{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0001","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[-2,-1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-2, 1/2], [-1, 1/2]], -3/2, 1/4, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[-2, 1/2], [-1, 1/2]], -3/2, 1/4, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181001,"parameters":{"values":[-2,-1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[-1,[1,2]],[-2,[1,2]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0002","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[-1,0],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-1, 1/2], [0, 1/2]], -1/2, 1/4, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[-1, 1/2], [0, 1/2]], -1/2, 1/4, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181002,"parameters":{"values":[-1,0],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[-1,[1,2]],[0,[1,2]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0003","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[0,1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/2], [1, 1/2]], 1/2, 1/4, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[0, 1/2], [1, 1/2]], 1/2, 1/4, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181003,"parameters":{"values":[0,1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[0,[1,2]],[1,[1,2]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0004","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[1,2],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1/2], [2, 1/2]], 3/2, 1/4, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[1, 1/2], [2, 1/2]], 3/2, 1/4, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181004,"parameters":{"values":[1,2],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[1,[1,2]],[2,[1,2]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0005","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[2,3],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[2, 1/2], [3, 1/2]], 5/2, 1/4, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[2, 1/2], [3, 1/2]], 5/2, 1/4, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181005,"parameters":{"values":[2,3],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[2,[1,2]],[3,[1,2]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0006","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[-2,-1],\"probabilities\":[[1,3],[2,3]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-2, 1/3], [-1, 2/3]], -4/3, 2/9, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[-2, 1/3], [-1, 2/3]], -4/3, 2/9, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181006,"parameters":{"values":[-2,-1],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[-1,[2,3]],[-2,[1,3]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0007","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[-1,0],\"probabilities\":[[1,3],[2,3]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-1, 1/3], [0, 2/3]], -1/3, 2/9, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[-1, 1/3], [0, 2/3]], -1/3, 2/9, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181007,"parameters":{"values":[-1,0],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[-1,[1,3]],[0,[2,3]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0008","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[0,1],\"probabilities\":[[1,3],[2,3]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/3], [1, 2/3]], 2/3, 2/9, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[0, 1/3], [1, 2/3]], 2/3, 2/9, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181008,"parameters":{"values":[0,1],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[0,[1,3]],[1,[2,3]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0009","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[1,2],\"probabilities\":[[1,3],[2,3]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1/3], [2, 2/3]], 5/3, 2/9, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[1, 1/3], [2, 2/3]], 5/3, 2/9, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181009,"parameters":{"values":[1,2],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[1,[1,3]],[2,[2,3]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0010","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[2,3],\"probabilities\":[[1,3],[2,3]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[2, 1/3], [3, 2/3]], 8/3, 2/9, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[2, 1/3], [3, 2/3]], 8/3, 2/9, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181010,"parameters":{"values":[2,3],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[2,[1,3]],[3,[2,3]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0011","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[-2,-1],\"probabilities\":[[1,4],[3,4]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-2, 1/4], [-1, 3/4]], -5/4, 3/16, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[-2, 1/4], [-1, 3/4]], -5/4, 3/16, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181011,"parameters":{"values":[-2,-1],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[-1,[3,4]],[-2,[1,4]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0012","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[-1,0],\"probabilities\":[[1,4],[3,4]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-1, 1/4], [0, 3/4]], -1/4, 3/16, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[-1, 1/4], [0, 3/4]], -1/4, 3/16, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181012,"parameters":{"values":[-1,0],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[-1,[1,4]],[0,[3,4]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0013","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[0,1],\"probabilities\":[[1,4],[3,4]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/4], [1, 3/4]], 3/4, 3/16, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[0, 1/4], [1, 3/4]], 3/4, 3/16, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181013,"parameters":{"values":[0,1],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[0,[1,4]],[1,[3,4]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0014","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[1,2],\"probabilities\":[[1,4],[3,4]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1/4], [2, 3/4]], 7/4, 3/16, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[1, 1/4], [2, 3/4]], 7/4, 3/16, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181014,"parameters":{"values":[1,2],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[1,[1,4]],[2,[3,4]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0015","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[2,3],\"probabilities\":[[1,4],[3,4]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[2, 1/4], [3, 3/4]], 11/4, 3/16, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[2, 1/4], [3, 3/4]], 11/4, 3/16, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181015,"parameters":{"values":[2,3],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[2,[1,4]],[3,[3,4]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0016","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[-2,-1],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-2, 2/5], [-1, 3/5]], -7/5, 6/25, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[-2, 2/5], [-1, 3/5]], -7/5, 6/25, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181016,"parameters":{"values":[-2,-1],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[-1,[3,5]],[-2,[2,5]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0017","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[-1,0],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-1, 2/5], [0, 3/5]], -2/5, 6/25, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[-1, 2/5], [0, 3/5]], -2/5, 6/25, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181017,"parameters":{"values":[-1,0],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[-1,[2,5]],[0,[3,5]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0018","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[0,1],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 2/5], [1, 3/5]], 3/5, 6/25, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[0, 2/5], [1, 3/5]], 3/5, 6/25, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181018,"parameters":{"values":[0,1],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[0,[2,5]],[1,[3,5]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0019","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[1,2],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 2/5], [2, 3/5]], 8/5, 6/25, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[1, 2/5], [2, 3/5]], 8/5, 6/25, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181019,"parameters":{"values":[1,2],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[1,[2,5]],[2,[3,5]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0020","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[2,3],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[2, 2/5], [3, 3/5]], 13/5, 6/25, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[2, 2/5], [3, 3/5]], 13/5, 6/25, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181020,"parameters":{"values":[2,3],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[2,[2,5]],[3,[3,5]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0021","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[-2,-1],\"probabilities\":[[1,5],[4,5]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-2, 1/5], [-1, 4/5]], -6/5, 4/25, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[-2, 1/5], [-1, 4/5]], -6/5, 4/25, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181021,"parameters":{"values":[-2,-1],"probabilities":[[1,5],[4,5]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[-1,[4,5]],[-2,[1,5]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0022","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[-1,0],\"probabilities\":[[1,5],[4,5]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-1, 1/5], [0, 4/5]], -1/5, 4/25, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[-1, 1/5], [0, 4/5]], -1/5, 4/25, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181022,"parameters":{"values":[-1,0],"probabilities":[[1,5],[4,5]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[-1,[1,5]],[0,[4,5]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0023","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[0,1],\"probabilities\":[[1,5],[4,5]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/5], [1, 4/5]], 4/5, 4/25, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[0, 1/5], [1, 4/5]], 4/5, 4/25, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181023,"parameters":{"values":[0,1],"probabilities":[[1,5],[4,5]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[0,[1,5]],[1,[4,5]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0024","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[1,2],\"probabilities\":[[1,5],[4,5]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 1/5], [2, 4/5]], 9/5, 4/25, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[1, 1/5], [2, 4/5]], 9/5, 4/25, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181024,"parameters":{"values":[1,2],"probabilities":[[1,5],[4,5]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[1,[1,5]],[2,[4,5]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0025","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-finite-law-moments；資料={\"values\":[2,3],\"probabilities\":[[1,5],[4,5]],\"process_certificate\":\"infinite_iid\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[2, 1/5], [3, 4/5]], 14/5, 4/25, 是, infinite_iid, CLT_hypotheses_qualified)","explanation_zh":"Wave6第十二層核對=([[2, 1/5], [3, 4/5]], 14/5, 4/25, 是, infinite_iid, CLT_hypotheses_qualified)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-finite-law-moments","seed":202611181025,"parameters":{"values":[2,3],"probabilities":[[1,5],[4,5]],"process_certificate":"infinite_iid","domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-finite-law-moments","semantic_template":"clt-finite-law-moments|{law:[[2,[1,5]],[3,[4,5]]],process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0026","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[-2,-1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[-2, 1/2], [-1, 1/2]], 1, -3/2, 1/4, -3/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(1, [[-2, 1/2], [-1, 1/2]], 1, -3/2, 1/4, -3/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181026,"parameters":{"values":[-2,-1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":1,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[-2,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0027","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[-1,0],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[-1, 1/2], [0, 1/2]], 1, -1/2, 1/4, -1/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(1, [[-1, 1/2], [0, 1/2]], 1, -1/2, 1/4, -1/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181027,"parameters":{"values":[-1,0],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":1,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[0,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0028","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[0,1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[0, 1/2], [1, 1/2]], 1, 1/2, 1/4, 1/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(1, [[0, 1/2], [1, 1/2]], 1, 1/2, 1/4, 1/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181028,"parameters":{"values":[0,1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":1,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[0,[1,2]],[1,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0029","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[1,2],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[1, 1/2], [2, 1/2]], 1, 3/2, 1/4, 3/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(1, [[1, 1/2], [2, 1/2]], 1, 3/2, 1/4, 3/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181029,"parameters":{"values":[1,2],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":1,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[1,[1,2]],[2,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0030","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[2,3],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[2, 1/2], [3, 1/2]], 1, 5/2, 1/4, 5/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(1, [[2, 1/2], [3, 1/2]], 1, 5/2, 1/4, 5/2, 1/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181030,"parameters":{"values":[2,3],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":1,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[2,[1,2]],[3,[1,2]]],n:1,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0031","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[-2,-1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1, -3, 1/2, -3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(2, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1, -3, 1/2, -3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181031,"parameters":{"values":[-2,-1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":2,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[-2,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0032","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[-1,0],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[-2, 1/4], [-1, 1/2], [0, 1/4]], 1, -1, 1/2, -1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(2, [[-2, 1/4], [-1, 1/2], [0, 1/4]], 1, -1, 1/2, -1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181032,"parameters":{"values":[-1,0],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":2,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[0,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0033","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[0,1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 1/2, 1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 1/2, 1, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181033,"parameters":{"values":[0,1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":2,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[0,[1,2]],[1,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0034","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[1,2],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[2, 1/4], [3, 1/2], [4, 1/4]], 1, 3, 1/2, 3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(2, [[2, 1/4], [3, 1/2], [4, 1/4]], 1, 3, 1/2, 3, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181034,"parameters":{"values":[1,2],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":2,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[1,[1,2]],[2,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0035","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[2,3],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[4, 1/4], [5, 1/2], [6, 1/4]], 1, 5, 1/2, 5, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(2, [[4, 1/4], [5, 1/2], [6, 1/4]], 1, 5, 1/2, 5, 1/2, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181035,"parameters":{"values":[2,3],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":2,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[2,[1,2]],[3,[1,2]]],n:2,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0036","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[-2,-1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[-6, 1/8], [-5, 3/8], [-4, 3/8], [-3, 1/8]], 1, -9/2, 3/4, -9/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(3, [[-6, 1/8], [-5, 3/8], [-4, 3/8], [-3, 1/8]], 1, -9/2, 3/4, -9/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181036,"parameters":{"values":[-2,-1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":3,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[-2,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0037","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[-1,0],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[-3, 1/8], [-2, 3/8], [-1, 3/8], [0, 1/8]], 1, -3/2, 3/4, -3/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(3, [[-3, 1/8], [-2, 3/8], [-1, 3/8], [0, 1/8]], 1, -3/2, 3/4, -3/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181037,"parameters":{"values":[-1,0],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":3,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[0,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0038","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[0,1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 3/2, 3/4, 3/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(3, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 3/2, 3/4, 3/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181038,"parameters":{"values":[0,1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":3,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[0,[1,2]],[1,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0039","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[1,2],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[3, 1/8], [4, 3/8], [5, 3/8], [6, 1/8]], 1, 9/2, 3/4, 9/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(3, [[3, 1/8], [4, 3/8], [5, 3/8], [6, 1/8]], 1, 9/2, 3/4, 9/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181039,"parameters":{"values":[1,2],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":3,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[1,[1,2]],[2,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0040","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[2,3],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[6, 1/8], [7, 3/8], [8, 3/8], [9, 1/8]], 1, 15/2, 3/4, 15/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(3, [[6, 1/8], [7, 3/8], [8, 3/8], [9, 1/8]], 1, 15/2, 3/4, 15/2, 3/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181040,"parameters":{"values":[2,3],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":3,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[2,[1,2]],[3,[1,2]]],n:3,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0041","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[-2,-1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[-8, 1/16], [-7, 1/4], [-6, 3/8], [-5, 1/4], [-4, 1/16]], 1, -6, 1, -6, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(4, [[-8, 1/16], [-7, 1/4], [-6, 3/8], [-5, 1/4], [-4, 1/16]], 1, -6, 1, -6, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181041,"parameters":{"values":[-2,-1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":4,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[-2,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0042","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[-1,0],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[-4, 1/16], [-3, 1/4], [-2, 3/8], [-1, 1/4], [0, 1/16]], 1, -2, 1, -2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(4, [[-4, 1/16], [-3, 1/4], [-2, 3/8], [-1, 1/4], [0, 1/16]], 1, -2, 1, -2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181042,"parameters":{"values":[-1,0],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":4,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[0,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0043","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[0,1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 2, 1, 2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(4, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 2, 1, 2, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181043,"parameters":{"values":[0,1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":4,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[0,[1,2]],[1,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0044","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[1,2],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[4, 1/16], [5, 1/4], [6, 3/8], [7, 1/4], [8, 1/16]], 1, 6, 1, 6, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(4, [[4, 1/16], [5, 1/4], [6, 3/8], [7, 1/4], [8, 1/16]], 1, 6, 1, 6, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181044,"parameters":{"values":[1,2],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":4,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[1,[1,2]],[2,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0045","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[2,3],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[8, 1/16], [9, 1/4], [10, 3/8], [11, 1/4], [12, 1/16]], 1, 10, 1, 10, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(4, [[8, 1/16], [9, 1/4], [10, 3/8], [11, 1/4], [12, 1/16]], 1, 10, 1, 10, 1, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181045,"parameters":{"values":[2,3],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":4,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[2,[1,2]],[3,[1,2]]],n:4,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0046","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[-2,-1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[-10, 1/32], [-9, 5/32], [-8, 5/16], [-7, 5/16], [-6, 5/32], [-5, 1/32]], 1, -15/2, 5/4, -15/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(5, [[-10, 1/32], [-9, 5/32], [-8, 5/16], [-7, 5/16], [-6, 5/32], [-5, 1/32]], 1, -15/2, 5/4, -15/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181046,"parameters":{"values":[-2,-1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":5,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[-2,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0047","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[-1,0],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[-5, 1/32], [-4, 5/32], [-3, 5/16], [-2, 5/16], [-1, 5/32], [0, 1/32]], 1, -5/2, 5/4, -5/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(5, [[-5, 1/32], [-4, 5/32], [-3, 5/16], [-2, 5/16], [-1, 5/32], [0, 1/32]], 1, -5/2, 5/4, -5/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181047,"parameters":{"values":[-1,0],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":5,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[-1,[1,2]],[0,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0048","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[0,1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 5/2, 5/4, 5/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(5, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 5/2, 5/4, 5/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181048,"parameters":{"values":[0,1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":5,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[0,[1,2]],[1,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0049","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[1,2],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[5, 1/32], [6, 5/32], [7, 5/16], [8, 5/16], [9, 5/32], [10, 1/32]], 1, 15/2, 5/4, 15/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(5, [[5, 1/32], [6, 5/32], [7, 5/16], [8, 5/16], [9, 5/32], [10, 1/32]], 1, 15/2, 5/4, 15/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181049,"parameters":{"values":[1,2],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":5,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[1,[1,2]],[2,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0050","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-standardized-sum-moments；資料={\"values\":[2,3],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[10, 1/32], [11, 5/32], [12, 5/16], [13, 5/16], [14, 5/32], [15, 1/32]], 1, 25/2, 5/4, 25/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)","explanation_zh":"Wave6第十二層核對=(5, [[10, 1/32], [11, 5/32], [12, 5/16], [13, 5/16], [14, 5/32], [15, 1/32]], 1, 25/2, 5/4, 25/2, 5/4, 0, 1, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), complete_joint_law_moments_do_not_prove_distributional_convergence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-standardized-sum-moments","seed":202611181050,"parameters":{"values":[2,3],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":5,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-standardized-sum-moments","semantic_template":"clt-standardized-sum-moments|{law:[[2,[1,2]],[3,[1,2]]],n:5,process_certificate:\"infinite_iid\"}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0051","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":2,\"threshold\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 0, 1/4, 是, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 0, 1/4, 是, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181051,"parameters":{"success_probability":[1,2],"n":2,"threshold":0,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:2,success_probability:[1,2],threshold:0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0052","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":2,\"threshold\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 3/4, 是, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 3/4, 是, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181052,"parameters":{"success_probability":[1,2],"n":2,"threshold":1,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:2,success_probability:[1,2],threshold:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0053","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":2,\"threshold\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 2, 1, 是, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(2, 1/2, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 2, 1, 是, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181053,"parameters":{"success_probability":[1,2],"n":2,"threshold":2,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:2,success_probability:[1,2],threshold:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0054","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":3,\"threshold\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 0, 1/8, 是, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 0, 1/8, 是, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181054,"parameters":{"success_probability":[1,2],"n":3,"threshold":0,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:3,success_probability:[1,2],threshold:0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0055","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":3,\"threshold\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 1, 1/2, 是, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 1, 1/2, 是, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181055,"parameters":{"success_probability":[1,2],"n":3,"threshold":1,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:3,success_probability:[1,2],threshold:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0056","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":3,\"threshold\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 2, 7/8, 是, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 2, 7/8, 是, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181056,"parameters":{"success_probability":[1,2],"n":3,"threshold":2,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:3,success_probability:[1,2],threshold:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0057","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":3,\"threshold\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 3, 1, 是, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(3, 1/2, [[0, 1/8], [1, 3/8], [2, 3/8], [3, 1/8]], 1, 3, 1, 是, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181057,"parameters":{"success_probability":[1,2],"n":3,"threshold":3,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:3,success_probability:[1,2],threshold:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0058","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":4,\"threshold\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 0, 1/16, 是, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 0, 1/16, 是, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181058,"parameters":{"success_probability":[1,2],"n":4,"threshold":0,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:4,success_probability:[1,2],threshold:0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0059","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":4,\"threshold\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 1, 5/16, 是, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 1, 5/16, 是, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181059,"parameters":{"success_probability":[1,2],"n":4,"threshold":1,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:4,success_probability:[1,2],threshold:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0060","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":4,\"threshold\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 2, 11/16, 是, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 2, 11/16, 是, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181060,"parameters":{"success_probability":[1,2],"n":4,"threshold":2,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:4,success_probability:[1,2],threshold:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0061","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":4,\"threshold\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 3, 15/16, 是, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 3, 15/16, 是, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181061,"parameters":{"success_probability":[1,2],"n":4,"threshold":3,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:4,success_probability:[1,2],threshold:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0062","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":4,\"threshold\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 4, 1, 是, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(4, 1/2, [[0, 1/16], [1, 1/4], [2, 3/8], [3, 1/4], [4, 1/16]], 1, 4, 1, 是, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181062,"parameters":{"success_probability":[1,2],"n":4,"threshold":4,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:4,success_probability:[1,2],threshold:4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0063","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":5,\"threshold\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 0, 1/32, 是, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 0, 1/32, 是, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181063,"parameters":{"success_probability":[1,2],"n":5,"threshold":0,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:5,success_probability:[1,2],threshold:0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0064","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":5,\"threshold\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 1, 3/16, 是, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 1, 3/16, 是, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181064,"parameters":{"success_probability":[1,2],"n":5,"threshold":1,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:5,success_probability:[1,2],threshold:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0065","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":5,\"threshold\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 2, 1/2, 是, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 2, 1/2, 是, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181065,"parameters":{"success_probability":[1,2],"n":5,"threshold":2,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:5,success_probability:[1,2],threshold:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0066","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":5,\"threshold\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 3, 13/16, 是, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 3, 13/16, 是, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181066,"parameters":{"success_probability":[1,2],"n":5,"threshold":3,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:5,success_probability:[1,2],threshold:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0067","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":5,\"threshold\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 4, 31/32, 是, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 4, 31/32, 是, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181067,"parameters":{"success_probability":[1,2],"n":5,"threshold":4,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:5,success_probability:[1,2],threshold:4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0068","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":5,\"threshold\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 5, 1, 是, z_threshold=(5-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(5, 1/2, [[0, 1/32], [1, 5/32], [2, 5/16], [3, 5/16], [4, 5/32], [5, 1/32]], 1, 5, 1, 是, z_threshold=(5-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181068,"parameters":{"success_probability":[1,2],"n":5,"threshold":5,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:5,success_probability:[1,2],threshold:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0069","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":6,\"threshold\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 0, 1/64, 是, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 0, 1/64, 是, z_threshold=(0-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181069,"parameters":{"success_probability":[1,2],"n":6,"threshold":0,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0070","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":6,\"threshold\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 1, 7/64, 是, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 1, 7/64, 是, z_threshold=(1-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181070,"parameters":{"success_probability":[1,2],"n":6,"threshold":1,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0071","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":6,\"threshold\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 2, 11/32, 是, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 2, 11/32, 是, z_threshold=(2-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181071,"parameters":{"success_probability":[1,2],"n":6,"threshold":2,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0072","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":6,\"threshold\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 3, 21/32, 是, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 3, 21/32, 是, z_threshold=(3-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181072,"parameters":{"success_probability":[1,2],"n":6,"threshold":3,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0073","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":6,\"threshold\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 4, 57/64, 是, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 4, 57/64, 是, z_threshold=(4-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181073,"parameters":{"success_probability":[1,2],"n":6,"threshold":4,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0074","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":6,\"threshold\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 5, 63/64, 是, z_threshold=(5-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 5, 63/64, 是, z_threshold=(5-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181074,"parameters":{"success_probability":[1,2],"n":6,"threshold":5,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0075","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-binomial-lattice；資料={\"success_probability\":[1,2],\"n\":6,\"threshold\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 6, 1, 是, z_threshold=(6-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)","explanation_zh":"Wave6第十二層核對=(6, 1/2, [[0, 1/64], [1, 3/32], [2, 15/64], [3, 5/16], [4, 15/64], [5, 3/32], [6, 1/64]], 1, 6, 1, 是, z_threshold=(6-n*p)/sqrt(n*p*(1-p)), exact_lattice_not_normal_approximation_or_convergence_evidence)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-binomial-lattice","seed":202611181075,"parameters":{"success_probability":[1,2],"n":6,"threshold":6,"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-binomial-lattice","semantic_template":"clt-binomial-lattice|{n:6,success_probability:[1,2],threshold:6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0076","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"symbolic_application\",\"model\":{\"values\":[-2,-1],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":2}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(symbolic_application, infinite_iid, -3/2, 1/4, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), 是)","explanation_zh":"Wave6第十二層核對=(symbolic_application, infinite_iid, -3/2, 1/4, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), 是)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181076,"parameters":{"mode":"symbolic_application","model":{"values":[-2,-1],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":2},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"symbolic_application\",model:{law:[[-1,[1,2]],[-2,[1,2]]],n:2,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0077","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"non_iid\",\"model\":{\"values_a\":[-1,0],\"probabilities_a\":[[1,2],[1,2]],\"values_b\":[-1,0],\"probabilities_b\":[[1,7],[6,7]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_iid, [[-1, 1/2], [0, 1/2]], [[-1, 1/7], [0, 6/7]], 否, 否, this_IID_CLT_not_applicable_not_a_general_counterexample)","explanation_zh":"Wave6第十二層核對=(non_iid, [[-1, 1/2], [0, 1/2]], [[-1, 1/7], [0, 6/7]], 否, 否, this_IID_CLT_not_applicable_not_a_general_counterexample)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181077,"parameters":{"mode":"non_iid","model":{"values_a":[-1,0],"probabilities_a":[[1,2],[1,2]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"non_iid\",model:{law_a:[[-1,[1,2]],[0,[1,2]]],law_b:[[-1,[1,7]],[0,[6,7]]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0078","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"zero_variance\",\"model\":{\"value\":-1,\"n\":4}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_variance, -1, 4, 0, sigma=0_makes_standardization_undefined, 否)","explanation_zh":"Wave6第十二層核對=(zero_variance, -1, 4, 0, sigma=0_makes_standardization_undefined, 否)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181078,"parameters":{"mode":"zero_variance","model":{"value":-1,"n":4},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"zero_variance\",model:{n:4,value:-1}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0079","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"wrong_scaling\",\"model\":{\"values\":[1,2],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":5}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_scaling, 3/2, 1/4, 5, denominator=n*sigma, required=sigma*sqrt(n), 否)","explanation_zh":"Wave6第十二層核對=(wrong_scaling, 3/2, 1/4, 5, denominator=n*sigma, required=sigma*sqrt(n), 否)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181079,"parameters":{"mode":"wrong_scaling","model":{"values":[1,2],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":5},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"wrong_scaling\",model:{law:[[1,[1,2]],[2,[1,2]]],n:5,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0080","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"finite_histogram\",\"model\":{\"values\":[2,3],\"probabilities\":[[1,2],[1,2]],\"process_certificate\":\"infinite_iid\",\"n\":6}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_histogram, 5/2, 1/4, 6, one_finite_histogram, 否, finite_histogram_does_not_prove_distributional_limit)","explanation_zh":"Wave6第十二層核對=(finite_histogram, 5/2, 1/4, 6, one_finite_histogram, 否, finite_histogram_does_not_prove_distributional_limit)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181080,"parameters":{"mode":"finite_histogram","model":{"values":[2,3],"probabilities":[[1,2],[1,2]],"process_certificate":"infinite_iid","n":6},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"finite_histogram\",model:{law:[[2,[1,2]],[3,[1,2]]],n:6,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0081","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"symbolic_application\",\"model\":{\"values\":[-2,-1],\"probabilities\":[[1,3],[2,3]],\"process_certificate\":\"infinite_iid\",\"n\":2}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(symbolic_application, infinite_iid, -4/3, 2/9, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), 是)","explanation_zh":"Wave6第十二層核對=(symbolic_application, infinite_iid, -4/3, 2/9, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), 是)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181081,"parameters":{"mode":"symbolic_application","model":{"values":[-2,-1],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","n":2},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"symbolic_application\",model:{law:[[-1,[2,3]],[-2,[1,3]]],n:2,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0082","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"non_iid\",\"model\":{\"values_a\":[-1,0],\"probabilities_a\":[[1,3],[2,3]],\"values_b\":[-1,0],\"probabilities_b\":[[1,7],[6,7]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_iid, [[-1, 1/3], [0, 2/3]], [[-1, 1/7], [0, 6/7]], 否, 否, this_IID_CLT_not_applicable_not_a_general_counterexample)","explanation_zh":"Wave6第十二層核對=(non_iid, [[-1, 1/3], [0, 2/3]], [[-1, 1/7], [0, 6/7]], 否, 否, this_IID_CLT_not_applicable_not_a_general_counterexample)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181082,"parameters":{"mode":"non_iid","model":{"values_a":[-1,0],"probabilities_a":[[1,3],[2,3]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"non_iid\",model:{law_a:[[-1,[1,3]],[0,[2,3]]],law_b:[[-1,[1,7]],[0,[6,7]]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0083","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"zero_variance\",\"model\":{\"value\":-3,\"n\":4}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_variance, -3, 4, 0, sigma=0_makes_standardization_undefined, 否)","explanation_zh":"Wave6第十二層核對=(zero_variance, -3, 4, 0, sigma=0_makes_standardization_undefined, 否)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181083,"parameters":{"mode":"zero_variance","model":{"value":-3,"n":4},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"zero_variance\",model:{n:4,value:-3}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0084","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"wrong_scaling\",\"model\":{\"values\":[1,2],\"probabilities\":[[1,3],[2,3]],\"process_certificate\":\"infinite_iid\",\"n\":5}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_scaling, 5/3, 2/9, 5, denominator=n*sigma, required=sigma*sqrt(n), 否)","explanation_zh":"Wave6第十二層核對=(wrong_scaling, 5/3, 2/9, 5, denominator=n*sigma, required=sigma*sqrt(n), 否)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181084,"parameters":{"mode":"wrong_scaling","model":{"values":[1,2],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","n":5},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"wrong_scaling\",model:{law:[[1,[1,3]],[2,[2,3]]],n:5,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0085","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"finite_histogram\",\"model\":{\"values\":[2,3],\"probabilities\":[[1,3],[2,3]],\"process_certificate\":\"infinite_iid\",\"n\":6}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_histogram, 8/3, 2/9, 6, one_finite_histogram, 否, finite_histogram_does_not_prove_distributional_limit)","explanation_zh":"Wave6第十二層核對=(finite_histogram, 8/3, 2/9, 6, one_finite_histogram, 否, finite_histogram_does_not_prove_distributional_limit)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181085,"parameters":{"mode":"finite_histogram","model":{"values":[2,3],"probabilities":[[1,3],[2,3]],"process_certificate":"infinite_iid","n":6},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"finite_histogram\",model:{law:[[2,[1,3]],[3,[2,3]]],n:6,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0086","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"symbolic_application\",\"model\":{\"values\":[-2,-1],\"probabilities\":[[1,4],[3,4]],\"process_certificate\":\"infinite_iid\",\"n\":2}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(symbolic_application, infinite_iid, -5/4, 3/16, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), 是)","explanation_zh":"Wave6第十二層核對=(symbolic_application, infinite_iid, -5/4, 3/16, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), 是)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181086,"parameters":{"mode":"symbolic_application","model":{"values":[-2,-1],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","n":2},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"symbolic_application\",model:{law:[[-1,[3,4]],[-2,[1,4]]],n:2,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0087","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"non_iid\",\"model\":{\"values_a\":[-1,0],\"probabilities_a\":[[1,4],[3,4]],\"values_b\":[-1,0],\"probabilities_b\":[[1,7],[6,7]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_iid, [[-1, 1/4], [0, 3/4]], [[-1, 1/7], [0, 6/7]], 否, 否, this_IID_CLT_not_applicable_not_a_general_counterexample)","explanation_zh":"Wave6第十二層核對=(non_iid, [[-1, 1/4], [0, 3/4]], [[-1, 1/7], [0, 6/7]], 否, 否, this_IID_CLT_not_applicable_not_a_general_counterexample)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181087,"parameters":{"mode":"non_iid","model":{"values_a":[-1,0],"probabilities_a":[[1,4],[3,4]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"non_iid\",model:{law_a:[[-1,[1,4]],[0,[3,4]]],law_b:[[-1,[1,7]],[0,[6,7]]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0088","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"zero_variance\",\"model\":{\"value\":2,\"n\":4}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_variance, 2, 4, 0, sigma=0_makes_standardization_undefined, 否)","explanation_zh":"Wave6第十二層核對=(zero_variance, 2, 4, 0, sigma=0_makes_standardization_undefined, 否)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181088,"parameters":{"mode":"zero_variance","model":{"value":2,"n":4},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"zero_variance\",model:{n:4,value:2}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0089","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"wrong_scaling\",\"model\":{\"values\":[1,2],\"probabilities\":[[1,4],[3,4]],\"process_certificate\":\"infinite_iid\",\"n\":5}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_scaling, 7/4, 3/16, 5, denominator=n*sigma, required=sigma*sqrt(n), 否)","explanation_zh":"Wave6第十二層核對=(wrong_scaling, 7/4, 3/16, 5, denominator=n*sigma, required=sigma*sqrt(n), 否)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181089,"parameters":{"mode":"wrong_scaling","model":{"values":[1,2],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","n":5},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"wrong_scaling\",model:{law:[[1,[1,4]],[2,[3,4]]],n:5,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0090","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"finite_histogram\",\"model\":{\"values\":[2,3],\"probabilities\":[[1,4],[3,4]],\"process_certificate\":\"infinite_iid\",\"n\":6}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_histogram, 11/4, 3/16, 6, one_finite_histogram, 否, finite_histogram_does_not_prove_distributional_limit)","explanation_zh":"Wave6第十二層核對=(finite_histogram, 11/4, 3/16, 6, one_finite_histogram, 否, finite_histogram_does_not_prove_distributional_limit)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181090,"parameters":{"mode":"finite_histogram","model":{"values":[2,3],"probabilities":[[1,4],[3,4]],"process_certificate":"infinite_iid","n":6},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"finite_histogram\",model:{law:[[2,[1,4]],[3,[3,4]]],n:6,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0091","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"symbolic_application\",\"model\":{\"values\":[-2,-1],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\",\"n\":2}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(symbolic_application, infinite_iid, -7/5, 6/25, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), 是)","explanation_zh":"Wave6第十二層核對=(symbolic_application, infinite_iid, -7/5, 6/25, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), 是)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181091,"parameters":{"mode":"symbolic_application","model":{"values":[-2,-1],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","n":2},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"symbolic_application\",model:{law:[[-1,[3,5]],[-2,[2,5]]],n:2,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0092","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"non_iid\",\"model\":{\"values_a\":[-1,0],\"probabilities_a\":[[2,5],[3,5]],\"values_b\":[-1,0],\"probabilities_b\":[[1,7],[6,7]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_iid, [[-1, 2/5], [0, 3/5]], [[-1, 1/7], [0, 6/7]], 否, 否, this_IID_CLT_not_applicable_not_a_general_counterexample)","explanation_zh":"Wave6第十二層核對=(non_iid, [[-1, 2/5], [0, 3/5]], [[-1, 1/7], [0, 6/7]], 否, 否, this_IID_CLT_not_applicable_not_a_general_counterexample)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181092,"parameters":{"mode":"non_iid","model":{"values_a":[-1,0],"probabilities_a":[[2,5],[3,5]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"non_iid\",model:{law_a:[[-1,[2,5]],[0,[3,5]]],law_b:[[-1,[1,7]],[0,[6,7]]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0093","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"zero_variance\",\"model\":{\"value\":0,\"n\":4}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_variance, 0, 4, 0, sigma=0_makes_standardization_undefined, 否)","explanation_zh":"Wave6第十二層核對=(zero_variance, 0, 4, 0, sigma=0_makes_standardization_undefined, 否)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181093,"parameters":{"mode":"zero_variance","model":{"value":0,"n":4},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"zero_variance\",model:{n:4,value:0}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0094","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"wrong_scaling\",\"model\":{\"values\":[1,2],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\",\"n\":5}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_scaling, 8/5, 6/25, 5, denominator=n*sigma, required=sigma*sqrt(n), 否)","explanation_zh":"Wave6第十二層核對=(wrong_scaling, 8/5, 6/25, 5, denominator=n*sigma, required=sigma*sqrt(n), 否)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181094,"parameters":{"mode":"wrong_scaling","model":{"values":[1,2],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","n":5},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"wrong_scaling\",model:{law:[[1,[2,5]],[2,[3,5]]],n:5,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0095","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"finite_histogram\",\"model\":{\"values\":[2,3],\"probabilities\":[[2,5],[3,5]],\"process_certificate\":\"infinite_iid\",\"n\":6}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_histogram, 13/5, 6/25, 6, one_finite_histogram, 否, finite_histogram_does_not_prove_distributional_limit)","explanation_zh":"Wave6第十二層核對=(finite_histogram, 13/5, 6/25, 6, one_finite_histogram, 否, finite_histogram_does_not_prove_distributional_limit)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181095,"parameters":{"mode":"finite_histogram","model":{"values":[2,3],"probabilities":[[2,5],[3,5]],"process_certificate":"infinite_iid","n":6},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"finite_histogram\",model:{law:[[2,[2,5]],[3,[3,5]]],n:6,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0096","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"symbolic_application\",\"model\":{\"values\":[-2,-1],\"probabilities\":[[1,5],[4,5]],\"process_certificate\":\"infinite_iid\",\"n\":2}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(symbolic_application, infinite_iid, -6/5, 4/25, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), 是)","explanation_zh":"Wave6第十二層核對=(symbolic_application, infinite_iid, -6/5, 4/25, 2, Z_n=(S_n-n*mu)/(sigma*sqrt(n)), Z_n_converges_in_distribution_to_N(0,1), 是)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181096,"parameters":{"mode":"symbolic_application","model":{"values":[-2,-1],"probabilities":[[1,5],[4,5]],"process_certificate":"infinite_iid","n":2},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"symbolic_application\",model:{law:[[-1,[4,5]],[-2,[1,5]]],n:2,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0097","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"non_iid\",\"model\":{\"values_a\":[-1,0],\"probabilities_a\":[[1,5],[4,5]],\"values_b\":[-1,0],\"probabilities_b\":[[1,7],[6,7]]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(non_iid, [[-1, 1/5], [0, 4/5]], [[-1, 1/7], [0, 6/7]], 否, 否, this_IID_CLT_not_applicable_not_a_general_counterexample)","explanation_zh":"Wave6第十二層核對=(non_iid, [[-1, 1/5], [0, 4/5]], [[-1, 1/7], [0, 6/7]], 否, 否, this_IID_CLT_not_applicable_not_a_general_counterexample)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181097,"parameters":{"mode":"non_iid","model":{"values_a":[-1,0],"probabilities_a":[[1,5],[4,5]],"values_b":[-1,0],"probabilities_b":[[1,7],[6,7]]},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"non_iid\",model:{law_a:[[-1,[1,5]],[0,[4,5]]],law_b:[[-1,[1,7]],[0,[6,7]]]}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0098","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"zero_variance\",\"model\":{\"value\":-2,\"n\":4}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(zero_variance, -2, 4, 0, sigma=0_makes_standardization_undefined, 否)","explanation_zh":"Wave6第十二層核對=(zero_variance, -2, 4, 0, sigma=0_makes_standardization_undefined, 否)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181098,"parameters":{"mode":"zero_variance","model":{"value":-2,"n":4},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"zero_variance\",model:{n:4,value:-2}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0099","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"wrong_scaling\",\"model\":{\"values\":[1,2],\"probabilities\":[[1,5],[4,5]],\"process_certificate\":\"infinite_iid\",\"n\":5}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_scaling, 9/5, 4/25, 5, denominator=n*sigma, required=sigma*sqrt(n), 否)","explanation_zh":"Wave6第十二層核對=(wrong_scaling, 9/5, 4/25, 5, denominator=n*sigma, required=sigma*sqrt(n), 否)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181099,"parameters":{"mode":"wrong_scaling","model":{"values":[1,2],"probabilities":[[1,5],[4,5]],"process_certificate":"infinite_iid","n":5},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"wrong_scaling\",model:{law:[[1,[1,5]],[2,[4,5]]],n:5,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-0100","batch_id":"batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations","language":"zh-Hant","source_mko_ids":["mko-central-limit-theorem-finite-variance-iid"],"title_zh":"有限變異數IID中央極限定理","stem_zh":"明示無限IID、共同mu、0<sigma^2<infinity與sigma*sqrt(n)標準化責任。操作=clt-application-boundary；資料={\"mode\":\"finite_histogram\",\"model\":{\"values\":[2,3],\"probabilities\":[[1,5],[4,5]],\"process_certificate\":\"infinite_iid\",\"n\":6}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_histogram, 14/5, 4/25, 6, one_finite_histogram, 否, finite_histogram_does_not_prove_distributional_limit)","explanation_zh":"Wave6第十二層核對=(finite_histogram, 14/5, 4/25, 6, one_finite_histogram, 否, finite_histogram_does_not_prove_distributional_limit)；依完整明示資料獨立重算；finite moments與binomial lattice完整重算但不作normal approximation或limit evidence；non-IID為不適用，zero variance、wrong scaling與finite histogram不得冒充CLT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限變異數IID中央極限定理"]},"variant":{"family_id":"qf-mod49-clt-application-boundary","seed":202611181100,"parameters":{"mode":"finite_histogram","model":{"values":[2,3],"probabilities":[[1,5],[4,5]],"process_certificate":"infinite_iid","n":6},"domain":"Exact_finite_IID_limit_theorem_boundary","operation":"clt-application-boundary","semantic_template":"clt-application-boundary|{mode:\"finite_histogram\",model:{law:[[2,[1,5]],[3,[4,5]]],n:6,process_certificate:\"infinite_iid\"}}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}