{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0001","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":12,\"factors\":[2,3,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [2, 3, 2], 是, 12, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(12, [2, 3, 2], 是, 12, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086001,"parameters":{"n":12,"factors":[2,3,2],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,2,3],\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0002","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":18,\"factors\":[2,3,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, [2, 3, 3], 是, 18, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(18, [2, 3, 3], 是, 18, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086002,"parameters":{"n":18,"factors":[2,3,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,3],\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0003","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":30,\"factors\":[2,3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(30, [2, 3, 5], 是, 30, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(30, [2, 3, 5], 是, 30, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086003,"parameters":{"n":30,"factors":[2,3,5],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,5],\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0004","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":42,\"factors\":[2,3,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(42, [2, 3, 7], 是, 42, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(42, [2, 3, 7], 是, 42, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086004,"parameters":{"n":42,"factors":[2,3,7],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,7],\"n\":42}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0005","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":66,\"factors\":[2,3,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(66, [2, 3, 11], 是, 66, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(66, [2, 3, 11], 是, 66, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086005,"parameters":{"n":66,"factors":[2,3,11],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,11],\"n\":66}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0006","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":78,\"factors\":[2,3,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(78, [2, 3, 13], 是, 78, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(78, [2, 3, 13], 是, 78, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086006,"parameters":{"n":78,"factors":[2,3,13],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,13],\"n\":78}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0007","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":102,\"factors\":[2,3,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(102, [2, 3, 17], 是, 102, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(102, [2, 3, 17], 是, 102, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086007,"parameters":{"n":102,"factors":[2,3,17],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,17],\"n\":102}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0008","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":114,\"factors\":[2,3,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(114, [2, 3, 19], 是, 114, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(114, [2, 3, 19], 是, 114, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086008,"parameters":{"n":114,"factors":[2,3,19],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,19],\"n\":114}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0009","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":138,\"factors\":[2,3,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(138, [2, 3, 23], 是, 138, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(138, [2, 3, 23], 是, 138, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086009,"parameters":{"n":138,"factors":[2,3,23],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,23],\"n\":138}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0010","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":174,\"factors\":[2,3,29]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(174, [2, 3, 29], 是, 174, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(174, [2, 3, 29], 是, 174, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086010,"parameters":{"n":174,"factors":[2,3,29],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,29],\"n\":174}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0011","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":186,\"factors\":[2,3,31]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(186, [2, 3, 31], 是, 186, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(186, [2, 3, 31], 是, 186, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086011,"parameters":{"n":186,"factors":[2,3,31],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,31],\"n\":186}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0012","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":222,\"factors\":[2,3,37]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(222, [2, 3, 37], 是, 222, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(222, [2, 3, 37], 是, 222, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086012,"parameters":{"n":222,"factors":[2,3,37],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,37],\"n\":222}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0013","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":246,\"factors\":[2,3,41]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(246, [2, 3, 41], 是, 246, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(246, [2, 3, 41], 是, 246, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086013,"parameters":{"n":246,"factors":[2,3,41],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,41],\"n\":246}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0014","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":258,\"factors\":[2,3,43]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(258, [2, 3, 43], 是, 258, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(258, [2, 3, 43], 是, 258, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086014,"parameters":{"n":258,"factors":[2,3,43],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,43],\"n\":258}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0015","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":282,\"factors\":[2,3,47]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(282, [2, 3, 47], 是, 282, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(282, [2, 3, 47], 是, 282, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086015,"parameters":{"n":282,"factors":[2,3,47],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,47],\"n\":282}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0016","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":318,\"factors\":[2,3,53]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(318, [2, 3, 53], 是, 318, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(318, [2, 3, 53], 是, 318, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086016,"parameters":{"n":318,"factors":[2,3,53],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,53],\"n\":318}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0017","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":354,\"factors\":[2,3,59]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(354, [2, 3, 59], 是, 354, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(354, [2, 3, 59], 是, 354, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086017,"parameters":{"n":354,"factors":[2,3,59],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,59],\"n\":354}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0018","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":366,\"factors\":[2,3,61]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(366, [2, 3, 61], 是, 366, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(366, [2, 3, 61], 是, 366, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086018,"parameters":{"n":366,"factors":[2,3,61],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,61],\"n\":366}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0019","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":402,\"factors\":[2,3,67]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(402, [2, 3, 67], 是, 402, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(402, [2, 3, 67], 是, 402, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086019,"parameters":{"n":402,"factors":[2,3,67],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,67],\"n\":402}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0020","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":426,\"factors\":[2,3,71]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(426, [2, 3, 71], 是, 426, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(426, [2, 3, 71], 是, 426, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086020,"parameters":{"n":426,"factors":[2,3,71],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,71],\"n\":426}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0021","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":438,\"factors\":[2,3,73]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(438, [2, 3, 73], 是, 438, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(438, [2, 3, 73], 是, 438, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086021,"parameters":{"n":438,"factors":[2,3,73],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,73],\"n\":438}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0022","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":474,\"factors\":[2,3,79]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(474, [2, 3, 79], 是, 474, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(474, [2, 3, 79], 是, 474, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086022,"parameters":{"n":474,"factors":[2,3,79],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,79],\"n\":474}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0023","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":498,\"factors\":[2,3,83]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(498, [2, 3, 83], 是, 498, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(498, [2, 3, 83], 是, 498, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086023,"parameters":{"n":498,"factors":[2,3,83],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,83],\"n\":498}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0024","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":534,\"factors\":[2,3,89]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(534, [2, 3, 89], 是, 534, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(534, [2, 3, 89], 是, 534, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086024,"parameters":{"n":534,"factors":[2,3,89],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,89],\"n\":534}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0025","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={\"n\":582,\"factors\":[2,3,97]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(582, [2, 3, 97], 是, 582, 是, existence_requires_complete_prime_product)","explanation_zh":"Wave6第二層核對=(582, [2, 3, 97], 是, 582, 是, existence_requires_complete_prime_product)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-factorization-audit","seed":202611086025,"parameters":{"n":582,"factors":[2,3,97],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-factorization-audit","semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,97],\"n\":582}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0026","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [2, 2, 3], [[2, 2], [3, 1]], 12, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(12, [2, 2, 3], [[2, 2], [3, 1]], 12, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086026,"parameters":{"n":12,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0027","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, [2, 3, 3], [[2, 1], [3, 2]], 18, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(18, [2, 3, 3], [[2, 1], [3, 2]], 18, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086027,"parameters":{"n":18,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0028","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":30}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(30, [2, 3, 5], [[2, 1], [3, 1], [5, 1]], 30, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(30, [2, 3, 5], [[2, 1], [3, 1], [5, 1]], 30, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086028,"parameters":{"n":30,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0029","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":42}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(42, [2, 3, 7], [[2, 1], [3, 1], [7, 1]], 42, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(42, [2, 3, 7], [[2, 1], [3, 1], [7, 1]], 42, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086029,"parameters":{"n":42,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":42}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0030","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":66}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(66, [2, 3, 11], [[2, 1], [3, 1], [11, 1]], 66, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(66, [2, 3, 11], [[2, 1], [3, 1], [11, 1]], 66, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086030,"parameters":{"n":66,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":66}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0031","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":78}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(78, [2, 3, 13], [[2, 1], [3, 1], [13, 1]], 78, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(78, [2, 3, 13], [[2, 1], [3, 1], [13, 1]], 78, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086031,"parameters":{"n":78,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":78}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0032","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":102}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(102, [2, 3, 17], [[2, 1], [3, 1], [17, 1]], 102, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(102, [2, 3, 17], [[2, 1], [3, 1], [17, 1]], 102, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086032,"parameters":{"n":102,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":102}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0033","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":114}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(114, [2, 3, 19], [[2, 1], [3, 1], [19, 1]], 114, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(114, [2, 3, 19], [[2, 1], [3, 1], [19, 1]], 114, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086033,"parameters":{"n":114,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":114}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0034","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":138}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(138, [2, 3, 23], [[2, 1], [3, 1], [23, 1]], 138, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(138, [2, 3, 23], [[2, 1], [3, 1], [23, 1]], 138, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086034,"parameters":{"n":138,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":138}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0035","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":174}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(174, [2, 3, 29], [[2, 1], [3, 1], [29, 1]], 174, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(174, [2, 3, 29], [[2, 1], [3, 1], [29, 1]], 174, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086035,"parameters":{"n":174,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":174}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0036","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":186}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(186, [2, 3, 31], [[2, 1], [3, 1], [31, 1]], 186, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(186, [2, 3, 31], [[2, 1], [3, 1], [31, 1]], 186, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086036,"parameters":{"n":186,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":186}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0037","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":222}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(222, [2, 3, 37], [[2, 1], [3, 1], [37, 1]], 222, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(222, [2, 3, 37], [[2, 1], [3, 1], [37, 1]], 222, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086037,"parameters":{"n":222,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":222}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0038","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":246}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(246, [2, 3, 41], [[2, 1], [3, 1], [41, 1]], 246, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(246, [2, 3, 41], [[2, 1], [3, 1], [41, 1]], 246, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086038,"parameters":{"n":246,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":246}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0039","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":258}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(258, [2, 3, 43], [[2, 1], [3, 1], [43, 1]], 258, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(258, [2, 3, 43], [[2, 1], [3, 1], [43, 1]], 258, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086039,"parameters":{"n":258,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":258}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0040","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":282}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(282, [2, 3, 47], [[2, 1], [3, 1], [47, 1]], 282, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(282, [2, 3, 47], [[2, 1], [3, 1], [47, 1]], 282, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086040,"parameters":{"n":282,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":282}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0041","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":318}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(318, [2, 3, 53], [[2, 1], [3, 1], [53, 1]], 318, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(318, [2, 3, 53], [[2, 1], [3, 1], [53, 1]], 318, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086041,"parameters":{"n":318,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":318}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0042","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":354}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086042,"parameters":{"n":354,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":354}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0043","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":366}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(366, [2, 3, 61], [[2, 1], [3, 1], [61, 1]], 366, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(366, [2, 3, 61], [[2, 1], [3, 1], [61, 1]], 366, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086043,"parameters":{"n":366,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":366}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0044","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":402}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(402, [2, 3, 67], [[2, 1], [3, 1], [67, 1]], 402, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(402, [2, 3, 67], [[2, 1], [3, 1], [67, 1]], 402, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086044,"parameters":{"n":402,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":402}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0045","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":426}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(426, [2, 3, 71], [[2, 1], [3, 1], [71, 1]], 426, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(426, [2, 3, 71], [[2, 1], [3, 1], [71, 1]], 426, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086045,"parameters":{"n":426,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":426}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0046","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":438}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(438, [2, 3, 73], [[2, 1], [3, 1], [73, 1]], 438, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(438, [2, 3, 73], [[2, 1], [3, 1], [73, 1]], 438, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086046,"parameters":{"n":438,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":438}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0047","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":474}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(474, [2, 3, 79], [[2, 1], [3, 1], [79, 1]], 474, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(474, [2, 3, 79], [[2, 1], [3, 1], [79, 1]], 474, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086047,"parameters":{"n":474,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":474}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0048","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":498}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(498, [2, 3, 83], [[2, 1], [3, 1], [83, 1]], 498, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(498, [2, 3, 83], [[2, 1], [3, 1], [83, 1]], 498, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086048,"parameters":{"n":498,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":498}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0049","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":534}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(534, [2, 3, 89], [[2, 1], [3, 1], [89, 1]], 534, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(534, [2, 3, 89], [[2, 1], [3, 1], [89, 1]], 534, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086049,"parameters":{"n":534,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":534}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0050","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-canonical-factorization；資料={\"n\":582}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(582, [2, 3, 97], [[2, 1], [3, 1], [97, 1]], 582, 是, complete_trial_division_range)","explanation_zh":"Wave6第二層核對=(582, [2, 3, 97], [[2, 1], [3, 1], [97, 1]], 582, 是, complete_trial_division_range)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-canonical-factorization","seed":202611086050,"parameters":{"n":582,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-canonical-factorization","semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":582}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0051","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":12,\"factors_left\":[2,3,2],\"factors_right\":[2,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 2, 3], [2, 2, 3], [[2, 是, 0, 2, 是], [2, 是, 0, 2, 是], [3, 是, 2, 3, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 2, 3], [2, 2, 3], [[2, 是, 0, 2, 是], [2, 是, 0, 2, 是], [3, 是, 2, 3, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086051,"parameters":{"n":12,"factors_left":[2,3,2],"factors_right":[2,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,2,3],\"factors_right\":[2,2,3],\"n\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0052","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":18,\"factors_left\":[2,3,3],\"factors_right\":[3,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 3], [2, 3, 3], [[2, 是, 1, 2, 是], [3, 是, 0, 3, 是], [3, 是, 0, 3, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 3], [2, 3, 3], [[2, 是, 1, 2, 是], [3, 是, 0, 3, 是], [3, 是, 0, 3, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086052,"parameters":{"n":18,"factors_left":[2,3,3],"factors_right":[3,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,3],\"factors_right\":[2,3,3],\"n\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0053","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":30,\"factors_left\":[2,3,5],\"factors_right\":[5,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 5], [2, 3, 5], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [5, 是, 0, 5, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 5], [2, 3, 5], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [5, 是, 0, 5, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086053,"parameters":{"n":30,"factors_left":[2,3,5],"factors_right":[5,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,5],\"factors_right\":[2,3,5],\"n\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0054","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":42,\"factors_left\":[2,3,7],\"factors_right\":[7,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 7], [2, 3, 7], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [7, 是, 0, 7, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 7], [2, 3, 7], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [7, 是, 0, 7, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086054,"parameters":{"n":42,"factors_left":[2,3,7],"factors_right":[7,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,7],\"factors_right\":[2,3,7],\"n\":42}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0055","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":66,\"factors_left\":[2,3,11],\"factors_right\":[11,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 11], [2, 3, 11], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [11, 是, 0, 11, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 11], [2, 3, 11], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [11, 是, 0, 11, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086055,"parameters":{"n":66,"factors_left":[2,3,11],"factors_right":[11,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,11],\"factors_right\":[2,3,11],\"n\":66}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0056","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":78,\"factors_left\":[2,3,13],\"factors_right\":[13,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 13], [2, 3, 13], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [13, 是, 0, 13, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 13], [2, 3, 13], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [13, 是, 0, 13, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086056,"parameters":{"n":78,"factors_left":[2,3,13],"factors_right":[13,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,13],\"factors_right\":[2,3,13],\"n\":78}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0057","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":102,\"factors_left\":[2,3,17],\"factors_right\":[17,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 17], [2, 3, 17], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [17, 是, 0, 17, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 17], [2, 3, 17], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [17, 是, 0, 17, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086057,"parameters":{"n":102,"factors_left":[2,3,17],"factors_right":[17,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,17],\"factors_right\":[2,3,17],\"n\":102}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0058","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":114,\"factors_left\":[2,3,19],\"factors_right\":[19,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 19], [2, 3, 19], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [19, 是, 0, 19, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 19], [2, 3, 19], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [19, 是, 0, 19, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086058,"parameters":{"n":114,"factors_left":[2,3,19],"factors_right":[19,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,19],\"factors_right\":[2,3,19],\"n\":114}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0059","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":138,\"factors_left\":[2,3,23],\"factors_right\":[23,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 23], [2, 3, 23], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [23, 是, 0, 23, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 23], [2, 3, 23], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [23, 是, 0, 23, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086059,"parameters":{"n":138,"factors_left":[2,3,23],"factors_right":[23,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,23],\"factors_right\":[2,3,23],\"n\":138}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0060","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":174,\"factors_left\":[2,3,29],\"factors_right\":[29,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 29], [2, 3, 29], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [29, 是, 0, 29, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 29], [2, 3, 29], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [29, 是, 0, 29, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086060,"parameters":{"n":174,"factors_left":[2,3,29],"factors_right":[29,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,29],\"factors_right\":[2,3,29],\"n\":174}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0061","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":186,\"factors_left\":[2,3,31],\"factors_right\":[31,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 31], [2, 3, 31], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [31, 是, 0, 31, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 31], [2, 3, 31], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [31, 是, 0, 31, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086061,"parameters":{"n":186,"factors_left":[2,3,31],"factors_right":[31,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,31],\"factors_right\":[2,3,31],\"n\":186}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0062","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":222,\"factors_left\":[2,3,37],\"factors_right\":[37,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 37], [2, 3, 37], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [37, 是, 0, 37, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 37], [2, 3, 37], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [37, 是, 0, 37, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086062,"parameters":{"n":222,"factors_left":[2,3,37],"factors_right":[37,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,37],\"factors_right\":[2,3,37],\"n\":222}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0063","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":246,\"factors_left\":[2,3,41],\"factors_right\":[41,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 41], [2, 3, 41], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [41, 是, 0, 41, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 41], [2, 3, 41], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [41, 是, 0, 41, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086063,"parameters":{"n":246,"factors_left":[2,3,41],"factors_right":[41,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,41],\"factors_right\":[2,3,41],\"n\":246}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0064","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":258,\"factors_left\":[2,3,43],\"factors_right\":[43,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 43], [2, 3, 43], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [43, 是, 0, 43, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 43], [2, 3, 43], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [43, 是, 0, 43, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086064,"parameters":{"n":258,"factors_left":[2,3,43],"factors_right":[43,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,43],\"factors_right\":[2,3,43],\"n\":258}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0065","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":282,\"factors_left\":[2,3,47],\"factors_right\":[47,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 47], [2, 3, 47], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [47, 是, 0, 47, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 47], [2, 3, 47], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [47, 是, 0, 47, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086065,"parameters":{"n":282,"factors_left":[2,3,47],"factors_right":[47,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,47],\"factors_right\":[2,3,47],\"n\":282}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0066","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":318,\"factors_left\":[2,3,53],\"factors_right\":[53,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 53], [2, 3, 53], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [53, 是, 0, 53, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 53], [2, 3, 53], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [53, 是, 0, 53, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086066,"parameters":{"n":318,"factors_left":[2,3,53],"factors_right":[53,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,53],\"factors_right\":[2,3,53],\"n\":318}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0067","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":354,\"factors_left\":[2,3,59],\"factors_right\":[59,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 59], [2, 3, 59], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [59, 是, 0, 59, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 59], [2, 3, 59], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [59, 是, 0, 59, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086067,"parameters":{"n":354,"factors_left":[2,3,59],"factors_right":[59,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,59],\"factors_right\":[2,3,59],\"n\":354}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0068","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":366,\"factors_left\":[2,3,61],\"factors_right\":[61,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 61], [2, 3, 61], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [61, 是, 0, 61, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 61], [2, 3, 61], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [61, 是, 0, 61, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086068,"parameters":{"n":366,"factors_left":[2,3,61],"factors_right":[61,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,61],\"factors_right\":[2,3,61],\"n\":366}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0069","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":402,\"factors_left\":[2,3,67],\"factors_right\":[67,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 67], [2, 3, 67], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [67, 是, 0, 67, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 67], [2, 3, 67], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [67, 是, 0, 67, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086069,"parameters":{"n":402,"factors_left":[2,3,67],"factors_right":[67,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,67],\"factors_right\":[2,3,67],\"n\":402}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0070","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":426,\"factors_left\":[2,3,71],\"factors_right\":[71,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 71], [2, 3, 71], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [71, 是, 0, 71, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 71], [2, 3, 71], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [71, 是, 0, 71, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086070,"parameters":{"n":426,"factors_left":[2,3,71],"factors_right":[71,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,71],\"factors_right\":[2,3,71],\"n\":426}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0071","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":438,\"factors_left\":[2,3,73],\"factors_right\":[73,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 73], [2, 3, 73], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [73, 是, 0, 73, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 73], [2, 3, 73], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [73, 是, 0, 73, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086071,"parameters":{"n":438,"factors_left":[2,3,73],"factors_right":[73,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,73],\"factors_right\":[2,3,73],\"n\":438}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0072","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":474,\"factors_left\":[2,3,79],\"factors_right\":[79,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 79], [2, 3, 79], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [79, 是, 0, 79, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 79], [2, 3, 79], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [79, 是, 0, 79, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086072,"parameters":{"n":474,"factors_left":[2,3,79],"factors_right":[79,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,79],\"factors_right\":[2,3,79],\"n\":474}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0073","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":498,\"factors_left\":[2,3,83],\"factors_right\":[83,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 83], [2, 3, 83], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [83, 是, 0, 83, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 83], [2, 3, 83], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [83, 是, 0, 83, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086073,"parameters":{"n":498,"factors_left":[2,3,83],"factors_right":[83,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,83],\"factors_right\":[2,3,83],\"n\":498}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0074","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":534,\"factors_left\":[2,3,89],\"factors_right\":[89,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 89], [2, 3, 89], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [89, 是, 0, 89, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 89], [2, 3, 89], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [89, 是, 0, 89, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086074,"parameters":{"n":534,"factors_left":[2,3,89],"factors_right":[89,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,89],\"factors_right\":[2,3,89],\"n\":534}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0075","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-uniqueness-audit；資料={\"n\":582,\"factors_left\":[2,3,97],\"factors_right\":[97,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(是, 是, [2, 3, 97], [2, 3, 97], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [97, 是, 0, 97, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation_zh":"Wave6第二層核對=(是, 是, [2, 3, 97], [2, 3, 97], [[2, 是, 1, 2, 是], [3, 是, 2, 3, 是], [97, 是, 0, 97, 是]], 是, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086075,"parameters":{"n":582,"factors_left":[2,3,97],"factors_right":[97,2,3],"domain":"Exact_integer_unique_prime_factorization","operation":"upf-uniqueness-audit","semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,97],\"factors_right\":[2,3,97],\"n\":582}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0076","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 否, null, [], zero_has_no_prime_factorization)","explanation_zh":"Wave6第二層核對=(0, 否, null, [], zero_has_no_prime_factorization)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086076,"parameters":{"n":0,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0077","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 否, 1, [], units_have_empty_prime_multiset)","explanation_zh":"Wave6第二層核對=(1, 否, 1, [], units_have_empty_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086077,"parameters":{"n":1,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0078","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 否, -1, [], units_have_empty_prime_multiset)","explanation_zh":"Wave6第二層核對=(-1, 否, -1, [], units_have_empty_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086078,"parameters":{"n":-1,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0079","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-42}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-42, 否, -1, [2, 3, 7], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-42, 否, -1, [2, 3, 7], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086079,"parameters":{"n":-42,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-42}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0080","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-66}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-66, 否, -1, [2, 3, 11], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-66, 否, -1, [2, 3, 11], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086080,"parameters":{"n":-66,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-66}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0081","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-78}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-78, 否, -1, [2, 3, 13], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-78, 否, -1, [2, 3, 13], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086081,"parameters":{"n":-78,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-78}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0082","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-102}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-102, 否, -1, [2, 3, 17], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-102, 否, -1, [2, 3, 17], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086082,"parameters":{"n":-102,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-102}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0083","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-114}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-114, 否, -1, [2, 3, 19], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-114, 否, -1, [2, 3, 19], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086083,"parameters":{"n":-114,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-114}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0084","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-138}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-138, 否, -1, [2, 3, 23], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-138, 否, -1, [2, 3, 23], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086084,"parameters":{"n":-138,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-138}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0085","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-174}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-174, 否, -1, [2, 3, 29], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-174, 否, -1, [2, 3, 29], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086085,"parameters":{"n":-174,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-174}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0086","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-186}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-186, 否, -1, [2, 3, 31], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-186, 否, -1, [2, 3, 31], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086086,"parameters":{"n":-186,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-186}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0087","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-222}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-222, 否, -1, [2, 3, 37], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-222, 否, -1, [2, 3, 37], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086087,"parameters":{"n":-222,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-222}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0088","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-246}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-246, 否, -1, [2, 3, 41], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-246, 否, -1, [2, 3, 41], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086088,"parameters":{"n":-246,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-246}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0089","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-258}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-258, 否, -1, [2, 3, 43], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-258, 否, -1, [2, 3, 43], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086089,"parameters":{"n":-258,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-258}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0090","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-282}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-282, 否, -1, [2, 3, 47], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-282, 否, -1, [2, 3, 47], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086090,"parameters":{"n":-282,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-282}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0091","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-318}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-318, 否, -1, [2, 3, 53], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-318, 否, -1, [2, 3, 53], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086091,"parameters":{"n":-318,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-318}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0092","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-354}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-354, 否, -1, [2, 3, 59], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-354, 否, -1, [2, 3, 59], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086092,"parameters":{"n":-354,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-354}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0093","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-366}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-366, 否, -1, [2, 3, 61], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-366, 否, -1, [2, 3, 61], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086093,"parameters":{"n":-366,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-366}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0094","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-402}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-402, 否, -1, [2, 3, 67], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-402, 否, -1, [2, 3, 67], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086094,"parameters":{"n":-402,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-402}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0095","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-426}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-426, 否, -1, [2, 3, 71], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-426, 否, -1, [2, 3, 71], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086095,"parameters":{"n":-426,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-426}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0096","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-438}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-438, 否, -1, [2, 3, 73], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-438, 否, -1, [2, 3, 73], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086096,"parameters":{"n":-438,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-438}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0097","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-474}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-474, 否, -1, [2, 3, 79], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-474, 否, -1, [2, 3, 79], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086097,"parameters":{"n":-474,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-474}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0098","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-498}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-498, 否, -1, [2, 3, 83], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-498, 否, -1, [2, 3, 83], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086098,"parameters":{"n":-498,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-498}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0099","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-534}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-534, 否, -1, [2, 3, 89], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-534, 否, -1, [2, 3, 89], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086099,"parameters":{"n":-534,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-534}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0100","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-unique-prime-factorization"],"title_zh":"整數唯一質因數分解","stem_zh":"完整bounded integer factorization與exact primality。操作=upf-unit-sign-boundary；資料={\"n\":-582}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-582, 否, -1, [2, 3, 97], integer_sign_separated_from_positive_prime_multiset)","explanation_zh":"Wave6第二層核對=(-582, 否, -1, [2, 3, 97], integer_sign_separated_from_positive_prime_multiset)；依完整明示資料獨立重算；positive n大於1的prime multiset唯一至多差順序；existence與uniqueness分開；uniqueness保留Euclid-lemma witness；負數分離minus one，units空乘積，zero無此factorization。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數唯一質因數分解"]},"variant":{"family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086100,"parameters":{"n":-582,"domain":"Exact_integer_unique_prime_factorization","operation":"upf-unit-sign-boundary","semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-582}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}