{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0001","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":1,\"a\":1,\"b\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[1, -11]], -11, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(1, [[1, -11]], -11, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106001,"parameters":{"n":1,"a":1,"b":-12,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":1,\"b\":-12,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0002","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":2,\"a\":2,\"b\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, -9], [2, -7]], -16, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(2, [[1, -9], [2, -7]], -16, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106002,"parameters":{"n":2,"a":2,"b":-11,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":2,\"b\":-11,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0003","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":3,\"a\":3,\"b\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, -7], [3, -1]], -8, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(3, [[1, -7], [3, -1]], -8, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106003,"parameters":{"n":3,"a":3,"b":-10,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":3,\"b\":-10,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0004","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":4,\"a\":4,\"b\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[1, -5], [2, -1], [4, 7]], 1, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(4, [[1, -5], [2, -1], [4, 7]], 1, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106004,"parameters":{"n":4,"a":4,"b":-9,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":4,\"b\":-9,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0005","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":5,\"a\":5,\"b\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[1, -3], [5, 17]], 14, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(5, [[1, -3], [5, 17]], 14, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106005,"parameters":{"n":5,"a":5,"b":-8,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":5,\"b\":-8,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0006","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":6,\"a\":1,\"b\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[1, -6], [2, -5], [3, -4], [6, -1]], -16, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(6, [[1, -6], [2, -5], [3, -4], [6, -1]], -16, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106006,"parameters":{"n":6,"a":1,"b":-7,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":1,\"b\":-7,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0007","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":7,\"a\":2,\"b\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, [[1, -4], [7, 8]], 4, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(7, [[1, -4], [7, 8]], 4, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106007,"parameters":{"n":7,"a":2,"b":-6,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":2,\"b\":-6,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0008","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":8,\"a\":3,\"b\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[1, -2], [2, 1], [4, 7], [8, 19]], 25, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(8, [[1, -2], [2, 1], [4, 7], [8, 19]], 25, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106008,"parameters":{"n":8,"a":3,"b":-5,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":3,\"b\":-5,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0009","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":9,\"a\":4,\"b\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, [[1, 0], [3, 8], [9, 32]], 40, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(9, [[1, 0], [3, 8], [9, 32]], 40, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106009,"parameters":{"n":9,"a":4,"b":-4,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":4,\"b\":-4,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0010","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":10,\"a\":5,\"b\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[1, 2], [2, 7], [5, 22], [10, 47]], 78, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(10, [[1, 2], [2, 7], [5, 22], [10, 47]], 78, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106010,"parameters":{"n":10,"a":5,"b":-3,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":5,\"b\":-3,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0011","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":11,\"a\":1,\"b\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, [[1, -1], [11, 9]], 8, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(11, [[1, -1], [11, 9]], 8, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106011,"parameters":{"n":11,"a":1,"b":-2,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":1,\"b\":-2,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0012","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":12,\"a\":2,\"b\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[1, 1], [2, 3], [3, 5], [4, 7], [6, 11], [12, 23]], 50, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(12, [[1, 1], [2, 3], [3, 5], [4, 7], [6, 11], [12, 23]], 50, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106012,"parameters":{"n":12,"a":2,"b":-1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":2,\"b\":-1,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0013","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":13,\"a\":3,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, [[1, 3], [13, 39]], 42, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(13, [[1, 3], [13, 39]], 42, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106013,"parameters":{"n":13,"a":3,"b":0,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":3,\"b\":0,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0014","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":14,\"a\":4,\"b\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [[1, 5], [2, 9], [7, 29], [14, 57]], 100, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(14, [[1, 5], [2, 9], [7, 29], [14, 57]], 100, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106014,"parameters":{"n":14,"a":4,"b":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":4,\"b\":1,\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0015","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":15,\"a\":5,\"b\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, [[1, 7], [3, 17], [5, 27], [15, 77]], 128, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(15, [[1, 7], [3, 17], [5, 27], [15, 77]], 128, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106015,"parameters":{"n":15,"a":5,"b":2,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":5,\"b\":2,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0016","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":16,\"a\":1,\"b\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, [[1, 4], [2, 5], [4, 7], [8, 11], [16, 19]], 46, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(16, [[1, 4], [2, 5], [4, 7], [8, 11], [16, 19]], 46, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106016,"parameters":{"n":16,"a":1,"b":3,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":1,\"b\":3,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0017","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":17,\"a\":2,\"b\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, [[1, 6], [17, 38]], 44, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(17, [[1, 6], [17, 38]], 44, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106017,"parameters":{"n":17,"a":2,"b":4,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":2,\"b\":4,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0018","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":18,\"a\":3,\"b\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, [[1, 8], [2, 11], [3, 14], [6, 23], [9, 32], [18, 59]], 147, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(18, [[1, 8], [2, 11], [3, 14], [6, 23], [9, 32], [18, 59]], 147, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106018,"parameters":{"n":18,"a":3,"b":5,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":3,\"b\":5,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0019","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":19,\"a\":4,\"b\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, [[1, 10], [19, 82]], 92, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(19, [[1, 10], [19, 82]], 92, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106019,"parameters":{"n":19,"a":4,"b":6,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":4,\"b\":6,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0020","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":20,\"a\":5,\"b\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, [[1, 12], [2, 17], [4, 27], [5, 32], [10, 57], [20, 107]], 252, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(20, [[1, 12], [2, 17], [4, 27], [5, 32], [10, 57], [20, 107]], 252, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106020,"parameters":{"n":20,"a":5,"b":7,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":5,\"b\":7,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0021","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":21,\"a\":1,\"b\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, [[1, 9], [3, 11], [7, 15], [21, 29]], 64, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(21, [[1, 9], [3, 11], [7, 15], [21, 29]], 64, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106021,"parameters":{"n":21,"a":1,"b":8,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":1,\"b\":8,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0022","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":22,\"a\":2,\"b\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, [[1, 11], [2, 13], [11, 31], [22, 53]], 108, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(22, [[1, 11], [2, 13], [11, 31], [22, 53]], 108, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106022,"parameters":{"n":22,"a":2,"b":9,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":2,\"b\":9,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0023","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":23,\"a\":3,\"b\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, [[1, 13], [23, 79]], 92, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(23, [[1, 13], [23, 79]], 92, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106023,"parameters":{"n":23,"a":3,"b":10,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":3,\"b\":10,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0024","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":24,\"a\":4,\"b\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, [[1, 15], [2, 19], [3, 23], [4, 27], [6, 35], [8, 43], [12, 59], [24, 107]], 328, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(24, [[1, 15], [2, 19], [3, 23], [4, 27], [6, 35], [8, 43], [12, 59], [24, 107]], 328, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106024,"parameters":{"n":24,"a":4,"b":11,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":4,\"b\":11,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0025","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-forward-divisor-transform；資料={\"n\":25,\"a\":5,\"b\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, [[1, 17], [5, 37], [25, 137]], 191, sum_over_all_positive_divisors)","explanation_zh":"Wave6第四層核對=(25, [[1, 17], [5, 37], [25, 137]], 191, sum_over_all_positive_divisors)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106025,"parameters":{"n":25,"a":5,"b":12,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-forward-divisor-transform","semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":5,\"b\":12,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0026","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":1,\"a\":1,\"b\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[1, 1, 1, -11, -11]], -11, -11, 是)","explanation_zh":"Wave6第四層核對=(1, [[1, 1, 1, -11, -11]], -11, -11, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106026,"parameters":{"n":1,"a":1,"b":-12,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":1,\"b\":-12,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0027","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":2,\"a\":2,\"b\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, 1, 2, -16, -16], [2, -1, 1, -9, 9]], -7, -7, 是)","explanation_zh":"Wave6第四層核對=(2, [[1, 1, 2, -16, -16], [2, -1, 1, -9, 9]], -7, -7, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106027,"parameters":{"n":2,"a":2,"b":-11,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":2,\"b\":-11,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0028","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":3,\"a\":3,\"b\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, 1, 3, -8, -8], [3, -1, 1, -7, 7]], -1, -1, 是)","explanation_zh":"Wave6第四層核對=(3, [[1, 1, 3, -8, -8], [3, -1, 1, -7, 7]], -1, -1, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106028,"parameters":{"n":3,"a":3,"b":-10,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":3,\"b\":-10,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0029","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":4,\"a\":4,\"b\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[1, 1, 4, 1, 1], [2, -1, 2, -6, 6], [4, 0, 1, -5, 0]], 7, 7, 是)","explanation_zh":"Wave6第四層核對=(4, [[1, 1, 4, 1, 1], [2, -1, 2, -6, 6], [4, 0, 1, -5, 0]], 7, 7, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106029,"parameters":{"n":4,"a":4,"b":-9,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":4,\"b\":-9,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0030","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":5,\"a\":5,\"b\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[1, 1, 5, 14, 14], [5, -1, 1, -3, 3]], 17, 17, 是)","explanation_zh":"Wave6第四層核對=(5, [[1, 1, 5, 14, 14], [5, -1, 1, -3, 3]], 17, 17, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106030,"parameters":{"n":5,"a":5,"b":-8,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":5,\"b\":-8,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0031","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":6,\"a\":1,\"b\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[1, 1, 6, -16, -16], [2, -1, 3, -10, 10], [3, -1, 2, -11, 11], [6, 1, 1, -6, -6]], -1, -1, 是)","explanation_zh":"Wave6第四層核對=(6, [[1, 1, 6, -16, -16], [2, -1, 3, -10, 10], [3, -1, 2, -11, 11], [6, 1, 1, -6, -6]], -1, -1, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106031,"parameters":{"n":6,"a":1,"b":-7,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":1,\"b\":-7,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0032","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":7,\"a\":2,\"b\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, [[1, 1, 7, 4, 4], [7, -1, 1, -4, 4]], 8, 8, 是)","explanation_zh":"Wave6第四層核對=(7, [[1, 1, 7, 4, 4], [7, -1, 1, -4, 4]], 8, 8, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106032,"parameters":{"n":7,"a":2,"b":-6,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":2,\"b\":-6,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0033","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":8,\"a\":3,\"b\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[1, 1, 8, 25, 25], [2, -1, 4, 6, -6], [4, 0, 2, -1, 0], [8, 0, 1, -2, 0]], 19, 19, 是)","explanation_zh":"Wave6第四層核對=(8, [[1, 1, 8, 25, 25], [2, -1, 4, 6, -6], [4, 0, 2, -1, 0], [8, 0, 1, -2, 0]], 19, 19, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106033,"parameters":{"n":8,"a":3,"b":-5,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":3,\"b\":-5,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0034","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":9,\"a\":4,\"b\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, [[1, 1, 9, 40, 40], [3, -1, 3, 8, -8], [9, 0, 1, 0, 0]], 32, 32, 是)","explanation_zh":"Wave6第四層核對=(9, [[1, 1, 9, 40, 40], [3, -1, 3, 8, -8], [9, 0, 1, 0, 0]], 32, 32, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106034,"parameters":{"n":9,"a":4,"b":-4,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":4,\"b\":-4,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0035","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":10,\"a\":5,\"b\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[1, 1, 10, 78, 78], [2, -1, 5, 24, -24], [5, -1, 2, 9, -9], [10, 1, 1, 2, 2]], 47, 47, 是)","explanation_zh":"Wave6第四層核對=(10, [[1, 1, 10, 78, 78], [2, -1, 5, 24, -24], [5, -1, 2, 9, -9], [10, 1, 1, 2, 2]], 47, 47, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106035,"parameters":{"n":10,"a":5,"b":-3,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":5,\"b\":-3,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0036","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":11,\"a\":1,\"b\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, [[1, 1, 11, 8, 8], [11, -1, 1, -1, 1]], 9, 9, 是)","explanation_zh":"Wave6第四層核對=(11, [[1, 1, 11, 8, 8], [11, -1, 1, -1, 1]], 9, 9, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106036,"parameters":{"n":11,"a":1,"b":-2,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":1,\"b\":-2,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0037","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":12,\"a\":2,\"b\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[1, 1, 12, 50, 50], [2, -1, 6, 20, -20], [3, -1, 4, 11, -11], [4, 0, 3, 6, 0], [6, 1, 2, 4, 4], [12, 0, 1, 1, 0]], 23, 23, 是)","explanation_zh":"Wave6第四層核對=(12, [[1, 1, 12, 50, 50], [2, -1, 6, 20, -20], [3, -1, 4, 11, -11], [4, 0, 3, 6, 0], [6, 1, 2, 4, 4], [12, 0, 1, 1, 0]], 23, 23, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106037,"parameters":{"n":12,"a":2,"b":-1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":2,\"b\":-1,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0038","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":13,\"a\":3,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, [[1, 1, 13, 42, 42], [13, -1, 1, 3, -3]], 39, 39, 是)","explanation_zh":"Wave6第四層核對=(13, [[1, 1, 13, 42, 42], [13, -1, 1, 3, -3]], 39, 39, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106038,"parameters":{"n":13,"a":3,"b":0,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":3,\"b\":0,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0039","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":14,\"a\":4,\"b\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [[1, 1, 14, 100, 100], [2, -1, 7, 34, -34], [7, -1, 2, 14, -14], [14, 1, 1, 5, 5]], 57, 57, 是)","explanation_zh":"Wave6第四層核對=(14, [[1, 1, 14, 100, 100], [2, -1, 7, 34, -34], [7, -1, 2, 14, -14], [14, 1, 1, 5, 5]], 57, 57, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106039,"parameters":{"n":14,"a":4,"b":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":4,\"b\":1,\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0040","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":15,\"a\":5,\"b\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, [[1, 1, 15, 128, 128], [3, -1, 5, 34, -34], [5, -1, 3, 24, -24], [15, 1, 1, 7, 7]], 77, 77, 是)","explanation_zh":"Wave6第四層核對=(15, [[1, 1, 15, 128, 128], [3, -1, 5, 34, -34], [5, -1, 3, 24, -24], [15, 1, 1, 7, 7]], 77, 77, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106040,"parameters":{"n":15,"a":5,"b":2,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":5,\"b\":2,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0041","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":16,\"a\":1,\"b\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, [[1, 1, 16, 46, 46], [2, -1, 8, 27, -27], [4, 0, 4, 16, 0], [8, 0, 2, 9, 0], [16, 0, 1, 4, 0]], 19, 19, 是)","explanation_zh":"Wave6第四層核對=(16, [[1, 1, 16, 46, 46], [2, -1, 8, 27, -27], [4, 0, 4, 16, 0], [8, 0, 2, 9, 0], [16, 0, 1, 4, 0]], 19, 19, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106041,"parameters":{"n":16,"a":1,"b":3,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":1,\"b\":3,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0042","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":17,\"a\":2,\"b\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, [[1, 1, 17, 44, 44], [17, -1, 1, 6, -6]], 38, 38, 是)","explanation_zh":"Wave6第四層核對=(17, [[1, 1, 17, 44, 44], [17, -1, 1, 6, -6]], 38, 38, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106042,"parameters":{"n":17,"a":2,"b":4,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":2,\"b\":4,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0043","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":18,\"a\":3,\"b\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, [[1, 1, 18, 147, 147], [2, -1, 9, 54, -54], [3, -1, 6, 56, -56], [6, 1, 3, 22, 22], [9, 0, 2, 19, 0], [18, 0, 1, 8, 0]], 59, 59, 是)","explanation_zh":"Wave6第四層核對=(18, [[1, 1, 18, 147, 147], [2, -1, 9, 54, -54], [3, -1, 6, 56, -56], [6, 1, 3, 22, 22], [9, 0, 2, 19, 0], [18, 0, 1, 8, 0]], 59, 59, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106043,"parameters":{"n":18,"a":3,"b":5,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":3,\"b\":5,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0044","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":19,\"a\":4,\"b\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, [[1, 1, 19, 92, 92], [19, -1, 1, 10, -10]], 82, 82, 是)","explanation_zh":"Wave6第四層核對=(19, [[1, 1, 19, 92, 92], [19, -1, 1, 10, -10]], 82, 82, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106044,"parameters":{"n":19,"a":4,"b":6,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":4,\"b\":6,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0045","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":20,\"a\":5,\"b\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, [[1, 1, 20, 252, 252], [2, -1, 10, 118, -118], [4, 0, 5, 44, 0], [5, -1, 4, 56, -56], [10, 1, 2, 29, 29], [20, 0, 1, 12, 0]], 107, 107, 是)","explanation_zh":"Wave6第四層核對=(20, [[1, 1, 20, 252, 252], [2, -1, 10, 118, -118], [4, 0, 5, 44, 0], [5, -1, 4, 56, -56], [10, 1, 2, 29, 29], [20, 0, 1, 12, 0]], 107, 107, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106045,"parameters":{"n":20,"a":5,"b":7,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":5,\"b\":7,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0046","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":21,\"a\":1,\"b\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, [[1, 1, 21, 64, 64], [3, -1, 7, 24, -24], [7, -1, 3, 20, -20], [21, 1, 1, 9, 9]], 29, 29, 是)","explanation_zh":"Wave6第四層核對=(21, [[1, 1, 21, 64, 64], [3, -1, 7, 24, -24], [7, -1, 3, 20, -20], [21, 1, 1, 9, 9]], 29, 29, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106046,"parameters":{"n":21,"a":1,"b":8,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":1,\"b\":8,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0047","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":22,\"a\":2,\"b\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, [[1, 1, 22, 108, 108], [2, -1, 11, 42, -42], [11, -1, 2, 24, -24], [22, 1, 1, 11, 11]], 53, 53, 是)","explanation_zh":"Wave6第四層核對=(22, [[1, 1, 22, 108, 108], [2, -1, 11, 42, -42], [11, -1, 2, 24, -24], [22, 1, 1, 11, 11]], 53, 53, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106047,"parameters":{"n":22,"a":2,"b":9,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":2,\"b\":9,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0048","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":23,\"a\":3,\"b\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, [[1, 1, 23, 92, 92], [23, -1, 1, 13, -13]], 79, 79, 是)","explanation_zh":"Wave6第四層核對=(23, [[1, 1, 23, 92, 92], [23, -1, 1, 13, -13]], 79, 79, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106048,"parameters":{"n":23,"a":3,"b":10,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":3,\"b\":10,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0049","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":24,\"a\":4,\"b\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, [[1, 1, 24, 328, 328], [2, -1, 12, 178, -178], [3, -1, 8, 104, -104], [4, 0, 6, 92, 0], [6, 1, 4, 61, 61], [8, 0, 3, 38, 0], [12, 0, 2, 34, 0], [24, 0, 1, 15, 0]], 107, 107, 是)","explanation_zh":"Wave6第四層核對=(24, [[1, 1, 24, 328, 328], [2, -1, 12, 178, -178], [3, -1, 8, 104, -104], [4, 0, 6, 92, 0], [6, 1, 4, 61, 61], [8, 0, 3, 38, 0], [12, 0, 2, 34, 0], [24, 0, 1, 15, 0]], 107, 107, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106049,"parameters":{"n":24,"a":4,"b":11,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":4,\"b\":11,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0050","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-inverse-reconstruction；資料={\"n\":25,\"a\":5,\"b\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, [[1, 1, 25, 191, 191], [5, -1, 5, 54, -54], [25, 0, 1, 17, 0]], 137, 137, 是)","explanation_zh":"Wave6第四層核對=(25, [[1, 1, 25, 191, 191], [5, -1, 5, 54, -54], [25, 0, 1, 17, 0]], 137, 137, 是)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106050,"parameters":{"n":25,"a":5,"b":12,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-inverse-reconstruction","semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":5,\"b\":12,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0051","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":1,\"a\":1,\"b\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[1, -11, -11, -11]], 是, mu_one_boundary_preserved)","explanation_zh":"Wave6第四層核對=(1, [[1, -11, -11, -11]], 是, mu_one_boundary_preserved)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106051,"parameters":{"n":1,"a":1,"b":-12,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":1,\"b\":-12,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0052","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":2,\"a\":2,\"b\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, -9, -9, -9], [2, -16, -7, -7]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(2, [[1, -9, -9, -9], [2, -16, -7, -7]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106052,"parameters":{"n":2,"a":2,"b":-11,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":2,\"b\":-11,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0053","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":3,\"a\":3,\"b\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, -7, -7, -7], [3, -8, -1, -1]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(3, [[1, -7, -7, -7], [3, -8, -1, -1]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106053,"parameters":{"n":3,"a":3,"b":-10,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":3,\"b\":-10,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0054","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":4,\"a\":4,\"b\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[1, -5, -5, -5], [2, -6, -1, -1], [4, 1, 7, 7]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(4, [[1, -5, -5, -5], [2, -6, -1, -1], [4, 1, 7, 7]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106054,"parameters":{"n":4,"a":4,"b":-9,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":4,\"b\":-9,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0055","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":5,\"a\":5,\"b\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[1, -3, -3, -3], [5, 14, 17, 17]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(5, [[1, -3, -3, -3], [5, 14, 17, 17]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106055,"parameters":{"n":5,"a":5,"b":-8,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":5,\"b\":-8,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0056","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":6,\"a\":1,\"b\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[1, -6, -6, -6], [2, -11, -5, -5], [3, -10, -4, -4], [6, -16, -1, -1]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(6, [[1, -6, -6, -6], [2, -11, -5, -5], [3, -10, -4, -4], [6, -16, -1, -1]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106056,"parameters":{"n":6,"a":1,"b":-7,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":1,\"b\":-7,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0057","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":7,\"a\":2,\"b\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, [[1, -4, -4, -4], [7, 4, 8, 8]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(7, [[1, -4, -4, -4], [7, 4, 8, 8]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106057,"parameters":{"n":7,"a":2,"b":-6,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":2,\"b\":-6,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0058","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":8,\"a\":3,\"b\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[1, -2, -2, -2], [2, -1, 1, 1], [4, 6, 7, 7], [8, 25, 19, 19]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(8, [[1, -2, -2, -2], [2, -1, 1, 1], [4, 6, 7, 7], [8, 25, 19, 19]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106058,"parameters":{"n":8,"a":3,"b":-5,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":3,\"b\":-5,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0059","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":9,\"a\":4,\"b\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, [[1, 0, 0, 0], [3, 8, 8, 8], [9, 40, 32, 32]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(9, [[1, 0, 0, 0], [3, 8, 8, 8], [9, 40, 32, 32]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106059,"parameters":{"n":9,"a":4,"b":-4,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":4,\"b\":-4,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0060","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":10,\"a\":5,\"b\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[1, 2, 2, 2], [2, 9, 7, 7], [5, 24, 22, 22], [10, 78, 47, 47]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(10, [[1, 2, 2, 2], [2, 9, 7, 7], [5, 24, 22, 22], [10, 78, 47, 47]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106060,"parameters":{"n":10,"a":5,"b":-3,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":5,\"b\":-3,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0061","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":11,\"a\":1,\"b\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, [[1, -1, -1, -1], [11, 8, 9, 9]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(11, [[1, -1, -1, -1], [11, 8, 9, 9]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106061,"parameters":{"n":11,"a":1,"b":-2,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":1,\"b\":-2,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0062","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":12,\"a\":2,\"b\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[1, 1, 1, 1], [2, 4, 3, 3], [3, 6, 5, 5], [4, 11, 7, 7], [6, 20, 11, 11], [12, 50, 23, 23]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(12, [[1, 1, 1, 1], [2, 4, 3, 3], [3, 6, 5, 5], [4, 11, 7, 7], [6, 20, 11, 11], [12, 50, 23, 23]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106062,"parameters":{"n":12,"a":2,"b":-1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":2,\"b\":-1,\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0063","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":13,\"a\":3,\"b\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, [[1, 3, 3, 3], [13, 42, 39, 39]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(13, [[1, 3, 3, 3], [13, 42, 39, 39]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106063,"parameters":{"n":13,"a":3,"b":0,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":3,\"b\":0,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0064","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":14,\"a\":4,\"b\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, [[1, 5, 5, 5], [2, 14, 9, 9], [7, 34, 29, 29], [14, 100, 57, 57]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(14, [[1, 5, 5, 5], [2, 14, 9, 9], [7, 34, 29, 29], [14, 100, 57, 57]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106064,"parameters":{"n":14,"a":4,"b":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":4,\"b\":1,\"n\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0065","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":15,\"a\":5,\"b\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, [[1, 7, 7, 7], [3, 24, 17, 17], [5, 34, 27, 27], [15, 128, 77, 77]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(15, [[1, 7, 7, 7], [3, 24, 17, 17], [5, 34, 27, 27], [15, 128, 77, 77]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106065,"parameters":{"n":15,"a":5,"b":2,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":5,\"b\":2,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0066","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":16,\"a\":1,\"b\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, [[1, 4, 4, 4], [2, 9, 5, 5], [4, 16, 7, 7], [8, 27, 11, 11], [16, 46, 19, 19]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(16, [[1, 4, 4, 4], [2, 9, 5, 5], [4, 16, 7, 7], [8, 27, 11, 11], [16, 46, 19, 19]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106066,"parameters":{"n":16,"a":1,"b":3,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":1,\"b\":3,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0067","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":17,\"a\":2,\"b\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, [[1, 6, 6, 6], [17, 44, 38, 38]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(17, [[1, 6, 6, 6], [17, 44, 38, 38]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106067,"parameters":{"n":17,"a":2,"b":4,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":2,\"b\":4,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0068","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":18,\"a\":3,\"b\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, [[1, 8, 8, 8], [2, 19, 11, 11], [3, 22, 14, 14], [6, 56, 23, 23], [9, 54, 32, 32], [18, 147, 59, 59]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(18, [[1, 8, 8, 8], [2, 19, 11, 11], [3, 22, 14, 14], [6, 56, 23, 23], [9, 54, 32, 32], [18, 147, 59, 59]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106068,"parameters":{"n":18,"a":3,"b":5,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":3,\"b\":5,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0069","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":19,\"a\":4,\"b\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, [[1, 10, 10, 10], [19, 92, 82, 82]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(19, [[1, 10, 10, 10], [19, 92, 82, 82]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106069,"parameters":{"n":19,"a":4,"b":6,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":4,\"b\":6,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0070","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":20,\"a\":5,\"b\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, [[1, 12, 12, 12], [2, 29, 17, 17], [4, 56, 27, 27], [5, 44, 32, 32], [10, 118, 57, 57], [20, 252, 107, 107]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(20, [[1, 12, 12, 12], [2, 29, 17, 17], [4, 56, 27, 27], [5, 44, 32, 32], [10, 118, 57, 57], [20, 252, 107, 107]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106070,"parameters":{"n":20,"a":5,"b":7,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":5,\"b\":7,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0071","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":21,\"a\":1,\"b\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, [[1, 9, 9, 9], [3, 20, 11, 11], [7, 24, 15, 15], [21, 64, 29, 29]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(21, [[1, 9, 9, 9], [3, 20, 11, 11], [7, 24, 15, 15], [21, 64, 29, 29]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106071,"parameters":{"n":21,"a":1,"b":8,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":1,\"b\":8,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0072","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":22,\"a\":2,\"b\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, [[1, 11, 11, 11], [2, 24, 13, 13], [11, 42, 31, 31], [22, 108, 53, 53]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(22, [[1, 11, 11, 11], [2, 24, 13, 13], [11, 42, 31, 31], [22, 108, 53, 53]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106072,"parameters":{"n":22,"a":2,"b":9,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":2,\"b\":9,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0073","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":23,\"a\":3,\"b\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, [[1, 13, 13, 13], [23, 92, 79, 79]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(23, [[1, 13, 13, 13], [23, 92, 79, 79]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106073,"parameters":{"n":23,"a":3,"b":10,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":3,\"b\":10,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0074","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":24,\"a\":4,\"b\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, [[1, 15, 15, 15], [2, 34, 19, 19], [3, 38, 23, 23], [4, 61, 27, 27], [6, 92, 35, 35], [8, 104, 43, 43], [12, 178, 59, 59], [24, 328, 107, 107]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(24, [[1, 15, 15, 15], [2, 34, 19, 19], [3, 38, 23, 23], [4, 61, 27, 27], [6, 92, 35, 35], [8, 104, 43, 43], [12, 178, 59, 59], [24, 328, 107, 107]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106074,"parameters":{"n":24,"a":4,"b":11,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":4,\"b\":11,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0075","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-roundtrip-table；資料={\"n\":25,\"a\":5,\"b\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, [[1, 17, 17, 17], [5, 54, 37, 37], [25, 191, 137, 137]], 是, finite_divisor_poset_roundtrip)","explanation_zh":"Wave6第四層核對=(25, [[1, 17, 17, 17], [5, 54, 37, 37], [25, 191, 137, 137]], 是, finite_divisor_poset_roundtrip)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106075,"parameters":{"n":25,"a":5,"b":12,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-roundtrip-table","semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":5,\"b\":12,\"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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0076","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":6,\"a\":1,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [1, 2, 3, 6], 1, [2, 3, 6], 7, -9, 7, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(6, [1, 2, 3, 6], 1, [2, 3, 6], 7, -9, 7, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106076,"parameters":{"n":6,"a":1,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":1,\"b\":1,\"n\":6,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0077","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":12,\"a\":2,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [1, 2, 3, 4, 6, 12], 1, [2, 3, 4, 6, 12], 25, -37, 25, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(12, [1, 2, 3, 4, 6, 12], 1, [2, 3, 4, 6, 12], 25, -37, 25, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106077,"parameters":{"n":12,"a":2,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":2,\"b\":1,\"n\":12,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0078","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":20,\"a\":3,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, [1, 2, 4, 5, 10, 20], 1, [2, 4, 5, 10, 20], 61, -71, 61, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(20, [1, 2, 4, 5, 10, 20], 1, [2, 4, 5, 10, 20], 61, -71, 61, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106078,"parameters":{"n":20,"a":3,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":3,\"b\":1,\"n\":20,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0079","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":30,\"a\":4,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(30, [1, 2, 3, 5, 6, 10, 15, 30], 1, [2, 3, 5, 6, 10, 15, 30], 121, -175, 121, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(30, [1, 2, 3, 5, 6, 10, 15, 30], 1, [2, 3, 5, 6, 10, 15, 30], 121, -175, 121, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106079,"parameters":{"n":30,"a":4,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":4,\"b\":1,\"n\":30,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0080","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":42,\"a\":5,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(42, [1, 2, 3, 6, 7, 14, 21, 42], 1, [2, 3, 6, 7, 14, 21, 42], 211, -277, 211, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(42, [1, 2, 3, 6, 7, 14, 21, 42], 1, [2, 3, 6, 7, 14, 21, 42], 211, -277, 211, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106080,"parameters":{"n":42,"a":5,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":5,\"b\":1,\"n\":42,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0081","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":56,\"a\":6,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(56, [1, 2, 4, 7, 8, 14, 28, 56], 1, [2, 4, 7, 8, 14, 28, 56], 337, -391, 337, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(56, [1, 2, 4, 7, 8, 14, 28, 56], 1, [2, 4, 7, 8, 14, 28, 56], 337, -391, 337, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106081,"parameters":{"n":56,"a":6,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":6,\"b\":1,\"n\":56,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0082","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":72,\"a\":7,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(72, [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], 1, [2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], 505, -872, 505, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(72, [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], 1, [2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], 505, -872, 505, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106082,"parameters":{"n":72,"a":7,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":7,\"b\":1,\"n\":72,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0083","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":90,\"a\":8,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(90, [1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 1, [2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 721, -1163, 721, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(90, [1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 1, [2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 721, -1163, 721, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106083,"parameters":{"n":90,"a":8,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":8,\"b\":1,\"n\":90,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0084","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":110,\"a\":9,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(110, [1, 2, 5, 10, 11, 22, 55, 110], 1, [2, 5, 10, 11, 22, 55, 110], 991, -961, 991, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(110, [1, 2, 5, 10, 11, 22, 55, 110], 1, [2, 5, 10, 11, 22, 55, 110], 991, -961, 991, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106084,"parameters":{"n":110,"a":9,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":9,\"b\":1,\"n\":110,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0085","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":132,\"a\":10,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(132, [1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1, [2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1321, -2051, 1321, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(132, [1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1, [2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1321, -2051, 1321, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106085,"parameters":{"n":132,"a":10,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":10,\"b\":1,\"n\":132,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0086","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":156,\"a\":11,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(156, [1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1, [2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1717, -2607, 1717, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(156, [1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1, [2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1717, -2607, 1717, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106086,"parameters":{"n":156,"a":11,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":11,\"b\":1,\"n\":156,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0087","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":182,\"a\":12,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(182, [1, 2, 7, 13, 14, 26, 91, 182], 1, [2, 7, 13, 14, 26, 91, 182], 2185, -1855, 2185, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(182, [1, 2, 7, 13, 14, 26, 91, 182], 1, [2, 7, 13, 14, 26, 91, 182], 2185, -1855, 2185, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106087,"parameters":{"n":182,"a":12,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":12,\"b\":1,\"n\":182,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0088","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":210,\"a\":13,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(210, [1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 1, [2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 2731, -4773, 2731, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(210, [1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 1, [2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 2731, -4773, 2731, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106088,"parameters":{"n":210,"a":13,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":13,\"b\":1,\"n\":210,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0089","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":240,\"a\":14,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(240, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240], 1, [2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240], 3361, -7075, 3361, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(240, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240], 1, [2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240], 3361, -7075, 3361, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106089,"parameters":{"n":240,"a":14,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":14,\"b\":1,\"n\":240,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0090","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":272,\"a\":15,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(272, [1, 2, 4, 8, 16, 17, 34, 68, 136, 272], 1, [2, 4, 8, 16, 17, 34, 68, 136, 272], 4081, -4299, 4081, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(272, [1, 2, 4, 8, 16, 17, 34, 68, 136, 272], 1, [2, 4, 8, 16, 17, 34, 68, 136, 272], 4081, -4299, 4081, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106090,"parameters":{"n":272,"a":15,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":15,\"b\":1,\"n\":272,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0091","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":306,\"a\":16,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(306, [1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 1, [2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 4897, -6347, 4897, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(306, [1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 1, [2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 4897, -6347, 4897, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106091,"parameters":{"n":306,"a":16,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":16,\"b\":1,\"n\":306,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0092","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":342,\"a\":17,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(342, [1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 1, [2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 5815, -7457, 5815, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(342, [1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 1, [2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 5815, -7457, 5815, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106092,"parameters":{"n":342,"a":17,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":17,\"b\":1,\"n\":342,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0093","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":380,\"a\":18,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(380, [1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 1, [2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 6841, -8291, 6841, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(380, [1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 1, [2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 6841, -8291, 6841, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106093,"parameters":{"n":380,"a":18,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":18,\"b\":1,\"n\":380,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0094","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":420,\"a\":19,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(420, [1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 1, [2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 7981, -17579, 7981, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(420, [1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 1, [2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 7981, -17579, 7981, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106094,"parameters":{"n":420,"a":19,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":19,\"b\":1,\"n\":420,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0095","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":462,\"a\":20,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 1, [2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 9241, -13815, 9241, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 1, [2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 9241, -13815, 9241, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106095,"parameters":{"n":462,"a":20,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":20,\"b\":1,\"n\":462,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0096","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":506,\"a\":21,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(506, [1, 2, 11, 22, 23, 46, 253, 506], 1, [2, 11, 22, 23, 46, 253, 506], 10627, -7525, 10627, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(506, [1, 2, 11, 22, 23, 46, 253, 506], 1, [2, 11, 22, 23, 46, 253, 506], 10627, -7525, 10627, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106096,"parameters":{"n":506,"a":21,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":21,\"b\":1,\"n\":506,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0097","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":552,\"a\":22,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(552, [1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 1, [2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 12145, -19551, 12145, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(552, [1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 1, [2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 12145, -19551, 12145, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106097,"parameters":{"n":552,"a":22,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":22,\"b\":1,\"n\":552,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0098","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":600,\"a\":23,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(600, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600], 1, [2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600], 13801, -29003, 13801, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(600, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600], 1, [2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600], 13801, -29003, 13801, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106098,"parameters":{"n":600,"a":23,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":23,\"b\":1,\"n\":600,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0099","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":650,\"a\":24,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(650, [1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 1, [2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 15601, -15659, 15601, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(650, [1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 1, [2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 15601, -15659, 15601, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106099,"parameters":{"n":650,"a":24,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":24,\"b\":1,\"n\":650,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0100","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","language":"zh-Hant","source_mko_ids":["mko-mobius-inversion-finite-divisors"],"title_zh":"有限因數格上的Möbius反演","stem_zh":"positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":702,\"a\":25,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(702, [1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 1, [2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 17551, -24465, 17551, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation_zh":"Wave6第四層核對=(702, [1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 1, [2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 17551, -24465, 17551, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限因數格上的Möbius反演"]},"variant":{"family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106100,"parameters":{"n":702,"a":25,"b":1,"omitted_divisor":1,"domain":"Exact_finite_divisor_Mobius_inversion","operation":"mobinv-incomplete-divisor-boundary","semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":25,\"b\":1,\"n\":702,\"omitted_divisor\":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、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}