{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0001","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [2], [2], -1, squarefree_parity_sign)","explanation_zh":"Wave6第三層核對=(2, [2], [2], -1, squarefree_parity_sign)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096001,"parameters":{"n":2,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0002","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [2, 3], [2, 3], 1, squarefree_parity_sign)","explanation_zh":"Wave6第三層核對=(6, [2, 3], [2, 3], 1, squarefree_parity_sign)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096002,"parameters":{"n":6,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0003","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [2, 2, 3], [2, 3], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(12, [2, 2, 3], [2, 3], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096003,"parameters":{"n":12,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0004","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, [2, 2, 5], [2, 5], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(20, [2, 2, 5], [2, 5], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096004,"parameters":{"n":20,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0005","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":30}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(30, [2, 3, 5], [2, 3, 5], -1, squarefree_parity_sign)","explanation_zh":"Wave6第三層核對=(30, [2, 3, 5], [2, 3, 5], -1, squarefree_parity_sign)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096005,"parameters":{"n":30,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0006","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":42}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(42, [2, 3, 7], [2, 3, 7], -1, squarefree_parity_sign)","explanation_zh":"Wave6第三層核對=(42, [2, 3, 7], [2, 3, 7], -1, squarefree_parity_sign)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096006,"parameters":{"n":42,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":42}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0007","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":56}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(56, [2, 2, 2, 7], [2, 7], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(56, [2, 2, 2, 7], [2, 7], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096007,"parameters":{"n":56,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0008","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":72}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(72, [2, 2, 2, 3, 3], [2, 3], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(72, [2, 2, 2, 3, 3], [2, 3], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096008,"parameters":{"n":72,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":72}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0009","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":90}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(90, [2, 3, 3, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(90, [2, 3, 3, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096009,"parameters":{"n":90,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":90}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0010","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":110}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(110, [2, 5, 11], [2, 5, 11], -1, squarefree_parity_sign)","explanation_zh":"Wave6第三層核對=(110, [2, 5, 11], [2, 5, 11], -1, squarefree_parity_sign)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096010,"parameters":{"n":110,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":110}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0011","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":132}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(132, [2, 2, 3, 11], [2, 3, 11], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(132, [2, 2, 3, 11], [2, 3, 11], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096011,"parameters":{"n":132,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":132}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0012","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":156}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(156, [2, 2, 3, 13], [2, 3, 13], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(156, [2, 2, 3, 13], [2, 3, 13], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096012,"parameters":{"n":156,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":156}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0013","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":182}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(182, [2, 7, 13], [2, 7, 13], -1, squarefree_parity_sign)","explanation_zh":"Wave6第三層核對=(182, [2, 7, 13], [2, 7, 13], -1, squarefree_parity_sign)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096013,"parameters":{"n":182,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":182}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0014","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":210}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(210, [2, 3, 5, 7], [2, 3, 5, 7], 1, squarefree_parity_sign)","explanation_zh":"Wave6第三層核對=(210, [2, 3, 5, 7], [2, 3, 5, 7], 1, squarefree_parity_sign)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096014,"parameters":{"n":210,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":210}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0015","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":240}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(240, [2, 2, 2, 2, 3, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(240, [2, 2, 2, 2, 3, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096015,"parameters":{"n":240,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":240}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0016","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":272}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(272, [2, 2, 2, 2, 17], [2, 17], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(272, [2, 2, 2, 2, 17], [2, 17], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096016,"parameters":{"n":272,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":272}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0017","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":306}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(306, [2, 3, 3, 17], [2, 3, 17], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(306, [2, 3, 3, 17], [2, 3, 17], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096017,"parameters":{"n":306,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":306}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0018","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":342}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(342, [2, 3, 3, 19], [2, 3, 19], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(342, [2, 3, 3, 19], [2, 3, 19], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096018,"parameters":{"n":342,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":342}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0019","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":380}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(380, [2, 2, 5, 19], [2, 5, 19], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(380, [2, 2, 5, 19], [2, 5, 19], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096019,"parameters":{"n":380,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":380}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0020","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":420}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(420, [2, 2, 3, 5, 7], [2, 3, 5, 7], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(420, [2, 2, 3, 5, 7], [2, 3, 5, 7], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096020,"parameters":{"n":420,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":420}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0021","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":462}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(462, [2, 3, 7, 11], [2, 3, 7, 11], 1, squarefree_parity_sign)","explanation_zh":"Wave6第三層核對=(462, [2, 3, 7, 11], [2, 3, 7, 11], 1, squarefree_parity_sign)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096021,"parameters":{"n":462,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":462}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0022","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":506}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(506, [2, 11, 23], [2, 11, 23], -1, squarefree_parity_sign)","explanation_zh":"Wave6第三層核對=(506, [2, 11, 23], [2, 11, 23], -1, squarefree_parity_sign)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096022,"parameters":{"n":506,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":506}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0023","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":552}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(552, [2, 2, 2, 3, 23], [2, 3, 23], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(552, [2, 2, 2, 3, 23], [2, 3, 23], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096023,"parameters":{"n":552,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":552}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0024","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":600}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(600, [2, 2, 2, 3, 5, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(600, [2, 2, 2, 3, 5, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096024,"parameters":{"n":600,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":600}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0025","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={\"n\":650}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(650, [2, 5, 5, 13], [2, 5, 13], 0, square_prime_factor_gives_zero)","explanation_zh":"Wave6第三層核對=(650, [2, 5, 5, 13], [2, 5, 13], 0, square_prime_factor_gives_zero)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-canonical-value","seed":202611096025,"parameters":{"n":650,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-canonical-value","semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":650}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0026","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":4,\"factors\":[2,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [2, 2], 是, 0, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(4, [2, 2], 是, 0, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096026,"parameters":{"n":4,"factors":[2,2],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,2],\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0027","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":6,\"factors\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [2, 3], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(6, [2, 3], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096027,"parameters":{"n":6,"factors":[2,3],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,3],\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0028","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":10,\"factors\":[2,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [2, 5], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(10, [2, 5], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096028,"parameters":{"n":10,"factors":[2,5],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,5],\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0029","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":14,\"factors\":[2,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [2, 7], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(14, [2, 7], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096029,"parameters":{"n":14,"factors":[2,7],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,7],\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0030","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":22,\"factors\":[2,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, [2, 11], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(22, [2, 11], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096030,"parameters":{"n":22,"factors":[2,11],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,11],\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0031","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":26,\"factors\":[2,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(26, [2, 13], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(26, [2, 13], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096031,"parameters":{"n":26,"factors":[2,13],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,13],\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0032","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":34,\"factors\":[2,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(34, [2, 17], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(34, [2, 17], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096032,"parameters":{"n":34,"factors":[2,17],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,17],\"n\":34}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0033","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":38,\"factors\":[2,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(38, [2, 19], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(38, [2, 19], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096033,"parameters":{"n":38,"factors":[2,19],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,19],\"n\":38}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0034","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":46,\"factors\":[2,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(46, [2, 23], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(46, [2, 23], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096034,"parameters":{"n":46,"factors":[2,23],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,23],\"n\":46}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0035","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":58,\"factors\":[2,29]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(58, [2, 29], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(58, [2, 29], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096035,"parameters":{"n":58,"factors":[2,29],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,29],\"n\":58}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0036","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":62,\"factors\":[2,31]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(62, [2, 31], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(62, [2, 31], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096036,"parameters":{"n":62,"factors":[2,31],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,31],\"n\":62}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0037","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":74,\"factors\":[2,37]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(74, [2, 37], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(74, [2, 37], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096037,"parameters":{"n":74,"factors":[2,37],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,37],\"n\":74}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0038","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":82,\"factors\":[2,41]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(82, [2, 41], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(82, [2, 41], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096038,"parameters":{"n":82,"factors":[2,41],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,41],\"n\":82}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0039","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":86,\"factors\":[2,43]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(86, [2, 43], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(86, [2, 43], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096039,"parameters":{"n":86,"factors":[2,43],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,43],\"n\":86}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0040","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":94,\"factors\":[2,47]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(94, [2, 47], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(94, [2, 47], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096040,"parameters":{"n":94,"factors":[2,47],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,47],\"n\":94}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0041","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":106,\"factors\":[2,53]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(106, [2, 53], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(106, [2, 53], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096041,"parameters":{"n":106,"factors":[2,53],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,53],\"n\":106}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0042","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":118,\"factors\":[2,59]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(118, [2, 59], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(118, [2, 59], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096042,"parameters":{"n":118,"factors":[2,59],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,59],\"n\":118}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0043","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":122,\"factors\":[2,61]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(122, [2, 61], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(122, [2, 61], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096043,"parameters":{"n":122,"factors":[2,61],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,61],\"n\":122}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0044","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":134,\"factors\":[2,67]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(134, [2, 67], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(134, [2, 67], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096044,"parameters":{"n":134,"factors":[2,67],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,67],\"n\":134}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0045","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":142,\"factors\":[2,71]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(142, [2, 71], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(142, [2, 71], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096045,"parameters":{"n":142,"factors":[2,71],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,71],\"n\":142}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0046","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":146,\"factors\":[2,73]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(146, [2, 73], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(146, [2, 73], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096046,"parameters":{"n":146,"factors":[2,73],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,73],\"n\":146}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0047","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":158,\"factors\":[2,79]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(158, [2, 79], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(158, [2, 79], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096047,"parameters":{"n":158,"factors":[2,79],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,79],\"n\":158}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0048","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":166,\"factors\":[2,83]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(166, [2, 83], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(166, [2, 83], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096048,"parameters":{"n":166,"factors":[2,83],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,83],\"n\":166}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0049","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":178,\"factors\":[2,89]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(178, [2, 89], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(178, [2, 89], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096049,"parameters":{"n":178,"factors":[2,89],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,89],\"n\":178}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0050","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-factorization-audit；資料={\"n\":194,\"factors\":[2,97]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(194, [2, 97], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)","explanation_zh":"Wave6第三層核對=(194, [2, 97], 是, 1, 是, complete_UPF_multiplicity_and_distinct_prime_count)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-factorization-audit","seed":202611096050,"parameters":{"n":194,"factors":[2,97],"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-factorization-audit","semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,97],\"n\":194}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0051","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[1, 1]], 1, 1, 是)","explanation_zh":"Wave6第三層核對=(1, [[1, 1]], 1, 1, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096051,"parameters":{"n":1,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0052","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, 1], [2, -1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(2, [[1, 1], [2, -1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096052,"parameters":{"n":2,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0053","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, 1], [3, -1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(3, [[1, 1], [3, -1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096053,"parameters":{"n":3,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0054","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[1, 1], [2, -1], [4, 0]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(4, [[1, 1], [2, -1], [4, 0]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096054,"parameters":{"n":4,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0055","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[1, 1], [5, -1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(5, [[1, 1], [5, -1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096055,"parameters":{"n":5,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0056","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[1, 1], [2, -1], [3, -1], [6, 1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(6, [[1, 1], [2, -1], [3, -1], [6, 1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096056,"parameters":{"n":6,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0057","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, [[1, 1], [7, -1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(7, [[1, 1], [7, -1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096057,"parameters":{"n":7,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0058","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[1, 1], [2, -1], [4, 0], [8, 0]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(8, [[1, 1], [2, -1], [4, 0], [8, 0]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096058,"parameters":{"n":8,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0059","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, [[1, 1], [3, -1], [9, 0]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(9, [[1, 1], [3, -1], [9, 0]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096059,"parameters":{"n":9,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0060","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[1, 1], [2, -1], [5, -1], [10, 1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(10, [[1, 1], [2, -1], [5, -1], [10, 1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096060,"parameters":{"n":10,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0061","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, [[1, 1], [11, -1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(11, [[1, 1], [11, -1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096061,"parameters":{"n":11,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0062","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [12, 0]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(12, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [12, 0]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096062,"parameters":{"n":12,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0063","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, [[1, 1], [13, -1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(13, [[1, 1], [13, -1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096063,"parameters":{"n":13,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0064","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [[1, 1], [2, -1], [7, -1], [14, 1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(14, [[1, 1], [2, -1], [7, -1], [14, 1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096064,"parameters":{"n":14,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0065","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, [[1, 1], [3, -1], [5, -1], [15, 1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(15, [[1, 1], [3, -1], [5, -1], [15, 1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096065,"parameters":{"n":15,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0066","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, [[1, 1], [2, -1], [4, 0], [8, 0], [16, 0]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(16, [[1, 1], [2, -1], [4, 0], [8, 0], [16, 0]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096066,"parameters":{"n":16,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0067","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, [[1, 1], [17, -1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(17, [[1, 1], [17, -1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096067,"parameters":{"n":17,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0068","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, [[1, 1], [2, -1], [3, -1], [6, 1], [9, 0], [18, 0]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(18, [[1, 1], [2, -1], [3, -1], [6, 1], [9, 0], [18, 0]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096068,"parameters":{"n":18,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0069","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, [[1, 1], [19, -1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(19, [[1, 1], [19, -1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096069,"parameters":{"n":19,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0070","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, [[1, 1], [2, -1], [4, 0], [5, -1], [10, 1], [20, 0]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(20, [[1, 1], [2, -1], [4, 0], [5, -1], [10, 1], [20, 0]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096070,"parameters":{"n":20,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0071","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, [[1, 1], [3, -1], [7, -1], [21, 1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(21, [[1, 1], [3, -1], [7, -1], [21, 1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096071,"parameters":{"n":21,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0072","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, [[1, 1], [2, -1], [11, -1], [22, 1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(22, [[1, 1], [2, -1], [11, -1], [22, 1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096072,"parameters":{"n":22,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0073","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, [[1, 1], [23, -1]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(23, [[1, 1], [23, -1]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096073,"parameters":{"n":23,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0074","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [8, 0], [12, 0], [24, 0]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(24, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [8, 0], [12, 0], [24, 0]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096074,"parameters":{"n":24,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0075","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-divisor-sum；資料={\"n\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, [[1, 1], [5, -1], [25, 0]], 0, 0, 是)","explanation_zh":"Wave6第三層核對=(25, [[1, 1], [5, -1], [25, 0]], 0, 0, 是)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-divisor-sum","seed":202611096075,"parameters":{"n":25,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-divisor-sum","semantic_template":"qf-mod40-mobius-divisor-sum/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0076","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(0, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096076,"parameters":{"n":0,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0077","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-1, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096077,"parameters":{"n":-1,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0078","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-2, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096078,"parameters":{"n":-2,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0079","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-3, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096079,"parameters":{"n":-3,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0080","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-4, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096080,"parameters":{"n":-4,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0081","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-5, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096081,"parameters":{"n":-5,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0082","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-6, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096082,"parameters":{"n":-6,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0083","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-7, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096083,"parameters":{"n":-7,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0084","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-8, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096084,"parameters":{"n":-8,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0085","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-9, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096085,"parameters":{"n":-9,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0086","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-10, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096086,"parameters":{"n":-10,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0087","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-11, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096087,"parameters":{"n":-11,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0088","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-12, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096088,"parameters":{"n":-12,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0089","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-13, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096089,"parameters":{"n":-13,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0090","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-14, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-14, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096090,"parameters":{"n":-14,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0091","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-15, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-15, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096091,"parameters":{"n":-15,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0092","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-16, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-16, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096092,"parameters":{"n":-16,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0093","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-17, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-17, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096093,"parameters":{"n":-17,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0094","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-18, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-18, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096094,"parameters":{"n":-18,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0095","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-19, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-19, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096095,"parameters":{"n":-19,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0096","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-20, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-20, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096096,"parameters":{"n":-20,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0097","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-21, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-21, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096097,"parameters":{"n":-21,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0098","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-22, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-22, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096098,"parameters":{"n":-22,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0099","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-23, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-23, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096099,"parameters":{"n":-23,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r06-mobius-function-0100","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-function"],"title_zh":"Möbius函數","stem_zh":"positive integer domain與完整UPF multiplicities。操作=mobius-domain-boundary；資料={\"n\":-24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-24, 否, null, Mobius_domain_positive_integers_only)","explanation_zh":"Wave6第三層核對=(-24, 否, null, Mobius_domain_positive_integers_only)；依完整明示資料獨立重算；mu(1)=1，square prime factor gives0，squarefree sign由distinct prime count parity；factor order不影響；zero negative或incomplete trial result不可分類。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Möbius函數"]},"variant":{"family_id":"qf-mod40-mobius-domain-boundary","seed":202611096100,"parameters":{"n":-24,"domain":"Exact_positive_integer_Mobius_function","operation":"mobius-domain-boundary","semantic_template":"qf-mod40-mobius-domain-boundary/{\"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、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}