 13], [20, 1]], 20, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167048,"parameters":{"n":25,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0049","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":26,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 3], [2, 9], [3, 1]], 3, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 3], [2, 9], [3, 1]], 3, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167049,"parameters":{"n":26,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[3,1],\"n\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0050","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-power-first-return；資料={\"n\":27,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2], [2, 4], [3, 8], [4, 16], [5, 5], [6, 10], [7, 20], [8, 13], [9, 26], [10, 25], [11, 23], [12, 19], [13, 11], [14, 22], [15, 17], [16, 7], [17, 14], [18, 1]], 18, 是, 是, order_is_the_first_positive_return)","explanation_zh":"Wave6第十層核對=([[1, 2], [2, 4], [3, 8], [4, 16], [5, 5], [6, 10], [7, 20], [8, 13], [9, 26], [10, 25], [11, 23], [12, 19], [13, 11], [14, 22], [15, 17], [16, 7], [17, 14], [18, 1]], 18, 是, 是, order_is_the_first_positive_return)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-power-first-return","seed":202611167050,"parameters":{"n":27,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-power-first-return","semantic_template":"qf-mod47-morder-power-first-return/{\"a\":[2,1],\"n\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0051","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":3,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, [[1, 2], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(2, 2, 是, [[1, 2], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167051,"parameters":{"n":3,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0052","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":4,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, [[1, 3], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(2, 2, 是, [[1, 3], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167052,"parameters":{"n":4,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0053","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":5,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 是, [[1, 2], [2, 4], [3, 3], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(4, 4, 是, [[1, 2], [2, 4], [3, 3], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167053,"parameters":{"n":5,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0054","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":6,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 是, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(2, 2, 是, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167054,"parameters":{"n":6,"a":5,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[5,1],\"n\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0055","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":7,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 6, 是, [[1, 2], [2, 4], [3, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(3, 6, 是, [[1, 2], [2, 4], [3, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167055,"parameters":{"n":7,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0056","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":8,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 是, [[1, 3], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(2, 4, 是, [[1, 3], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167056,"parameters":{"n":8,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0057","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":9,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 6, 是, [[1, 2], [2, 4], [3, 8], [4, 7], [5, 5], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(6, 6, 是, [[1, 2], [2, 4], [3, 8], [4, 7], [5, 5], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167057,"parameters":{"n":9,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0058","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":10,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 是, [[1, 3], [2, 9], [3, 7], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(4, 4, 是, [[1, 3], [2, 9], [3, 7], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167058,"parameters":{"n":10,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0059","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":11,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 10, 是, [[1, 2], [2, 4], [3, 8], [4, 5], [5, 10], [6, 9], [7, 7], [8, 3], [9, 6], [10, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(10, 10, 是, [[1, 2], [2, 4], [3, 8], [4, 5], [5, 10], [6, 9], [7, 7], [8, 3], [9, 6], [10, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167059,"parameters":{"n":11,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0060","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":12,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 是, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(2, 4, 是, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167060,"parameters":{"n":12,"a":5,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[5,1],\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0061","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":13,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 12, 是, [[1, 2], [2, 4], [3, 8], [4, 3], [5, 6], [6, 12], [7, 11], [8, 9], [9, 5], [10, 10], [11, 7], [12, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(12, 12, 是, [[1, 2], [2, 4], [3, 8], [4, 3], [5, 6], [6, 12], [7, 11], [8, 9], [9, 5], [10, 10], [11, 7], [12, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167061,"parameters":{"n":13,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0062","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":14,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 6, 是, [[1, 3], [2, 9], [3, 13], [4, 11], [5, 5], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(6, 6, 是, [[1, 3], [2, 9], [3, 13], [4, 11], [5, 5], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167062,"parameters":{"n":14,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0063","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":15,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 8, 是, [[1, 2], [2, 4], [3, 8], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(4, 8, 是, [[1, 2], [2, 4], [3, 8], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167063,"parameters":{"n":15,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0064","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":16,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 8, 是, [[1, 3], [2, 9], [3, 11], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(4, 8, 是, [[1, 3], [2, 9], [3, 11], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167064,"parameters":{"n":16,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0065","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":17,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 16, 是, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(8, 16, 是, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 15], [6, 13], [7, 9], [8, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167065,"parameters":{"n":17,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0066","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":18,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 6, 是, [[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(6, 6, 是, [[1, 5], [2, 7], [3, 17], [4, 13], [5, 11], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167066,"parameters":{"n":18,"a":5,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[5,1],\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0067","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":19,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 18, 是, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 13], [6, 7], [7, 14], [8, 9], [9, 18], [10, 17], [11, 15], [12, 11], [13, 3], [14, 6], [15, 12], [16, 5], [17, 10], [18, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(18, 18, 是, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 13], [6, 7], [7, 14], [8, 9], [9, 18], [10, 17], [11, 15], [12, 11], [13, 3], [14, 6], [15, 12], [16, 5], [17, 10], [18, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167067,"parameters":{"n":19,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0068","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":20,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 8, 是, [[1, 3], [2, 9], [3, 7], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(4, 8, 是, [[1, 3], [2, 9], [3, 7], [4, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167068,"parameters":{"n":20,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0069","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":21,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 12, 是, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(6, 12, 是, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 11], [6, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167069,"parameters":{"n":21,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0070","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":22,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 10, 是, [[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(5, 10, 是, [[1, 3], [2, 9], [3, 5], [4, 15], [5, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167070,"parameters":{"n":22,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0071","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":23,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 22, 是, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 9], [6, 18], [7, 13], [8, 3], [9, 6], [10, 12], [11, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(11, 22, 是, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 9], [6, 18], [7, 13], [8, 3], [9, 6], [10, 12], [11, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167071,"parameters":{"n":23,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0072","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":24,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 8, 是, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(2, 8, 是, [[1, 5], [2, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167072,"parameters":{"n":24,"a":5,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[5,1],\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0073","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":25,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 20, 是, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 7], [6, 14], [7, 3], [8, 6], [9, 12], [10, 24], [11, 23], [12, 21], [13, 17], [14, 9], [15, 18], [16, 11], [17, 22], [18, 19], [19, 13], [20, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(20, 20, 是, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 7], [6, 14], [7, 3], [8, 6], [9, 12], [10, 24], [11, 23], [12, 21], [13, 17], [14, 9], [15, 18], [16, 11], [17, 22], [18, 19], [19, 13], [20, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167073,"parameters":{"n":25,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0074","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":26,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 12, 是, [[1, 3], [2, 9], [3, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(3, 12, 是, [[1, 3], [2, 9], [3, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167074,"parameters":{"n":26,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[3,1],\"n\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0075","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-phi-divisibility；資料={\"n\":27,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 18, 是, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 5], [6, 10], [7, 20], [8, 13], [9, 26], [10, 25], [11, 23], [12, 19], [13, 11], [14, 22], [15, 17], [16, 7], [17, 14], [18, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)","explanation_zh":"Wave6第十層核對=(18, 18, 是, [[1, 2], [2, 4], [3, 8], [4, 16], [5, 5], [6, 10], [7, 20], [8, 13], [9, 26], [10, 25], [11, 23], [12, 19], [13, 11], [14, 22], [15, 17], [16, 7], [17, 14], [18, 1]], ord_n_a_divides_phi_n_without_claiming_equality_or_primitive_root)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-phi-divisibility","seed":202611167075,"parameters":{"n":27,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-phi-divisibility","semantic_template":"qf-mod47-morder-phi-divisibility/{\"a\":[2,1],\"n\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0076","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"noncoprime\",\"n\":8,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncoprime, 8, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(noncoprime, 8, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167076,"parameters":{"mode":"noncoprime","n":8,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0077","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"invalid_modulus\",\"n\":1,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(invalid_modulus, 1, 3, null, null, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(invalid_modulus, 1, 3, null, null, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167077,"parameters":{"mode":"invalid_modulus","n":1,"a":3,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[3,1],\"mode\":\"invalid_modulus\",\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0078","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"order_one\",\"n\":5,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(order_one, 5, 1, 1, 1, 是, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(order_one, 5, 1, 1, 1, 是, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167078,"parameters":{"mode":"order_one","n":5,"a":1,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[1,1],\"mode\":\"order_one\",\"n\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0079","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"negative_normalization\",\"n\":8,\"a\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(negative_normalization, 8, -1, 7, 1, 是, [[1, 7], [2, 1]], 2, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(negative_normalization, 8, -1, 7, 1, 是, [[1, 7], [2, 1]], 2, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167079,"parameters":{"mode":"negative_normalization","n":8,"a":-1,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[-1,1],\"mode\":\"negative_normalization\",\"n\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0080","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"noncoprime\",\"n\":16,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncoprime, 16, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(noncoprime, 16, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167080,"parameters":{"mode":"noncoprime","n":16,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0081","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"invalid_modulus\",\"n\":1,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(invalid_modulus, 1, 7, null, null, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(invalid_modulus, 1, 7, null, null, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167081,"parameters":{"mode":"invalid_modulus","n":1,"a":7,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[7,1],\"mode\":\"invalid_modulus\",\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0082","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"order_one\",\"n\":9,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(order_one, 9, 1, 1, 1, 是, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(order_one, 9, 1, 1, 1, 是, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167082,"parameters":{"mode":"order_one","n":9,"a":1,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[1,1],\"mode\":\"order_one\",\"n\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0083","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"negative_normalization\",\"n\":12,\"a\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(negative_normalization, 12, -1, 11, 1, 是, [[1, 11], [2, 1]], 2, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(negative_normalization, 12, -1, 11, 1, 是, [[1, 11], [2, 1]], 2, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167083,"parameters":{"mode":"negative_normalization","n":12,"a":-1,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[-1,1],\"mode\":\"negative_normalization\",\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0084","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"noncoprime\",\"n\":24,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncoprime, 24, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(noncoprime, 24, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167084,"parameters":{"mode":"noncoprime","n":24,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0085","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"invalid_modulus\",\"n\":1,\"a\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(invalid_modulus, 1, 11, null, null, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(invalid_modulus, 1, 11, null, null, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167085,"parameters":{"mode":"invalid_modulus","n":1,"a":11,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[11,1],\"mode\":\"invalid_modulus\",\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0086","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"order_one\",\"n\":13,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(order_one, 13, 1, 1, 1, 是, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(order_one, 13, 1, 1, 1, 是, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167086,"parameters":{"mode":"order_one","n":13,"a":1,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[1,1],\"mode\":\"order_one\",\"n\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0087","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"negative_normalization\",\"n\":16,\"a\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(negative_normalization, 16, -1, 15, 1, 是, [[1, 15], [2, 1]], 2, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(negative_normalization, 16, -1, 15, 1, 是, [[1, 15], [2, 1]], 2, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167087,"parameters":{"mode":"negative_normalization","n":16,"a":-1,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[-1,1],\"mode\":\"negative_normalization\",\"n\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0088","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"noncoprime\",\"n\":32,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncoprime, 32, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(noncoprime, 32, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167088,"parameters":{"mode":"noncoprime","n":32,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":32}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0089","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"invalid_modulus\",\"n\":1,\"a\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(invalid_modulus, 1, 15, null, null, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(invalid_modulus, 1, 15, null, null, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167089,"parameters":{"mode":"invalid_modulus","n":1,"a":15,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[15,1],\"mode\":\"invalid_modulus\",\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0090","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"order_one\",\"n\":17,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(order_one, 17, 1, 1, 1, 是, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(order_one, 17, 1, 1, 1, 是, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167090,"parameters":{"mode":"order_one","n":17,"a":1,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[1,1],\"mode\":\"order_one\",\"n\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0091","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"negative_normalization\",\"n\":20,\"a\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(negative_normalization, 20, -1, 19, 1, 是, [[1, 19], [2, 1]], 2, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(negative_normalization, 20, -1, 19, 1, 是, [[1, 19], [2, 1]], 2, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167091,"parameters":{"mode":"negative_normalization","n":20,"a":-1,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[-1,1],\"mode\":\"negative_normalization\",\"n\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0092","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"noncoprime\",\"n\":40,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncoprime, 40, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(noncoprime, 40, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167092,"parameters":{"mode":"noncoprime","n":40,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":40}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0093","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"invalid_modulus\",\"n\":1,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(invalid_modulus, 1, 19, null, null, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(invalid_modulus, 1, 19, null, null, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167093,"parameters":{"mode":"invalid_modulus","n":1,"a":19,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[19,1],\"mode\":\"invalid_modulus\",\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0094","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"order_one\",\"n\":21,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(order_one, 21, 1, 1, 1, 是, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(order_one, 21, 1, 1, 1, 是, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167094,"parameters":{"mode":"order_one","n":21,"a":1,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[1,1],\"mode\":\"order_one\",\"n\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0095","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"negative_normalization\",\"n\":24,\"a\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(negative_normalization, 24, -1, 23, 1, 是, [[1, 23], [2, 1]], 2, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(negative_normalization, 24, -1, 23, 1, 是, [[1, 23], [2, 1]], 2, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167095,"parameters":{"mode":"negative_normalization","n":24,"a":-1,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[-1,1],\"mode\":\"negative_normalization\",\"n\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0096","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"noncoprime\",\"n\":48,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncoprime, 48, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(noncoprime, 48, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167096,"parameters":{"mode":"noncoprime","n":48,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":48}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0097","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"invalid_modulus\",\"n\":1,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(invalid_modulus, 1, 23, null, null, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(invalid_modulus, 1, 23, null, null, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167097,"parameters":{"mode":"invalid_modulus","n":1,"a":23,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[23,1],\"mode\":\"invalid_modulus\",\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0098","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"order_one\",\"n\":25,\"a\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(order_one, 25, 1, 1, 1, 是, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(order_one, 25, 1, 1, 1, 是, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167098,"parameters":{"mode":"order_one","n":25,"a":1,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[1,1],\"mode\":\"order_one\",\"n\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0099","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"negative_normalization\",\"n\":28,\"a\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(negative_normalization, 28, -1, 27, 1, 是, [[1, 27], [2, 1]], 2, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(negative_normalization, 28, -1, 27, 1, 是, [[1, 27], [2, 1]], 2, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167099,"parameters":{"mode":"negative_normalization","n":28,"a":-1,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[-1,1],\"mode\":\"negative_normalization\",\"n\":28}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、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-20261116-r07-multiplicative-order-mod-n-0100","batch_id":"batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations","language":"zh-Hant","source_mko_ids":["mko-multiplicative-order-mod-n"],"title_zh":"模n乘法階","stem_zh":"n>=2、normalized coprime residue與first positive power return責任。操作=morder-boundary；資料={\"mode\":\"noncoprime\",\"n\":56,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(noncoprime, 56, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)","explanation_zh":"Wave6第十層核對=(noncoprime, 56, 2, 2, 2, 否, null, undefined, mode_specific_multiplicative_order_boundary)；依完整明示資料獨立重算；complete power table與phi enumeration核order divides phi；noncoprime/invalid modulus undefined；不宣稱order=phi、primitive root或additive order。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["模n乘法階"]},"variant":{"family_id":"qf-mod47-morder-boundary","seed":202611167100,"parameters":{"mode":"noncoprime","n":56,"a":2,"domain":"Exact_multiplicative_order_mod_n","operation":"morder-boundary","semantic_template":"qf-mod47-morder-boundary/{\"a\":[2,1],\"mode\":\"noncoprime\",\"n\":56}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}