{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0001","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[1,29],\"word\":[0,0,0,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 0, 0], 0, 4, 1/29, 614656/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 0, 0, 0], 0, 4, 1/29, 614656/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296001,"parameters":{"p":[1,29],"word":[0,0,0,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[1,29],\"word\":[0,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0002","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[2,29],\"word\":[1,0,0,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 0, 0], 1, 3, 2/29, 39366/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([1, 0, 0, 0], 1, 3, 2/29, 39366/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296002,"parameters":{"p":[2,29],"word":[1,0,0,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[2,29],\"word\":[1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0003","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[3,29],\"word\":[0,1,0,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 0], 1, 3, 3/29, 52728/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 1, 0, 0], 1, 3, 3/29, 52728/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296003,"parameters":{"p":[3,29],"word":[0,1,0,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[3,29],\"word\":[0,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0004","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[4,29],\"word\":[1,1,0,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0, 0], 2, 2, 4/29, 10000/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([1, 1, 0, 0], 2, 2, 4/29, 10000/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296004,"parameters":{"p":[4,29],"word":[1,1,0,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[4,29],\"word\":[1,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0005","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[5,29],\"word\":[0,0,1,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 1, 0], 1, 3, 5/29, 69120/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 0, 1, 0], 1, 3, 5/29, 69120/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296005,"parameters":{"p":[5,29],"word":[0,0,1,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[5,29],\"word\":[0,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0006","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[6,29],\"word\":[1,0,1,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0], 2, 2, 6/29, 19044/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([1, 0, 1, 0], 2, 2, 6/29, 19044/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296006,"parameters":{"p":[6,29],"word":[1,0,1,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[6,29],\"word\":[1,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0007","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[7,29],\"word\":[0,1,1,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 1, 0], 2, 2, 7/29, 23716/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 1, 1, 0], 2, 2, 7/29, 23716/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296007,"parameters":{"p":[7,29],"word":[0,1,1,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[7,29],\"word\":[0,1,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0008","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[8,29],\"word\":[1,1,1,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 0], 3, 1, 8/29, 10752/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([1, 1, 1, 0], 3, 1, 8/29, 10752/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296008,"parameters":{"p":[8,29],"word":[1,1,1,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[8,29],\"word\":[1,1,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0009","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[9,29],\"word\":[0,0,0,1]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 0, 1], 1, 3, 9/29, 72000/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 0, 0, 1], 1, 3, 9/29, 72000/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296009,"parameters":{"p":[9,29],"word":[0,0,0,1],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[9,29],\"word\":[0,0,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0010","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[10,29],\"word\":[1,0,0,1]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 0, 1], 2, 2, 10/29, 36100/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([1, 0, 0, 1], 2, 2, 10/29, 36100/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296010,"parameters":{"p":[10,29],"word":[1,0,0,1],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[10,29],\"word\":[1,0,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0011","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[11,29],\"word\":[0,1,0,1]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 1], 2, 2, 11/29, 39204/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 1, 0, 1], 2, 2, 11/29, 39204/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296011,"parameters":{"p":[11,29],"word":[0,1,0,1],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[11,29],\"word\":[0,1,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0012","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[12,29],\"word\":[1,1,0,1]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0, 1], 3, 1, 12/29, 29376/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([1, 1, 0, 1], 3, 1, 12/29, 29376/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296012,"parameters":{"p":[12,29],"word":[1,1,0,1],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[12,29],\"word\":[1,1,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0013","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[13,29],\"word\":[0,0,1,1]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 1, 1], 2, 2, 13/29, 43264/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 0, 1, 1], 2, 2, 13/29, 43264/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296013,"parameters":{"p":[13,29],"word":[0,0,1,1],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[13,29],\"word\":[0,0,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0014","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[14,29],\"word\":[1,0,1,1]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 1], 3, 1, 14/29, 41160/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([1, 0, 1, 1], 3, 1, 14/29, 41160/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296014,"parameters":{"p":[14,29],"word":[1,0,1,1],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[14,29],\"word\":[1,0,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0015","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[15,29],\"word\":[0,1,1,1]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 1, 1], 3, 1, 15/29, 47250/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 1, 1, 1], 3, 1, 15/29, 47250/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296015,"parameters":{"p":[15,29],"word":[0,1,1,1],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[15,29],\"word\":[0,1,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0016","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[16,29],\"word\":[1,1,1,1]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 1], 4, 0, 16/29, 65536/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([1, 1, 1, 1], 4, 0, 16/29, 65536/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296016,"parameters":{"p":[16,29],"word":[1,1,1,1],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[16,29],\"word\":[1,1,1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0017","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[17,29],\"word\":[0,0,0,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 0, 0], 0, 4, 17/29, 20736/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 0, 0, 0], 0, 4, 17/29, 20736/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296017,"parameters":{"p":[17,29],"word":[0,0,0,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[17,29],\"word\":[0,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0018","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[18,29],\"word\":[1,0,0,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 0, 0], 1, 3, 18/29, 23958/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([1, 0, 0, 0], 1, 3, 18/29, 23958/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296018,"parameters":{"p":[18,29],"word":[1,0,0,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[18,29],\"word\":[1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0019","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[19,29],\"word\":[0,1,0,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 0], 1, 3, 19/29, 19000/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 1, 0, 0], 1, 3, 19/29, 19000/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296019,"parameters":{"p":[19,29],"word":[0,1,0,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[19,29],\"word\":[0,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0020","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[20,29],\"word\":[1,1,0,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0, 0], 2, 2, 20/29, 32400/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([1, 1, 0, 0], 2, 2, 20/29, 32400/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296020,"parameters":{"p":[20,29],"word":[1,1,0,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[20,29],\"word\":[1,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0021","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[21,29],\"word\":[0,0,1,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 1, 0], 1, 3, 21/29, 10752/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 0, 1, 0], 1, 3, 21/29, 10752/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296021,"parameters":{"p":[21,29],"word":[0,0,1,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[21,29],\"word\":[0,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0022","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[22,29],\"word\":[1,0,1,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0], 2, 2, 22/29, 23716/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([1, 0, 1, 0], 2, 2, 22/29, 23716/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296022,"parameters":{"p":[22,29],"word":[1,0,1,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[22,29],\"word\":[1,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0023","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[23,29],\"word\":[0,1,1,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 1, 0], 2, 2, 23/29, 19044/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 1, 1, 0], 2, 2, 23/29, 19044/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296023,"parameters":{"p":[23,29],"word":[0,1,1,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[23,29],\"word\":[0,1,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0024","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[24,29],\"word\":[1,1,1,0]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 0], 3, 1, 24/29, 69120/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([1, 1, 1, 0], 3, 1, 24/29, 69120/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296024,"parameters":{"p":[24,29],"word":[1,1,1,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[24,29],\"word\":[1,1,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0025","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-bernoulli-cylinder；資料={\"p\":[25,29],\"word\":[0,0,0,1]}。請依 [word,ones,zeros,p,probability,certificate] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 0, 1], 1, 3, 25/29, 1600/707281, symbolic_product_law)","explanation_zh":"Wave4第三層核對=([0, 0, 0, 1], 1, 3, 25/29, 1600/707281, symbolic_product_law)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-bernoulli-cylinder","seed":202610296025,"parameters":{"p":[25,29],"word":[0,0,0,1],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-bernoulli-cylinder","semantic_template":"qf-mod29-iid-bernoulli-cylinder/{\"p\":[25,29],\"word\":[0,0,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0026","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[1,29],\"left_index\":0,\"right_index\":17}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, 17, [0, 28/29], [1, 1/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(0, 17, [0, 28/29], [1, 1/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296026,"parameters":{"p":[1,29],"left_index":0,"right_index":17,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":0,\"p\":[1,29],\"right_index\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0027","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[2,29],\"left_index\":1,\"right_index\":18}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, 18, [0, 27/29], [1, 2/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(1, 18, [0, 27/29], [1, 2/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296027,"parameters":{"p":[2,29],"left_index":1,"right_index":18,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":1,\"p\":[2,29],\"right_index\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0028","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[3,29],\"left_index\":2,\"right_index\":19}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, 19, [0, 26/29], [1, 3/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(2, 19, [0, 26/29], [1, 3/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296028,"parameters":{"p":[3,29],"left_index":2,"right_index":19,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":2,\"p\":[3,29],\"right_index\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0029","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[4,29],\"left_index\":3,\"right_index\":20}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, 20, [0, 25/29], [1, 4/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(3, 20, [0, 25/29], [1, 4/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296029,"parameters":{"p":[4,29],"left_index":3,"right_index":20,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":3,\"p\":[4,29],\"right_index\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0030","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[5,29],\"left_index\":4,\"right_index\":21}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, 21, [0, 24/29], [1, 5/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(4, 21, [0, 24/29], [1, 5/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296030,"parameters":{"p":[5,29],"left_index":4,"right_index":21,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":4,\"p\":[5,29],\"right_index\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0031","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[6,29],\"left_index\":5,\"right_index\":22}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, 22, [0, 23/29], [1, 6/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(5, 22, [0, 23/29], [1, 6/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296031,"parameters":{"p":[6,29],"left_index":5,"right_index":22,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":5,\"p\":[6,29],\"right_index\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0032","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[7,29],\"left_index\":6,\"right_index\":23}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(6, 23, [0, 22/29], [1, 7/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(6, 23, [0, 22/29], [1, 7/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296032,"parameters":{"p":[7,29],"left_index":6,"right_index":23,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":6,\"p\":[7,29],\"right_index\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0033","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[8,29],\"left_index\":7,\"right_index\":24}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(7, 24, [0, 21/29], [1, 8/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(7, 24, [0, 21/29], [1, 8/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296033,"parameters":{"p":[8,29],"left_index":7,"right_index":24,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":7,\"p\":[8,29],\"right_index\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0034","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[9,29],\"left_index\":8,\"right_index\":25}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(8, 25, [0, 20/29], [1, 9/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(8, 25, [0, 20/29], [1, 9/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296034,"parameters":{"p":[9,29],"left_index":8,"right_index":25,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":8,\"p\":[9,29],\"right_index\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0035","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[10,29],\"left_index\":9,\"right_index\":26}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(9, 26, [0, 19/29], [1, 10/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(9, 26, [0, 19/29], [1, 10/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296035,"parameters":{"p":[10,29],"left_index":9,"right_index":26,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":9,\"p\":[10,29],\"right_index\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0036","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[11,29],\"left_index\":10,\"right_index\":27}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(10, 27, [0, 18/29], [1, 11/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(10, 27, [0, 18/29], [1, 11/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296036,"parameters":{"p":[11,29],"left_index":10,"right_index":27,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":10,\"p\":[11,29],\"right_index\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0037","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[12,29],\"left_index\":11,\"right_index\":28}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(11, 28, [0, 17/29], [1, 12/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(11, 28, [0, 17/29], [1, 12/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296037,"parameters":{"p":[12,29],"left_index":11,"right_index":28,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":11,\"p\":[12,29],\"right_index\":28}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0038","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[13,29],\"left_index\":12,\"right_index\":29}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(12, 29, [0, 16/29], [1, 13/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(12, 29, [0, 16/29], [1, 13/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296038,"parameters":{"p":[13,29],"left_index":12,"right_index":29,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":12,\"p\":[13,29],\"right_index\":29}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0039","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[14,29],\"left_index\":13,\"right_index\":30}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(13, 30, [0, 15/29], [1, 14/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(13, 30, [0, 15/29], [1, 14/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296039,"parameters":{"p":[14,29],"left_index":13,"right_index":30,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":13,\"p\":[14,29],\"right_index\":30}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0040","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[15,29],\"left_index\":14,\"right_index\":31}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(14, 31, [0, 14/29], [1, 15/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(14, 31, [0, 14/29], [1, 15/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296040,"parameters":{"p":[15,29],"left_index":14,"right_index":31,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":14,\"p\":[15,29],\"right_index\":31}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0041","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[16,29],\"left_index\":15,\"right_index\":32}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(15, 32, [0, 13/29], [1, 16/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(15, 32, [0, 13/29], [1, 16/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296041,"parameters":{"p":[16,29],"left_index":15,"right_index":32,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":15,\"p\":[16,29],\"right_index\":32}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0042","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[17,29],\"left_index\":16,\"right_index\":33}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(16, 33, [0, 12/29], [1, 17/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(16, 33, [0, 12/29], [1, 17/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296042,"parameters":{"p":[17,29],"left_index":16,"right_index":33,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":16,\"p\":[17,29],\"right_index\":33}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0043","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[18,29],\"left_index\":17,\"right_index\":34}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(17, 34, [0, 11/29], [1, 18/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(17, 34, [0, 11/29], [1, 18/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296043,"parameters":{"p":[18,29],"left_index":17,"right_index":34,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":17,\"p\":[18,29],\"right_index\":34}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0044","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[19,29],\"left_index\":18,\"right_index\":35}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(18, 35, [0, 10/29], [1, 19/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(18, 35, [0, 10/29], [1, 19/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296044,"parameters":{"p":[19,29],"left_index":18,"right_index":35,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":18,\"p\":[19,29],\"right_index\":35}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0045","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[20,29],\"left_index\":19,\"right_index\":36}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(19, 36, [0, 9/29], [1, 20/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(19, 36, [0, 9/29], [1, 20/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296045,"parameters":{"p":[20,29],"left_index":19,"right_index":36,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":19,\"p\":[20,29],\"right_index\":36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0046","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[21,29],\"left_index\":20,\"right_index\":37}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(20, 37, [0, 8/29], [1, 21/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(20, 37, [0, 8/29], [1, 21/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296046,"parameters":{"p":[21,29],"left_index":20,"right_index":37,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":20,\"p\":[21,29],\"right_index\":37}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0047","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[22,29],\"left_index\":21,\"right_index\":38}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(21, 38, [0, 7/29], [1, 22/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(21, 38, [0, 7/29], [1, 22/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296047,"parameters":{"p":[22,29],"left_index":21,"right_index":38,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":21,\"p\":[22,29],\"right_index\":38}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0048","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[23,29],\"left_index\":22,\"right_index\":39}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(22, 39, [0, 6/29], [1, 23/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(22, 39, [0, 6/29], [1, 23/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296048,"parameters":{"p":[23,29],"left_index":22,"right_index":39,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":22,\"p\":[23,29],\"right_index\":39}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0049","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[24,29],\"left_index\":23,\"right_index\":40}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(23, 40, [0, 5/29], [1, 24/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(23, 40, [0, 5/29], [1, 24/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296049,"parameters":{"p":[24,29],"left_index":23,"right_index":40,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":23,\"p\":[24,29],\"right_index\":40}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0050","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[25,29],\"left_index\":24,\"right_index\":41}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。","answer":{"kind":"expression","value":"(24, 41, [0, 4/29], [1, 25/29], 是, full_Bernoulli_marginal_not_mean_only)","explanation_zh":"Wave4第三層核對=(24, 41, [0, 4/29], [1, 25/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-identical-marginal","seed":202610296050,"parameters":{"p":[25,29],"left_index":24,"right_index":41,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-identical-marginal","semantic_template":"qf-mod29-iid-identical-marginal/{\"left_index\":24,\"p\":[25,29],\"right_index\":41}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0051","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-finite-factorization；資料={\"p\":[1,29],\"word\":[0,0,0,0]}。請依 [word,atom_factors,product,pass] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 0, 0], [28/29, 28/29, 28/29, 28/29], 614656/707281, 是)","explanation_zh":"Wave4第三層核對=([0, 0, 0, 0], [28/29, 28/29, 28/29, 28/29], 614656/707281, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-finite-factorization","seed":202610296051,"parameters":{"p":[1,29],"word":[0,0,0,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-finite-factorization","semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[1,29],\"word\":[0,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0052","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-finite-factorization；資料={\"p\":[2,29],\"word\":[1,0,0,0]}。請依 [word,atom_factors,product,pass] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 0, 0], [2/29, 27/29, 27/29, 27/29], 39366/707281, 是)","explanation_zh":"Wave4第三層核對=([1, 0, 0, 0], [2/29, 27/29, 27/29, 27/29], 39366/707281, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-finite-factorization","seed":202610296052,"parameters":{"p":[2,29],"word":[1,0,0,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-finite-factorization","semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[2,29],\"word\":[1,0,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0062","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-finite-factorization；資料={\"p\":[12,29],\"word\":[1,1,0,1]}。請依 [word,atom_factors,product,pass] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0, 1], [12/29, 12/29, 17/29, 12/29], 29376/707281, 是)","explanation_zh":"Wave4第三層核對=([1, 1, 0, 1], [12/29, 12/29, 17/29, 12/29], 29376/707281, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-finite-factorization","seed":202610296062,"parameters":{"p":[12,29],"word":[1,1,0,1],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-finite-factorization","semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[12,29],\"word\":[1,1,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0069","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-finite-factorization；資料={\"p\":[19,29],\"word\":[0,1,0,0]}。請依 [word,atom_factors,product,pass] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 0], [10/29, 19/29, 10/29, 10/29], 19000/707281, 是)","explanation_zh":"Wave4第三層核對=([0, 1, 0, 0], [10/29, 19/29, 10/29, 10/29], 19000/707281, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-finite-factorization","seed":202610296069,"parameters":{"p":[19,29],"word":[0,1,0,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-finite-factorization","semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[19,29],\"word\":[0,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0070","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-finite-factorization；資料={\"p\":[20,29],\"word\":[1,1,0,0]}。請依 [word,atom_factors,product,pass] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 0, 0], [20/29, 20/29, 9/29, 9/29], 32400/707281, 是)","explanation_zh":"Wave4第三層核對=([1, 1, 0, 0], [20/29, 20/29, 9/29, 9/29], 32400/707281, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-finite-factorization","seed":202610296070,"parameters":{"p":[20,29],"word":[1,1,0,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-finite-factorization","semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[20,29],\"word\":[1,1,0,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0071","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-finite-factorization；資料={\"p\":[21,29],\"word\":[0,0,1,0]}。請依 [word,atom_factors,product,pass] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 1, 0], [8/29, 8/29, 21/29, 8/29], 10752/707281, 是)","explanation_zh":"Wave4第三層核對=([0, 0, 1, 0], [8/29, 8/29, 21/29, 8/29], 10752/707281, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-finite-factorization","seed":202610296071,"parameters":{"p":[21,29],"word":[0,0,1,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-finite-factorization","semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[21,29],\"word\":[0,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0072","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-finite-factorization；資料={\"p\":[22,29],\"word\":[1,0,1,0]}。請依 [word,atom_factors,product,pass] 欄序回答。","answer":{"kind":"expression","value":"([1, 0, 1, 0], [22/29, 7/29, 22/29, 7/29], 23716/707281, 是)","explanation_zh":"Wave4第三層核對=([1, 0, 1, 0], [22/29, 7/29, 22/29, 7/29], 23716/707281, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-finite-factorization","seed":202610296072,"parameters":{"p":[22,29],"word":[1,0,1,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-finite-factorization","semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[22,29],\"word\":[1,0,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0073","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-finite-factorization；資料={\"p\":[23,29],\"word\":[0,1,1,0]}。請依 [word,atom_factors,product,pass] 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 1, 0], [6/29, 23/29, 23/29, 6/29], 19044/707281, 是)","explanation_zh":"Wave4第三層核對=([0, 1, 1, 0], [6/29, 23/29, 23/29, 6/29], 19044/707281, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-finite-factorization","seed":202610296073,"parameters":{"p":[23,29],"word":[0,1,1,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-finite-factorization","semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[23,29],\"word\":[0,1,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0074","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-finite-factorization；資料={\"p\":[24,29],\"word\":[1,1,1,0]}。請依 [word,atom_factors,product,pass] 欄序回答。","answer":{"kind":"expression","value":"([1, 1, 1, 0], [24/29, 24/29, 24/29, 5/29], 69120/707281, 是)","explanation_zh":"Wave4第三層核對=([1, 1, 1, 0], [24/29, 24/29, 24/29, 5/29], 69120/707281, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-finite-factorization","seed":202610296074,"parameters":{"p":[24,29],"word":[1,1,1,0],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-finite-factorization","semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[24,29],\"word\":[1,1,1,0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0075","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-finite-factorization；資料={\"p\":[25,29],\"word\":[0,0,0,1]}。請依 [word,atom_factors,product,pass] 欄序回答。","answer":{"kind":"expression","value":"([0, 0, 0, 1], [4/29, 4/29, 4/29, 25/29], 1600/707281, 是)","explanation_zh":"Wave4第三層核對=([0, 0, 0, 1], [4/29, 4/29, 4/29, 25/29], 1600/707281, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-finite-factorization","seed":202610296075,"parameters":{"p":[25,29],"word":[0,0,0,1],"domain":"Symbolic_Bernoulli_product_law","operation":"iid-finite-factorization","semantic_template":"qf-mod29-iid-finite-factorization/{\"p\":[25,29],\"word\":[0,0,0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0076","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[1,29],\"evidence_mode\":\"finite_prefix_only\",\"checked_prefix\":1}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/29, finite_prefix_only, 1, 否, finite_prefix_only_not_infinite_IID)","explanation_zh":"Wave4第三層核對=(1/29, finite_prefix_only, 1, 否, finite_prefix_only_not_infinite_IID)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296076,"parameters":{"p":[1,29],"evidence_mode":"finite_prefix_only","checked_prefix":1,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":1,\"evidence_mode\":\"finite_prefix_only\",\"p\":[1,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0077","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[2,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":2}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/29, symbolic_product_law, 2, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(2/29, symbolic_product_law, 2, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296077,"parameters":{"p":[2,29],"evidence_mode":"symbolic_product_law","checked_prefix":2,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":2,\"evidence_mode\":\"symbolic_product_law\",\"p\":[2,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0078","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[3,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":3}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3/29, symbolic_product_law, 3, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(3/29, symbolic_product_law, 3, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296078,"parameters":{"p":[3,29],"evidence_mode":"symbolic_product_law","checked_prefix":3,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":3,\"evidence_mode\":\"symbolic_product_law\",\"p\":[3,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0079","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[4,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":4}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4/29, symbolic_product_law, 4, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(4/29, symbolic_product_law, 4, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296079,"parameters":{"p":[4,29],"evidence_mode":"symbolic_product_law","checked_prefix":4,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":4,\"evidence_mode\":\"symbolic_product_law\",\"p\":[4,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0080","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[5,29],\"evidence_mode\":\"finite_prefix_only\",\"checked_prefix\":5}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5/29, finite_prefix_only, 5, 否, finite_prefix_only_not_infinite_IID)","explanation_zh":"Wave4第三層核對=(5/29, finite_prefix_only, 5, 否, finite_prefix_only_not_infinite_IID)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296080,"parameters":{"p":[5,29],"evidence_mode":"finite_prefix_only","checked_prefix":5,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":5,\"evidence_mode\":\"finite_prefix_only\",\"p\":[5,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0081","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[6,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":6}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(6/29, symbolic_product_law, 6, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(6/29, symbolic_product_law, 6, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296081,"parameters":{"p":[6,29],"evidence_mode":"symbolic_product_law","checked_prefix":6,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":6,\"evidence_mode\":\"symbolic_product_law\",\"p\":[6,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0082","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[7,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":7}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(7/29, symbolic_product_law, 7, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(7/29, symbolic_product_law, 7, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296082,"parameters":{"p":[7,29],"evidence_mode":"symbolic_product_law","checked_prefix":7,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":7,\"evidence_mode\":\"symbolic_product_law\",\"p\":[7,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0083","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[8,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":8}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(8/29, symbolic_product_law, 8, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(8/29, symbolic_product_law, 8, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296083,"parameters":{"p":[8,29],"evidence_mode":"symbolic_product_law","checked_prefix":8,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":8,\"evidence_mode\":\"symbolic_product_law\",\"p\":[8,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0084","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[9,29],\"evidence_mode\":\"finite_prefix_only\",\"checked_prefix\":9}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(9/29, finite_prefix_only, 9, 否, finite_prefix_only_not_infinite_IID)","explanation_zh":"Wave4第三層核對=(9/29, finite_prefix_only, 9, 否, finite_prefix_only_not_infinite_IID)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296084,"parameters":{"p":[9,29],"evidence_mode":"finite_prefix_only","checked_prefix":9,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":9,\"evidence_mode\":\"finite_prefix_only\",\"p\":[9,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0085","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[10,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":10}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(10/29, symbolic_product_law, 10, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(10/29, symbolic_product_law, 10, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296085,"parameters":{"p":[10,29],"evidence_mode":"symbolic_product_law","checked_prefix":10,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":10,\"evidence_mode\":\"symbolic_product_law\",\"p\":[10,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0086","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[11,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":11}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(11/29, symbolic_product_law, 11, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(11/29, symbolic_product_law, 11, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296086,"parameters":{"p":[11,29],"evidence_mode":"symbolic_product_law","checked_prefix":11,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":11,\"evidence_mode\":\"symbolic_product_law\",\"p\":[11,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0087","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[12,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":12}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(12/29, symbolic_product_law, 12, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(12/29, symbolic_product_law, 12, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296087,"parameters":{"p":[12,29],"evidence_mode":"symbolic_product_law","checked_prefix":12,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":12,\"evidence_mode\":\"symbolic_product_law\",\"p\":[12,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0088","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[13,29],\"evidence_mode\":\"finite_prefix_only\",\"checked_prefix\":13}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(13/29, finite_prefix_only, 13, 否, finite_prefix_only_not_infinite_IID)","explanation_zh":"Wave4第三層核對=(13/29, finite_prefix_only, 13, 否, finite_prefix_only_not_infinite_IID)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296088,"parameters":{"p":[13,29],"evidence_mode":"finite_prefix_only","checked_prefix":13,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":13,\"evidence_mode\":\"finite_prefix_only\",\"p\":[13,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0089","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[14,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":14}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(14/29, symbolic_product_law, 14, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(14/29, symbolic_product_law, 14, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296089,"parameters":{"p":[14,29],"evidence_mode":"symbolic_product_law","checked_prefix":14,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":14,\"evidence_mode\":\"symbolic_product_law\",\"p\":[14,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0090","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[15,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":15}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(15/29, symbolic_product_law, 15, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(15/29, symbolic_product_law, 15, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296090,"parameters":{"p":[15,29],"evidence_mode":"symbolic_product_law","checked_prefix":15,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":15,\"evidence_mode\":\"symbolic_product_law\",\"p\":[15,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0091","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[16,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":16}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(16/29, symbolic_product_law, 16, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(16/29, symbolic_product_law, 16, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296091,"parameters":{"p":[16,29],"evidence_mode":"symbolic_product_law","checked_prefix":16,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":16,\"evidence_mode\":\"symbolic_product_law\",\"p\":[16,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0092","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[17,29],\"evidence_mode\":\"finite_prefix_only\",\"checked_prefix\":17}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(17/29, finite_prefix_only, 17, 否, finite_prefix_only_not_infinite_IID)","explanation_zh":"Wave4第三層核對=(17/29, finite_prefix_only, 17, 否, finite_prefix_only_not_infinite_IID)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296092,"parameters":{"p":[17,29],"evidence_mode":"finite_prefix_only","checked_prefix":17,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":17,\"evidence_mode\":\"finite_prefix_only\",\"p\":[17,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0093","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[18,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":18}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(18/29, symbolic_product_law, 18, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(18/29, symbolic_product_law, 18, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296093,"parameters":{"p":[18,29],"evidence_mode":"symbolic_product_law","checked_prefix":18,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":18,\"evidence_mode\":\"symbolic_product_law\",\"p\":[18,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0094","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[19,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":19}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(19/29, symbolic_product_law, 19, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(19/29, symbolic_product_law, 19, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296094,"parameters":{"p":[19,29],"evidence_mode":"symbolic_product_law","checked_prefix":19,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":19,\"evidence_mode\":\"symbolic_product_law\",\"p\":[19,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0095","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[20,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":20}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(20/29, symbolic_product_law, 20, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(20/29, symbolic_product_law, 20, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296095,"parameters":{"p":[20,29],"evidence_mode":"symbolic_product_law","checked_prefix":20,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":20,\"evidence_mode\":\"symbolic_product_law\",\"p\":[20,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0096","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[21,29],\"evidence_mode\":\"finite_prefix_only\",\"checked_prefix\":21}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(21/29, finite_prefix_only, 21, 否, finite_prefix_only_not_infinite_IID)","explanation_zh":"Wave4第三層核對=(21/29, finite_prefix_only, 21, 否, finite_prefix_only_not_infinite_IID)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296096,"parameters":{"p":[21,29],"evidence_mode":"finite_prefix_only","checked_prefix":21,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":21,\"evidence_mode\":\"finite_prefix_only\",\"p\":[21,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0097","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[22,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":22}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(22/29, symbolic_product_law, 22, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(22/29, symbolic_product_law, 22, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296097,"parameters":{"p":[22,29],"evidence_mode":"symbolic_product_law","checked_prefix":22,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":22,\"evidence_mode\":\"symbolic_product_law\",\"p\":[22,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0098","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[23,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":23}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(23/29, symbolic_product_law, 23, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(23/29, symbolic_product_law, 23, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296098,"parameters":{"p":[23,29],"evidence_mode":"symbolic_product_law","checked_prefix":23,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":23,\"evidence_mode\":\"symbolic_product_law\",\"p\":[23,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0099","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[24,29],\"evidence_mode\":\"symbolic_product_law\",\"checked_prefix\":24}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(24/29, symbolic_product_law, 24, 是, infinite_IID_certificate)","explanation_zh":"Wave4第三層核對=(24/29, symbolic_product_law, 24, 是, infinite_IID_certificate)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296099,"parameters":{"p":[24,29],"evidence_mode":"symbolic_product_law","checked_prefix":24,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":24,\"evidence_mode\":\"symbolic_product_law\",\"p\":[24,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261029-r06-iid-sequence-0100","batch_id":"batch-20261014-advance-20261029-r06-iid-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-iid-sequence"],"title_zh":"獨立同分布序列","stem_zh":"IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-evidence-mode；資料={\"p\":[25,29],\"evidence_mode\":\"finite_prefix_only\",\"checked_prefix\":25}。請依 [p,evidence_mode,checked_prefix,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(25/29, finite_prefix_only, 25, 否, finite_prefix_only_not_infinite_IID)","explanation_zh":"Wave4第三層核對=(25/29, finite_prefix_only, 25, 否, finite_prefix_only_not_infinite_IID)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["獨立同分布序列"]},"variant":{"family_id":"qf-mod29-iid-evidence-mode","seed":202610296100,"parameters":{"p":[25,29],"evidence_mode":"finite_prefix_only","checked_prefix":25,"domain":"Symbolic_Bernoulli_product_law","operation":"iid-evidence-mode","semantic_template":"qf-mod29-iid-evidence-mode/{\"checked_prefix\":25,\"evidence_mode\":\"finite_prefix_only\",\"p\":[25,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}