{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0001","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":2,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 2, 1, 1, [2], 是, 是)","explanation_zh":"Wave6第十二層核對=(2, 2, 2, 1, 1, [2], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185001,"parameters":{"p":2,"a":2,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:2,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0002","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":3,\"a\":-18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, -18, 18, 2, 2, [3, 9], 是, 是)","explanation_zh":"Wave6第十二層核對=(3, -18, 18, 2, 2, [3, 9], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185002,"parameters":{"p":3,"a":-18,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:-18,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0003","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":5,\"a\":375}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 375, 375, 3, 3, [5, 25, 125], 是, 是)","explanation_zh":"Wave6第十二層核對=(5, 375, 375, 3, 3, [5, 25, 125], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185003,"parameters":{"p":5,"a":375,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:375,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0004","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":2,\"a\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, -8, 8, 3, 1, [2, 4, 8], 是, 是)","explanation_zh":"Wave6第十二層核對=(2, -8, 8, 3, 1, [2, 4, 8], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185004,"parameters":{"p":2,"a":-8,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:-8,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0005","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":3,\"a\":45}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 45, 45, 2, 5, [3, 9], 是, 是)","explanation_zh":"Wave6第十二層核對=(3, 45, 45, 2, 5, [3, 9], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185005,"parameters":{"p":3,"a":45,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:45,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0006","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":5,\"a\":-750}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, -750, 750, 3, 6, [5, 25, 125], 是, 是)","explanation_zh":"Wave6第十二層核對=(5, -750, 750, 3, 6, [5, 25, 125], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185006,"parameters":{"p":5,"a":-750,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:-750,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0007","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":2,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 14, 14, 1, 7, [2], 是, 是)","explanation_zh":"Wave6第十二層核對=(2, 14, 14, 1, 7, [2], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185007,"parameters":{"p":2,"a":14,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:14,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0008","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":3,\"a\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, -9, 9, 2, 1, [3, 9], 是, 是)","explanation_zh":"Wave6第十二層核對=(3, -9, 9, 2, 1, [3, 9], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185008,"parameters":{"p":3,"a":-9,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:-9,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0009","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":5,\"a\":250}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 250, 250, 3, 2, [5, 25, 125], 是, 是)","explanation_zh":"Wave6第十二層核對=(5, 250, 250, 3, 2, [5, 25, 125], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185009,"parameters":{"p":5,"a":250,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:250,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0010","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":2,\"a\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, -6, 6, 1, 3, [2], 是, 是)","explanation_zh":"Wave6第十二層核對=(2, -6, 6, 1, 3, [2], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185010,"parameters":{"p":2,"a":-6,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:-6,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0011","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":3,\"a\":36}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 36, 36, 2, 4, [3, 9], 是, 是)","explanation_zh":"Wave6第十二層核對=(3, 36, 36, 2, 4, [3, 9], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185011,"parameters":{"p":3,"a":36,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:36,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0012","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":5,\"a\":-625}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, -625, 625, 4, 1, [5, 25, 125, 625], 是, 是)","explanation_zh":"Wave6第十二層核對=(5, -625, 625, 4, 1, [5, 25, 125, 625], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185012,"parameters":{"p":5,"a":-625,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:-625,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0013","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":2,\"a\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 12, 12, 2, 3, [2, 4], 是, 是)","explanation_zh":"Wave6第十二層核對=(2, 12, 12, 2, 3, [2, 4], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185013,"parameters":{"p":2,"a":12,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:12,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0014","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":3,\"a\":-63}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, -63, 63, 2, 7, [3, 9], 是, 是)","explanation_zh":"Wave6第十二層核對=(3, -63, 63, 2, 7, [3, 9], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185014,"parameters":{"p":3,"a":-63,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:-63,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0015","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":5,\"a\":125}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 125, 125, 3, 1, [5, 25, 125], 是, 是)","explanation_zh":"Wave6第十二層核對=(5, 125, 125, 3, 1, [5, 25, 125], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185015,"parameters":{"p":5,"a":125,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:125,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0016","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":2,\"a\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, -4, 4, 2, 1, [2, 4], 是, 是)","explanation_zh":"Wave6第十二層核對=(2, -4, 4, 2, 1, [2, 4], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185016,"parameters":{"p":2,"a":-4,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:-4,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0017","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":3,\"a\":27}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 27, 27, 3, 1, [3, 9, 27], 是, 是)","explanation_zh":"Wave6第十二層核對=(3, 27, 27, 3, 1, [3, 9, 27], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185017,"parameters":{"p":3,"a":27,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:27,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0018","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":5,\"a\":-500}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, -500, 500, 3, 4, [5, 25, 125], 是, 是)","explanation_zh":"Wave6第十二層核對=(5, -500, 500, 3, 4, [5, 25, 125], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185018,"parameters":{"p":5,"a":-500,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:-500,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0019","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":2,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 10, 10, 1, 5, [2], 是, 是)","explanation_zh":"Wave6第十二層核對=(2, 10, 10, 1, 5, [2], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185019,"parameters":{"p":2,"a":10,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:10,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0020","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":3,\"a\":-54}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, -54, 54, 3, 2, [3, 9, 27], 是, 是)","explanation_zh":"Wave6第十二層核對=(3, -54, 54, 3, 2, [3, 9, 27], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185020,"parameters":{"p":3,"a":-54,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:-54,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0021","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":5,\"a\":875}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 875, 875, 3, 7, [5, 25, 125], 是, 是)","explanation_zh":"Wave6第十二層核對=(5, 875, 875, 3, 7, [5, 25, 125], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185021,"parameters":{"p":5,"a":875,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:875,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0022","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":2,\"a\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, -2, 2, 1, 1, [2], 是, 是)","explanation_zh":"Wave6第十二層核對=(2, -2, 2, 1, 1, [2], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185022,"parameters":{"p":2,"a":-2,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:-2,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0023","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":3,\"a\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 18, 18, 2, 2, [3, 9], 是, 是)","explanation_zh":"Wave6第十二層核對=(3, 18, 18, 2, 2, [3, 9], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185023,"parameters":{"p":3,"a":18,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:18,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0024","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":5,\"a\":-375}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, -375, 375, 3, 3, [5, 25, 125], 是, 是)","explanation_zh":"Wave6第十二層核對=(5, -375, 375, 3, 3, [5, 25, 125], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185024,"parameters":{"p":5,"a":-375,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:-375,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0025","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-factor-extraction；資料={\"p\":2,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 8, 8, 3, 1, [2, 4, 8], 是, 是)","explanation_zh":"Wave6第十二層核對=(2, 8, 8, 3, 1, [2, 4, 8], 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-factor-extraction","seed":202611185025,"parameters":{"p":2,"a":8,"domain":"Exact_integer_p_adic_valuation","operation":"pval-factor-extraction","semantic_template":"pval-factor-extraction|{a:8,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0026","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":2,\"a\":1,\"b\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1, -4, 0, 2, -4, 2, 2, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(2, 1, -4, 0, 2, -4, 2, 2, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185026,"parameters":{"p":2,"a":1,"b":-4,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:1,b:-4,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0027","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":3,\"a\":6,\"b\":-27}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 6, -27, 1, 3, -162, 4, 4, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(3, 6, -27, 1, 3, -162, 4, 4, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185027,"parameters":{"p":3,"a":6,"b":-27,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:6,b:-27,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0028","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":5,\"a\":75,\"b\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 75, -4, 2, 0, -300, 2, 2, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(5, 75, -4, 2, 0, -300, 2, 2, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185028,"parameters":{"p":5,"a":75,"b":-4,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:75,b:-4,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0029","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":7,\"a\":4,\"b\":-35}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 4, -35, 0, 1, -140, 1, 1, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(7, 4, -35, 0, 1, -140, 1, 1, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185029,"parameters":{"p":7,"a":4,"b":-35,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:4,b:-35,p:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0030","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":11,\"a\":55,\"b\":-726}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 55, -726, 1, 2, -39930, 3, 3, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(11, 55, -726, 1, 2, -39930, 3, 3, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185030,"parameters":{"p":11,"a":55,"b":-726,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:55,b:-726,p:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0031","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":13,\"a\":1014,\"b\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 1014, -7, 2, 0, -7098, 2, 2, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(13, 1014, -7, 2, 0, -7098, 2, 2, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185031,"parameters":{"p":13,"a":1014,"b":-7,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:1014,b:-7,p:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0032","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":2,\"a\":7,\"b\":-16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 7, -16, 0, 4, -112, 4, 4, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(2, 7, -16, 0, 4, -112, 4, 4, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185032,"parameters":{"p":2,"a":7,"b":-16,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:7,b:-16,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0033","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":3,\"a\":24,\"b\":-81}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 24, -81, 1, 4, -1944, 5, 5, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(3, 24, -81, 1, 4, -1944, 5, 5, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185033,"parameters":{"p":3,"a":24,"b":-81,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:24,b:-81,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0034","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":5,\"a\":225,\"b\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 225, -10, 2, 1, -2250, 3, 3, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(5, 225, -10, 2, 1, -2250, 3, 3, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185034,"parameters":{"p":5,"a":225,"b":-10,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:225,b:-10,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0035","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":7,\"a\":10,\"b\":-77}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 10, -77, 0, 1, -770, 1, 1, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(7, 10, -77, 0, 1, -770, 1, 1, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185035,"parameters":{"p":7,"a":10,"b":-77,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:10,b:-77,p:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0036","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":11,\"a\":121,\"b\":-1452}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 121, -1452, 2, 2, -175692, 4, 4, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(11, 121, -1452, 2, 2, -175692, 4, 4, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185036,"parameters":{"p":11,"a":121,"b":-1452,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:121,b:-1452,p:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0037","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":13,\"a\":2028,\"b\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 2028, -13, 2, 1, -26364, 3, 3, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(13, 2028, -13, 2, 1, -26364, 3, 3, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185037,"parameters":{"p":13,"a":2028,"b":-13,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:2028,b:-13,p:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0038","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":2,\"a\":13,\"b\":-28}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 13, -28, 0, 2, -364, 2, 2, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(2, 13, -28, 0, 2, -364, 2, 2, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185038,"parameters":{"p":2,"a":13,"b":-28,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:13,b:-28,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0039","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":3,\"a\":42,\"b\":-135}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 42, -135, 1, 3, -5670, 4, 4, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(3, 42, -135, 1, 3, -5670, 4, 4, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185039,"parameters":{"p":3,"a":42,"b":-135,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:42,b:-135,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0040","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":5,\"a\":375,\"b\":-16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 375, -16, 3, 0, -6000, 3, 3, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(5, 375, -16, 3, 0, -6000, 3, 3, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185040,"parameters":{"p":5,"a":375,"b":-16,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:375,b:-16,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0041","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":7,\"a\":16,\"b\":-119}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 16, -119, 0, 1, -1904, 1, 1, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(7, 16, -119, 0, 1, -1904, 1, 1, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185041,"parameters":{"p":7,"a":16,"b":-119,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:16,b:-119,p:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0042","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":11,\"a\":187,\"b\":-2178}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 187, -2178, 1, 2, -407286, 3, 3, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(11, 187, -2178, 1, 2, -407286, 3, 3, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185042,"parameters":{"p":11,"a":187,"b":-2178,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:187,b:-2178,p:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0043","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":13,\"a\":3042,\"b\":-19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 3042, -19, 2, 0, -57798, 2, 2, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(13, 3042, -19, 2, 0, -57798, 2, 2, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185043,"parameters":{"p":13,"a":3042,"b":-19,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:3042,b:-19,p:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0044","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":2,\"a\":19,\"b\":-40}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 19, -40, 0, 3, -760, 3, 3, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(2, 19, -40, 0, 3, -760, 3, 3, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185044,"parameters":{"p":2,"a":19,"b":-40,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:19,b:-40,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0045","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":3,\"a\":60,\"b\":-189}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 60, -189, 1, 3, -11340, 4, 4, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(3, 60, -189, 1, 3, -11340, 4, 4, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185045,"parameters":{"p":3,"a":60,"b":-189,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:60,b:-189,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0046","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":5,\"a\":525,\"b\":-22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 525, -22, 2, 0, -11550, 2, 2, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(5, 525, -22, 2, 0, -11550, 2, 2, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185046,"parameters":{"p":5,"a":525,"b":-22,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:525,b:-22,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0047","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":7,\"a\":22,\"b\":-161}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 22, -161, 0, 1, -3542, 1, 1, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(7, 22, -161, 0, 1, -3542, 1, 1, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185047,"parameters":{"p":7,"a":22,"b":-161,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:22,b:-161,p:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0048","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":11,\"a\":253,\"b\":-2904}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 253, -2904, 1, 2, -734712, 3, 3, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(11, 253, -2904, 1, 2, -734712, 3, 3, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185048,"parameters":{"p":11,"a":253,"b":-2904,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:253,b:-2904,p:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0049","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":13,\"a\":4056,\"b\":-25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 4056, -25, 2, 0, -101400, 2, 2, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(13, 4056, -25, 2, 0, -101400, 2, 2, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185049,"parameters":{"p":13,"a":4056,"b":-25,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:4056,b:-25,p:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0050","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":2,\"a\":25,\"b\":-52}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 25, -52, 0, 2, -1300, 2, 2, 是, infinity_plus_k_equals_infinity)","explanation_zh":"Wave6第十二層核對=(2, 25, -52, 0, 2, -1300, 2, 2, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-product-additivity","seed":202611185050,"parameters":{"p":2,"a":25,"b":-52,"domain":"Exact_integer_p_adic_valuation","operation":"pval-product-additivity","semantic_template":"pval-product-additivity|{a:25,b:-52,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0051","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":2,\"a\":2,\"b\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 4, 1, 2, 6, 1, 1, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(2, 2, 4, 1, 2, 6, 1, 1, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185051,"parameters":{"p":2,"a":2,"b":4,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:2,b:4,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0052","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":3,\"a\":18,\"b\":27}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 18, 27, 2, 3, 45, 2, 2, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(3, 18, 27, 2, 3, 45, 2, 2, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185052,"parameters":{"p":3,"a":18,"b":27,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:18,b:27,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0053","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":5,\"a\":375,\"b\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 375, 4, 3, 0, 379, 0, 0, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(5, 375, 4, 3, 0, 379, 0, 0, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185053,"parameters":{"p":5,"a":375,"b":4,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:375,b:4,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0054","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":7,\"a\":28,\"b\":35}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 28, 35, 1, 1, 63, 1, 1, 是, 否, not_guaranteed)","explanation_zh":"Wave6第十二層核對=(7, 28, 35, 1, 1, 63, 1, 1, 是, 否, not_guaranteed)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185054,"parameters":{"p":7,"a":28,"b":35,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:28,b:35,p:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0055","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":11,\"a\":605,\"b\":726}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 605, 726, 2, 2, 1331, 3, 2, 是, 否, not_guaranteed)","explanation_zh":"Wave6第十二層核對=(11, 605, 726, 2, 2, 1331, 3, 2, 是, 否, not_guaranteed)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185055,"parameters":{"p":11,"a":605,"b":726,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:605,b:726,p:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0056","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":13,\"a\":13182,\"b\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 13182, 7, 3, 0, 13189, 0, 0, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(13, 13182, 7, 3, 0, 13189, 0, 0, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185056,"parameters":{"p":13,"a":13182,"b":7,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:13182,b:7,p:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0057","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":2,\"a\":14,\"b\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 14, 16, 1, 4, 30, 1, 1, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(2, 14, 16, 1, 4, 30, 1, 1, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185057,"parameters":{"p":2,"a":14,"b":16,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:14,b:16,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0058","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":3,\"a\":72,\"b\":81}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 72, 81, 2, 4, 153, 2, 2, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(3, 72, 81, 2, 4, 153, 2, 2, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185058,"parameters":{"p":3,"a":72,"b":81,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:72,b:81,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0059","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":5,\"a\":1125,\"b\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 1125, 10, 3, 1, 1135, 1, 1, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(5, 1125, 10, 3, 1, 1135, 1, 1, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185059,"parameters":{"p":5,"a":1125,"b":10,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:1125,b:10,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0060","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":7,\"a\":70,\"b\":77}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 70, 77, 1, 1, 147, 2, 1, 是, 否, not_guaranteed)","explanation_zh":"Wave6第十二層核對=(7, 70, 77, 1, 1, 147, 2, 1, 是, 否, not_guaranteed)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185060,"parameters":{"p":7,"a":70,"b":77,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:70,b:77,p:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0061","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":11,\"a\":1331,\"b\":1452}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 1331, 1452, 3, 2, 2783, 2, 2, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(11, 1331, 1452, 3, 2, 2783, 2, 2, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185061,"parameters":{"p":11,"a":1331,"b":1452,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:1331,b:1452,p:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0062","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":13,\"a\":26364,\"b\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 26364, 13, 3, 1, 26377, 1, 1, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(13, 26364, 13, 3, 1, 26377, 1, 1, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185062,"parameters":{"p":13,"a":26364,"b":13,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:26364,b:13,p:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0063","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":2,\"a\":26,\"b\":28}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 26, 28, 1, 2, 54, 1, 1, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(2, 26, 28, 1, 2, 54, 1, 1, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185063,"parameters":{"p":2,"a":26,"b":28,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:26,b:28,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0064","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":3,\"a\":126,\"b\":135}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 126, 135, 2, 3, 261, 2, 2, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(3, 126, 135, 2, 3, 261, 2, 2, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185064,"parameters":{"p":3,"a":126,"b":135,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:126,b:135,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0065","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":5,\"a\":1875,\"b\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 1875, 16, 4, 0, 1891, 0, 0, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(5, 1875, 16, 4, 0, 1891, 0, 0, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185065,"parameters":{"p":5,"a":1875,"b":16,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:1875,b:16,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0066","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":7,\"a\":112,\"b\":119}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 112, 119, 1, 1, 231, 1, 1, 是, 否, not_guaranteed)","explanation_zh":"Wave6第十二層核對=(7, 112, 119, 1, 1, 231, 1, 1, 是, 否, not_guaranteed)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185066,"parameters":{"p":7,"a":112,"b":119,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:112,b:119,p:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0067","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":11,\"a\":2057,\"b\":2178}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 2057, 2178, 2, 2, 4235, 2, 2, 是, 否, not_guaranteed)","explanation_zh":"Wave6第十二層核對=(11, 2057, 2178, 2, 2, 4235, 2, 2, 是, 否, not_guaranteed)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185067,"parameters":{"p":11,"a":2057,"b":2178,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:2057,b:2178,p:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0068","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":13,\"a\":39546,\"b\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 39546, 19, 3, 0, 39565, 0, 0, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(13, 39546, 19, 3, 0, 39565, 0, 0, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185068,"parameters":{"p":13,"a":39546,"b":19,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:39546,b:19,p:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0069","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":2,\"a\":38,\"b\":40}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 38, 40, 1, 3, 78, 1, 1, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(2, 38, 40, 1, 3, 78, 1, 1, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185069,"parameters":{"p":2,"a":38,"b":40,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:38,b:40,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0070","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":3,\"a\":180,\"b\":189}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 180, 189, 2, 3, 369, 2, 2, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(3, 180, 189, 2, 3, 369, 2, 2, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185070,"parameters":{"p":3,"a":180,"b":189,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:180,b:189,p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0071","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":5,\"a\":2625,\"b\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 2625, 22, 3, 0, 2647, 0, 0, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(5, 2625, 22, 3, 0, 2647, 0, 0, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185071,"parameters":{"p":5,"a":2625,"b":22,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:2625,b:22,p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0072","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":7,\"a\":154,\"b\":161}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 154, 161, 1, 1, 315, 1, 1, 是, 否, not_guaranteed)","explanation_zh":"Wave6第十二層核對=(7, 154, 161, 1, 1, 315, 1, 1, 是, 否, not_guaranteed)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185072,"parameters":{"p":7,"a":154,"b":161,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:154,b:161,p:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0073","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":11,\"a\":2783,\"b\":2904}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 2783, 2904, 2, 2, 5687, 2, 2, 是, 否, not_guaranteed)","explanation_zh":"Wave6第十二層核對=(11, 2783, 2904, 2, 2, 5687, 2, 2, 是, 否, not_guaranteed)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185073,"parameters":{"p":11,"a":2783,"b":2904,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:2783,b:2904,p:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0074","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":13,\"a\":52728,\"b\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 52728, 25, 3, 0, 52753, 0, 0, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(13, 52728, 25, 3, 0, 52753, 0, 0, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185074,"parameters":{"p":13,"a":52728,"b":25,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:52728,b:25,p:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0075","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":2,\"a\":50,\"b\":52}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 50, 52, 1, 2, 102, 1, 1, 是, 是, 是)","explanation_zh":"Wave6第十二層核對=(2, 50, 52, 1, 2, 102, 1, 1, 是, 是, 是)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185075,"parameters":{"p":2,"a":50,"b":52,"domain":"Exact_integer_p_adic_valuation","operation":"pval-sum-ultrametric","semantic_template":"pval-sum-ultrametric|{a:50,b:52,p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0076","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"zero_convention\",\"p\":2,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_convention, 2, 0, infinity, single_serializable_zero_convention)","explanation_zh":"Wave6第十二層核對=(zero_convention, 2, 0, infinity, single_serializable_zero_convention)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185076,"parameters":{"mode":"zero_convention","p":2,"a":0,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0077","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"nonprime_base\",\"p\":6,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonprime_base, 6, 2, 否, not_called_p_adic_valuation)","explanation_zh":"Wave6第十二層核對=(nonprime_base, 6, 2, 否, not_called_p_adic_valuation)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185077,"parameters":{"mode":"nonprime_base","p":6,"a":2,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:2,mode:\"nonprime_base\",p:6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0078","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"negative_normalization\",\"p\":2,\"a\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(negative_normalization, 2, -12, 12, 2, 是, negative_integer_is_valid)","explanation_zh":"Wave6第十二層核對=(negative_normalization, 2, -12, 12, 2, 是, negative_integer_is_valid)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185078,"parameters":{"mode":"negative_normalization","p":2,"a":-12,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:-12,mode:\"negative_normalization\",p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0079","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"equal_valuations_cancellation\",\"p\":2,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(equal_valuations_cancellation, 2, 8, -8, 3, 3, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)","explanation_zh":"Wave6第十二層核對=(equal_valuations_cancellation, 2, 8, -8, 3, 3, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185079,"parameters":{"mode":"equal_valuations_cancellation","p":2,"a":8,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:8,mode:\"equal_valuations_cancellation\",p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0080","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"zero_convention\",\"p\":3,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_convention, 3, 0, infinity, single_serializable_zero_convention)","explanation_zh":"Wave6第十二層核對=(zero_convention, 3, 0, infinity, single_serializable_zero_convention)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185080,"parameters":{"mode":"zero_convention","p":3,"a":0,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0081","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"nonprime_base\",\"p\":12,\"a\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonprime_base, 12, 6, 否, not_called_p_adic_valuation)","explanation_zh":"Wave6第十二層核對=(nonprime_base, 12, 6, 否, not_called_p_adic_valuation)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185081,"parameters":{"mode":"nonprime_base","p":12,"a":6,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:6,mode:\"nonprime_base\",p: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；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0082","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"negative_normalization\",\"p\":3,\"a\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(negative_normalization, 3, -7, 7, 0, 是, negative_integer_is_valid)","explanation_zh":"Wave6第十二層核對=(negative_normalization, 3, -7, 7, 0, 是, negative_integer_is_valid)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185082,"parameters":{"mode":"negative_normalization","p":3,"a":-7,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:-7,mode:\"negative_normalization\",p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0083","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"equal_valuations_cancellation\",\"p\":3,\"a\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(equal_valuations_cancellation, 3, 24, -24, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)","explanation_zh":"Wave6第十二層核對=(equal_valuations_cancellation, 3, 24, -24, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185083,"parameters":{"mode":"equal_valuations_cancellation","p":3,"a":24,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:24,mode:\"equal_valuations_cancellation\",p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0084","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"zero_convention\",\"p\":5,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_convention, 5, 0, infinity, single_serializable_zero_convention)","explanation_zh":"Wave6第十二層核對=(zero_convention, 5, 0, infinity, single_serializable_zero_convention)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185084,"parameters":{"mode":"zero_convention","p":5,"a":0,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0085","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"nonprime_base\",\"p\":18,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonprime_base, 18, 10, 否, not_called_p_adic_valuation)","explanation_zh":"Wave6第十二層核對=(nonprime_base, 18, 10, 否, not_called_p_adic_valuation)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185085,"parameters":{"mode":"nonprime_base","p":18,"a":10,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:10,mode:\"nonprime_base\",p: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；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0086","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"negative_normalization\",\"p\":5,\"a\":-55}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(negative_normalization, 5, -55, 55, 1, 是, negative_integer_is_valid)","explanation_zh":"Wave6第十二層核對=(negative_normalization, 5, -55, 55, 1, 是, negative_integer_is_valid)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185086,"parameters":{"mode":"negative_normalization","p":5,"a":-55,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:-55,mode:\"negative_normalization\",p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0087","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"equal_valuations_cancellation\",\"p\":5,\"a\":60}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(equal_valuations_cancellation, 5, 60, -60, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)","explanation_zh":"Wave6第十二層核對=(equal_valuations_cancellation, 5, 60, -60, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185087,"parameters":{"mode":"equal_valuations_cancellation","p":5,"a":60,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:60,mode:\"equal_valuations_cancellation\",p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0088","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"zero_convention\",\"p\":7,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_convention, 7, 0, infinity, single_serializable_zero_convention)","explanation_zh":"Wave6第十二層核對=(zero_convention, 7, 0, infinity, single_serializable_zero_convention)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185088,"parameters":{"mode":"zero_convention","p":7,"a":0,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0089","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"nonprime_base\",\"p\":24,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonprime_base, 24, 14, 否, not_called_p_adic_valuation)","explanation_zh":"Wave6第十二層核對=(nonprime_base, 24, 14, 否, not_called_p_adic_valuation)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185089,"parameters":{"mode":"nonprime_base","p":24,"a":14,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:14,mode:\"nonprime_base\",p:24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0090","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"negative_normalization\",\"p\":7,\"a\":-735}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(negative_normalization, 7, -735, 735, 2, 是, negative_integer_is_valid)","explanation_zh":"Wave6第十二層核對=(negative_normalization, 7, -735, 735, 2, 是, negative_integer_is_valid)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185090,"parameters":{"mode":"negative_normalization","p":7,"a":-735,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:-735,mode:\"negative_normalization\",p:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0091","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"equal_valuations_cancellation\",\"p\":7,\"a\":112}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)","explanation_zh":"Wave6第十二層核對=(equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185091,"parameters":{"mode":"equal_valuations_cancellation","p":7,"a":112,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:112,mode:\"equal_valuations_cancellation\",p:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0092","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"zero_convention\",\"p\":11,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_convention, 11, 0, infinity, single_serializable_zero_convention)","explanation_zh":"Wave6第十二層核對=(zero_convention, 11, 0, infinity, single_serializable_zero_convention)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185092,"parameters":{"mode":"zero_convention","p":11,"a":0,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0093","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"nonprime_base\",\"p\":28,\"a\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonprime_base, 28, 18, 否, not_called_p_adic_valuation)","explanation_zh":"Wave6第十二層核對=(nonprime_base, 28, 18, 否, not_called_p_adic_valuation)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185093,"parameters":{"mode":"nonprime_base","p":28,"a":18,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:18,mode:\"nonprime_base\",p:28}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0094","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"negative_normalization\",\"p\":11,\"a\":-19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(negative_normalization, 11, -19, 19, 0, 是, negative_integer_is_valid)","explanation_zh":"Wave6第十二層核對=(negative_normalization, 11, -19, 19, 0, 是, negative_integer_is_valid)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185094,"parameters":{"mode":"negative_normalization","p":11,"a":-19,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:-19,mode:\"negative_normalization\",p:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0095","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"equal_valuations_cancellation\",\"p\":11,\"a\":220}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(equal_valuations_cancellation, 11, 220, -220, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)","explanation_zh":"Wave6第十二層核對=(equal_valuations_cancellation, 11, 220, -220, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185095,"parameters":{"mode":"equal_valuations_cancellation","p":11,"a":220,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:220,mode:\"equal_valuations_cancellation\",p:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0096","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"zero_convention\",\"p\":13,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_convention, 13, 0, infinity, single_serializable_zero_convention)","explanation_zh":"Wave6第十二層核對=(zero_convention, 13, 0, infinity, single_serializable_zero_convention)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185096,"parameters":{"mode":"zero_convention","p":13,"a":0,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0097","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"nonprime_base\",\"p\":34,\"a\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonprime_base, 34, 22, 否, not_called_p_adic_valuation)","explanation_zh":"Wave6第十二層核對=(nonprime_base, 34, 22, 否, not_called_p_adic_valuation)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185097,"parameters":{"mode":"nonprime_base","p":34,"a":22,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:22,mode:\"nonprime_base\",p:34}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0098","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"negative_normalization\",\"p\":13,\"a\":-299}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(negative_normalization, 13, -299, 299, 1, 是, negative_integer_is_valid)","explanation_zh":"Wave6第十二層核對=(negative_normalization, 13, -299, 299, 1, 是, negative_integer_is_valid)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185098,"parameters":{"mode":"negative_normalization","p":13,"a":-299,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:-299,mode:\"negative_normalization\",p:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0099","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"equal_valuations_cancellation\",\"p\":13,\"a\":312}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(equal_valuations_cancellation, 13, 312, -312, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)","explanation_zh":"Wave6第十二層核對=(equal_valuations_cancellation, 13, 312, -312, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185099,"parameters":{"mode":"equal_valuations_cancellation","p":13,"a":312,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:312,mode:\"equal_valuations_cancellation\",p:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0100","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","language":"zh-Hant","source_mko_ids":["mko-p-adic-valuation-integers"],"title_zh":"整數p進賦值","stem_zh":"prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={\"mode\":\"zero_convention\",\"p\":17,\"a\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_convention, 17, 0, infinity, single_serializable_zero_convention)","explanation_zh":"Wave6第十二層核對=(zero_convention, 17, 0, infinity, single_serializable_zero_convention)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["整數p進賦值"]},"variant":{"family_id":"qf-mod49-pval-boundary","seed":202611185100,"parameters":{"mode":"zero_convention","p":17,"a":0,"domain":"Exact_integer_p_adic_valuation","operation":"pval-boundary","semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}