{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r03-bias-variance-decomposition-0001","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":-12,\"support\":[-12,-11,-10],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, [[-12, 1/4], [-11, 1/2], [-10, 1/4]], -11, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(-12, [[-12, 1/4], [-11, 1/2], [-10, 1/4]], -11, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153001,"parameters":{"theta":-12,"support":[-12,-11,-10],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-10,1],[-11,1],[-12,1]],\"theta\":[-12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0002","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":-11,\"support\":[-11,-9,-7],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, [[-11, 1/4], [-9, 1/2], [-7, 1/4]], -9, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(-11, [[-11, 1/4], [-9, 1/2], [-7, 1/4]], -9, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153002,"parameters":{"theta":-11,"support":[-11,-9,-7],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-11,1],[-7,1],[-9,1]],\"theta\":[-11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0003","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":-10,\"support\":[-10,-7,-4],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, [[-10, 1/4], [-7, 1/2], [-4, 1/4]], -7, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(-10, [[-10, 1/4], [-7, 1/2], [-4, 1/4]], -7, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153003,"parameters":{"theta":-10,"support":[-10,-7,-4],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-10,1],[-4,1],[-7,1]],\"theta\":[-10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0004","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":-9,\"support\":[-9,-8,-7],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, [[-9, 1/4], [-8, 1/2], [-7, 1/4]], -8, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(-9, [[-9, 1/4], [-8, 1/2], [-7, 1/4]], -8, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153004,"parameters":{"theta":-9,"support":[-9,-8,-7],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-7,1],[-8,1],[-9,1]],\"theta\":[-9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0005","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":-8,\"support\":[-8,-6,-4],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, [[-8, 1/4], [-6, 1/2], [-4, 1/4]], -6, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(-8, [[-8, 1/4], [-6, 1/2], [-4, 1/4]], -6, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153005,"parameters":{"theta":-8,"support":[-8,-6,-4],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-4,1],[-6,1],[-8,1]],\"theta\":[-8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0006","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":-7,\"support\":[-7,-4,-1],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, [[-7, 1/4], [-4, 1/2], [-1, 1/4]], -4, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(-7, [[-7, 1/4], [-4, 1/2], [-1, 1/4]], -4, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153006,"parameters":{"theta":-7,"support":[-7,-4,-1],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[-4,1],[-7,1]],\"theta\":[-7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0007","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":-6,\"support\":[-6,-5,-4],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, [[-6, 1/4], [-5, 1/2], [-4, 1/4]], -5, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(-6, [[-6, 1/4], [-5, 1/2], [-4, 1/4]], -5, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153007,"parameters":{"theta":-6,"support":[-6,-5,-4],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-4,1],[-5,1],[-6,1]],\"theta\":[-6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0008","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":-5,\"support\":[-5,-3,-1],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, [[-5, 1/4], [-3, 1/2], [-1, 1/4]], -3, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(-5, [[-5, 1/4], [-3, 1/2], [-1, 1/4]], -3, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153008,"parameters":{"theta":-5,"support":[-5,-3,-1],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[-3,1],[-5,1]],\"theta\":[-5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0009","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":-4,\"support\":[-4,-1,2],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, [[-4, 1/4], [-1, 1/2], [2, 1/4]], -1, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(-4, [[-4, 1/4], [-1, 1/2], [2, 1/4]], -1, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153009,"parameters":{"theta":-4,"support":[-4,-1,2],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-1,1],[-4,1],[2,1]],\"theta\":[-4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0010","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":-3,\"support\":[-3,-2,-1],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], -2, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(-3, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], -2, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153010,"parameters":{"theta":-3,"support":[-3,-2,-1],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[-2,1],[-3,1]],\"theta\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0011","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":-2,\"support\":[-2,0,2],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, [[-2, 1/4], [0, 1/2], [2, 1/4]], 0, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(-2, [[-2, 1/4], [0, 1/2], [2, 1/4]], 0, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153011,"parameters":{"theta":-2,"support":[-2,0,2],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-2,1],[0,1],[2,1]],\"theta\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0012","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":-1,\"support\":[-1,2,5],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, [[-1, 1/4], [2, 1/2], [5, 1/4]], 2, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(-1, [[-1, 1/4], [2, 1/2], [5, 1/4]], 2, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153012,"parameters":{"theta":-1,"support":[-1,2,5],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[2,1],[5,1]],\"theta\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0013","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":0,\"support\":[0,1,2],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(0, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153013,"parameters":{"theta":0,"support":[0,1,2],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[0,1],[1,1],[2,1]],\"theta\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0014","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":1,\"support\":[1,3,5],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[1, 1/4], [3, 1/2], [5, 1/4]], 3, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(1, [[1, 1/4], [3, 1/2], [5, 1/4]], 3, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153014,"parameters":{"theta":1,"support":[1,3,5],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[1,1],[3,1],[5,1]],\"theta\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0015","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":2,\"support\":[2,5,8],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[2, 1/4], [5, 1/2], [8, 1/4]], 5, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(2, [[2, 1/4], [5, 1/2], [8, 1/4]], 5, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153015,"parameters":{"theta":2,"support":[2,5,8],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[5,1],[8,1]],\"theta\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0016","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":3,\"support\":[3,4,5],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[3, 1/4], [4, 1/2], [5, 1/4]], 4, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(3, [[3, 1/4], [4, 1/2], [5, 1/4]], 4, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153016,"parameters":{"theta":3,"support":[3,4,5],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[3,1],[4,1],[5,1]],\"theta\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0017","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":4,\"support\":[4,6,8],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[4, 1/4], [6, 1/2], [8, 1/4]], 6, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(4, [[4, 1/4], [6, 1/2], [8, 1/4]], 6, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153017,"parameters":{"theta":4,"support":[4,6,8],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[4,1],[6,1],[8,1]],\"theta\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0018","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":5,\"support\":[5,8,11],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[5, 1/4], [8, 1/2], [11, 1/4]], 8, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(5, [[5, 1/4], [8, 1/2], [11, 1/4]], 8, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153018,"parameters":{"theta":5,"support":[5,8,11],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[11,1],[5,1],[8,1]],\"theta\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0019","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":6,\"support\":[6,7,8],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[6, 1/4], [7, 1/2], [8, 1/4]], 7, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(6, [[6, 1/4], [7, 1/2], [8, 1/4]], 7, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153019,"parameters":{"theta":6,"support":[6,7,8],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[6,1],[7,1],[8,1]],\"theta\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0020","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":7,\"support\":[7,9,11],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, [[7, 1/4], [9, 1/2], [11, 1/4]], 9, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(7, [[7, 1/4], [9, 1/2], [11, 1/4]], 9, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153020,"parameters":{"theta":7,"support":[7,9,11],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[11,1],[7,1],[9,1]],\"theta\":[7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0021","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":8,\"support\":[8,11,14],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[8, 1/4], [11, 1/2], [14, 1/4]], 11, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(8, [[8, 1/4], [11, 1/2], [14, 1/4]], 11, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153021,"parameters":{"theta":8,"support":[8,11,14],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[11,1],[14,1],[8,1]],\"theta\":[8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0022","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":9,\"support\":[9,10,11],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, [[9, 1/4], [10, 1/2], [11, 1/4]], 10, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(9, [[9, 1/4], [10, 1/2], [11, 1/4]], 10, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153022,"parameters":{"theta":9,"support":[9,10,11],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[10,1],[11,1],[9,1]],\"theta\":[9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0023","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":10,\"support\":[10,12,14],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[10, 1/4], [12, 1/2], [14, 1/4]], 12, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(10, [[10, 1/4], [12, 1/2], [14, 1/4]], 12, 2, 2, 6, 6, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153023,"parameters":{"theta":10,"support":[10,12,14],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[10,1],[12,1],[14,1]],\"theta\":[10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0024","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":11,\"support\":[11,14,17],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, [[11, 1/4], [14, 1/2], [17, 1/4]], 14, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(11, [[11, 1/4], [14, 1/2], [17, 1/4]], 14, 3, 9/2, 27/2, 27/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153024,"parameters":{"theta":11,"support":[11,14,17],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[11,1],[14,1],[17,1]],\"theta\":[11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0025","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-single-theta-identity；資料={\"theta\":12,\"support\":[12,13,14],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[12, 1/4], [13, 1/2], [14, 1/4]], 13, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)","explanation_zh":"Wave6第九層核對=(12, [[12, 1/4], [13, 1/2], [14, 1/4]], 13, 1, 1/2, 3/2, 3/2, 是, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-single-theta-identity","seed":202611153025,"parameters":{"theta":12,"support":[12,13,14],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-single-theta-identity","semantic_template":"qf-mod46-biasvar-single-theta-identity/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[12,1],[13,1],[14,1]],\"theta\":[12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0026","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":-12,\"support\":[-11,-10,-9],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(-10, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153026,"parameters":{"theta":-12,"support":[-11,-10,-9],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-10,1],[-11,1],[-9,1]],\"theta\":[-12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0027","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":-11,\"support\":[-10,-9,-8],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(-9, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153027,"parameters":{"theta":-11,"support":[-10,-9,-8],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-10,1],[-8,1],[-9,1]],\"theta\":[-11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0028","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":-10,\"support\":[-9,-8,-7],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(-8, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153028,"parameters":{"theta":-10,"support":[-9,-8,-7],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-7,1],[-8,1],[-9,1]],\"theta\":[-10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0029","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":-9,\"support\":[-8,-7,-6],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(-7, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153029,"parameters":{"theta":-9,"support":[-8,-7,-6],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-6,1],[-7,1],[-8,1]],\"theta\":[-9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0030","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":-8,\"support\":[-7,-6,-5],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(-6, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153030,"parameters":{"theta":-8,"support":[-7,-6,-5],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-5,1],[-6,1],[-7,1]],\"theta\":[-8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0031","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":-7,\"support\":[-6,-5,-4],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(-5, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153031,"parameters":{"theta":-7,"support":[-6,-5,-4],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-4,1],[-5,1],[-6,1]],\"theta\":[-7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0032","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":-6,\"support\":[-5,-4,-3],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(-4, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153032,"parameters":{"theta":-6,"support":[-5,-4,-3],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[-4,1],[-5,1]],\"theta\":[-6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0033","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":-5,\"support\":[-4,-3,-2],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(-3, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153033,"parameters":{"theta":-5,"support":[-4,-3,-2],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-2,1],[-3,1],[-4,1]],\"theta\":[-5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0034","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":-4,\"support\":[-3,-2,-1],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(-2, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153034,"parameters":{"theta":-4,"support":[-3,-2,-1],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[-2,1],[-3,1]],\"theta\":[-4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0035","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":-3,\"support\":[-2,-1,0],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(-1, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153035,"parameters":{"theta":-3,"support":[-2,-1,0],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-1,1],[-2,1],[0,1]],\"theta\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0036","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":-2,\"support\":[-1,0,1],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(0, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153036,"parameters":{"theta":-2,"support":[-1,0,1],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[0,1],[1,1]],\"theta\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0037","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":-1,\"support\":[0,1,2],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(1, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153037,"parameters":{"theta":-1,"support":[0,1,2],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[0,1],[1,1],[2,1]],\"theta\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0038","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":0,\"support\":[1,2,3],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(2, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153038,"parameters":{"theta":0,"support":[1,2,3],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[1,1],[2,1],[3,1]],\"theta\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0039","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":1,\"support\":[2,3,4],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(3, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153039,"parameters":{"theta":1,"support":[2,3,4],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[3,1],[4,1]],\"theta\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0040","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":2,\"support\":[3,4,5],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(4, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153040,"parameters":{"theta":2,"support":[3,4,5],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[3,1],[4,1],[5,1]],\"theta\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0041","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":3,\"support\":[4,5,6],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(5, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153041,"parameters":{"theta":3,"support":[4,5,6],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[4,1],[5,1],[6,1]],\"theta\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0042","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":4,\"support\":[5,6,7],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(6, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153042,"parameters":{"theta":4,"support":[5,6,7],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[5,1],[6,1],[7,1]],\"theta\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0043","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":5,\"support\":[6,7,8],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(7, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153043,"parameters":{"theta":5,"support":[6,7,8],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[6,1],[7,1],[8,1]],\"theta\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0044","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":6,\"support\":[7,8,9],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(8, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153044,"parameters":{"theta":6,"support":[7,8,9],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[7,1],[8,1],[9,1]],\"theta\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0045","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":7,\"support\":[8,9,10],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(9, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153045,"parameters":{"theta":7,"support":[8,9,10],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[10,1],[8,1],[9,1]],\"theta\":[7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0046","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":8,\"support\":[9,10,11],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(10, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153046,"parameters":{"theta":8,"support":[9,10,11],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[10,1],[11,1],[9,1]],\"theta\":[8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0047","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":9,\"support\":[10,11,12],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(11, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153047,"parameters":{"theta":9,"support":[10,11,12],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[10,1],[11,1],[12,1]],\"theta\":[9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0048","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":10,\"support\":[11,12,13],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(12, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153048,"parameters":{"theta":10,"support":[11,12,13],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[11,1],[12,1],[13,1]],\"theta\":[10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0049","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":11,\"support\":[12,13,14],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(13, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153049,"parameters":{"theta":11,"support":[12,13,14],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[12,1],[13,1],[14,1]],\"theta\":[11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0050","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-cross-term-cancellation；資料={\"theta\":12,\"support\":[13,14,15],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)","explanation_zh":"Wave6第九層核對=(14, 0, 2, 0, 是, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-cross-term-cancellation","seed":202611153050,"parameters":{"theta":12,"support":[13,14,15],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-cross-term-cancellation","semantic_template":"qf-mod46-biasvar-cross-term-cancellation/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[13,1],[14,1],[15,1]],\"theta\":[12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0051","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[-12,[-12,-11,-10],[[1,4],[1,2],[1,4]]],[-11,[-12,-11,-10],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-11, -11, 0, 1/2, 1/2, 1/2, 是], [-12, -11, 1, 1/2, 3/2, 3/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[-11, -11, 0, 1/2, 1/2, 1/2, 是], [-12, -11, 1, 1/2, 3/2, 3/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153051,"parameters":{"laws":[[-12,[-12,-11,-10],[[1,4],[1,2],[1,4]]],[-11,[-12,-11,-10],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[-11,1],[[-10,1],[-11,1],[-12,1]],[[1,4],[1,2],[1,4]]],[[-12,1],[[-10,1],[-11,1],[-12,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0052","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[-11,[-11,-9,-7],[[1,4],[1,2],[1,4]]],[-10,[-11,-10,-9],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-10, -10, 0, 1/2, 1/2, 1/2, 是], [-11, -9, 2, 2, 6, 6, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[-10, -10, 0, 1/2, 1/2, 1/2, 是], [-11, -9, 2, 2, 6, 6, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153052,"parameters":{"laws":[[-11,[-11,-9,-7],[[1,4],[1,2],[1,4]]],[-10,[-11,-10,-9],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[-10,1],[[-10,1],[-11,1],[-9,1]],[[1,2],[1,4],[1,4]]],[[-11,1],[[-11,1],[-7,1],[-9,1]],[[1,4],[1,4],[1,2]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0053","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[-10,[-10,-7,-4],[[1,4],[1,2],[1,4]]],[-9,[-10,-9,-8],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-10, -7, 3, 9/2, 27/2, 27/2, 是], [-9, -9, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[-10, -7, 3, 9/2, 27/2, 27/2, 是], [-9, -9, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153053,"parameters":{"laws":[[-10,[-10,-7,-4],[[1,4],[1,2],[1,4]]],[-9,[-10,-9,-8],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[-10,1],[[-10,1],[-4,1],[-7,1]],[[1,4],[1,4],[1,2]]],[[-9,1],[[-10,1],[-8,1],[-9,1]],[[1,4],[1,4],[1,2]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0054","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[-9,[-9,-8,-7],[[1,4],[1,2],[1,4]]],[-8,[-9,-8,-7],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-8, -8, 0, 1/2, 1/2, 1/2, 是], [-9, -8, 1, 1/2, 3/2, 3/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[-8, -8, 0, 1/2, 1/2, 1/2, 是], [-9, -8, 1, 1/2, 3/2, 3/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153054,"parameters":{"laws":[[-9,[-9,-8,-7],[[1,4],[1,2],[1,4]]],[-8,[-9,-8,-7],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[-8,1],[[-7,1],[-8,1],[-9,1]],[[1,4],[1,2],[1,4]]],[[-9,1],[[-7,1],[-8,1],[-9,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0055","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[-8,[-8,-6,-4],[[1,4],[1,2],[1,4]]],[-7,[-8,-7,-6],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-7, -7, 0, 1/2, 1/2, 1/2, 是], [-8, -6, 2, 2, 6, 6, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[-7, -7, 0, 1/2, 1/2, 1/2, 是], [-8, -6, 2, 2, 6, 6, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153055,"parameters":{"laws":[[-8,[-8,-6,-4],[[1,4],[1,2],[1,4]]],[-7,[-8,-7,-6],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[-7,1],[[-6,1],[-7,1],[-8,1]],[[1,4],[1,2],[1,4]]],[[-8,1],[[-4,1],[-6,1],[-8,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0056","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[-7,[-7,-4,-1],[[1,4],[1,2],[1,4]]],[-6,[-7,-6,-5],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-6, -6, 0, 1/2, 1/2, 1/2, 是], [-7, -4, 3, 9/2, 27/2, 27/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[-6, -6, 0, 1/2, 1/2, 1/2, 是], [-7, -4, 3, 9/2, 27/2, 27/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153056,"parameters":{"laws":[[-7,[-7,-4,-1],[[1,4],[1,2],[1,4]]],[-6,[-7,-6,-5],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[-6,1],[[-5,1],[-6,1],[-7,1]],[[1,4],[1,2],[1,4]]],[[-7,1],[[-1,1],[-4,1],[-7,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0057","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[-6,[-6,-5,-4],[[1,4],[1,2],[1,4]]],[-5,[-6,-5,-4],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-5, -5, 0, 1/2, 1/2, 1/2, 是], [-6, -5, 1, 1/2, 3/2, 3/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[-5, -5, 0, 1/2, 1/2, 1/2, 是], [-6, -5, 1, 1/2, 3/2, 3/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153057,"parameters":{"laws":[[-6,[-6,-5,-4],[[1,4],[1,2],[1,4]]],[-5,[-6,-5,-4],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[-5,1],[[-4,1],[-5,1],[-6,1]],[[1,4],[1,2],[1,4]]],[[-6,1],[[-4,1],[-5,1],[-6,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0058","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[-5,[-5,-3,-1],[[1,4],[1,2],[1,4]]],[-4,[-5,-4,-3],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-4, -4, 0, 1/2, 1/2, 1/2, 是], [-5, -3, 2, 2, 6, 6, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[-4, -4, 0, 1/2, 1/2, 1/2, 是], [-5, -3, 2, 2, 6, 6, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153058,"parameters":{"laws":[[-5,[-5,-3,-1],[[1,4],[1,2],[1,4]]],[-4,[-5,-4,-3],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[-4,1],[[-3,1],[-4,1],[-5,1]],[[1,4],[1,2],[1,4]]],[[-5,1],[[-1,1],[-3,1],[-5,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0059","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[-4,[-4,-1,2],[[1,4],[1,2],[1,4]]],[-3,[-4,-3,-2],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-3, -3, 0, 1/2, 1/2, 1/2, 是], [-4, -1, 3, 9/2, 27/2, 27/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[-3, -3, 0, 1/2, 1/2, 1/2, 是], [-4, -1, 3, 9/2, 27/2, 27/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153059,"parameters":{"laws":[[-4,[-4,-1,2],[[1,4],[1,2],[1,4]]],[-3,[-4,-3,-2],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[-3,1],[[-2,1],[-3,1],[-4,1]],[[1,4],[1,2],[1,4]]],[[-4,1],[[-1,1],[-4,1],[2,1]],[[1,2],[1,4],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0060","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[-3,[-3,-2,-1],[[1,4],[1,2],[1,4]]],[-2,[-3,-2,-1],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-2, -2, 0, 1/2, 1/2, 1/2, 是], [-3, -2, 1, 1/2, 3/2, 3/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[-2, -2, 0, 1/2, 1/2, 1/2, 是], [-3, -2, 1, 1/2, 3/2, 3/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153060,"parameters":{"laws":[[-3,[-3,-2,-1],[[1,4],[1,2],[1,4]]],[-2,[-3,-2,-1],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[-2,1],[[-1,1],[-2,1],[-3,1]],[[1,4],[1,2],[1,4]]],[[-3,1],[[-1,1],[-2,1],[-3,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0061","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[-2,[-2,0,2],[[1,4],[1,2],[1,4]]],[-1,[-2,-1,0],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-1, -1, 0, 1/2, 1/2, 1/2, 是], [-2, 0, 2, 2, 6, 6, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[-1, -1, 0, 1/2, 1/2, 1/2, 是], [-2, 0, 2, 2, 6, 6, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153061,"parameters":{"laws":[[-2,[-2,0,2],[[1,4],[1,2],[1,4]]],[-1,[-2,-1,0],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[-1,1],[[-1,1],[-2,1],[0,1]],[[1,2],[1,4],[1,4]]],[[-2,1],[[-2,1],[0,1],[2,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0062","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[-1,[-1,2,5],[[1,4],[1,2],[1,4]]],[0,[-1,0,1],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-1, 2, 3, 9/2, 27/2, 27/2, 是], [0, 0, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[-1, 2, 3, 9/2, 27/2, 27/2, 是], [0, 0, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153062,"parameters":{"laws":[[-1,[-1,2,5],[[1,4],[1,2],[1,4]]],[0,[-1,0,1],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[-1,1],[[-1,1],[2,1],[5,1]],[[1,4],[1,2],[1,4]]],[[0,1],[[-1,1],[0,1],[1,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0063","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[0,[0,1,2],[[1,4],[1,2],[1,4]]],[1,[0,1,2],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 1, 1/2, 3/2, 3/2, 是], [1, 1, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[0, 1, 1, 1/2, 3/2, 3/2, 是], [1, 1, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153063,"parameters":{"laws":[[0,[0,1,2],[[1,4],[1,2],[1,4]]],[1,[0,1,2],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[0,1],[[0,1],[1,1],[2,1]],[[1,4],[1,2],[1,4]]],[[1,1],[[0,1],[1,1],[2,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0064","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[1,[1,3,5],[[1,4],[1,2],[1,4]]],[2,[1,2,3],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, 3, 2, 2, 6, 6, 是], [2, 2, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[1, 3, 2, 2, 6, 6, 是], [2, 2, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153064,"parameters":{"laws":[[1,[1,3,5],[[1,4],[1,2],[1,4]]],[2,[1,2,3],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[1,1],[[1,1],[3,1],[5,1]],[[1,4],[1,2],[1,4]]],[[2,1],[[1,1],[2,1],[3,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0065","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[2,[2,5,8],[[1,4],[1,2],[1,4]]],[3,[2,3,4],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[2, 5, 3, 9/2, 27/2, 27/2, 是], [3, 3, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[2, 5, 3, 9/2, 27/2, 27/2, 是], [3, 3, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153065,"parameters":{"laws":[[2,[2,5,8],[[1,4],[1,2],[1,4]]],[3,[2,3,4],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[2,1],[[2,1],[5,1],[8,1]],[[1,4],[1,2],[1,4]]],[[3,1],[[2,1],[3,1],[4,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0066","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[3,[3,4,5],[[1,4],[1,2],[1,4]]],[4,[3,4,5],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[3, 4, 1, 1/2, 3/2, 3/2, 是], [4, 4, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[3, 4, 1, 1/2, 3/2, 3/2, 是], [4, 4, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153066,"parameters":{"laws":[[3,[3,4,5],[[1,4],[1,2],[1,4]]],[4,[3,4,5],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[3,1],[[3,1],[4,1],[5,1]],[[1,4],[1,2],[1,4]]],[[4,1],[[3,1],[4,1],[5,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0067","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[4,[4,6,8],[[1,4],[1,2],[1,4]]],[5,[4,5,6],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[4, 6, 2, 2, 6, 6, 是], [5, 5, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[4, 6, 2, 2, 6, 6, 是], [5, 5, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153067,"parameters":{"laws":[[4,[4,6,8],[[1,4],[1,2],[1,4]]],[5,[4,5,6],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[4,1],[[4,1],[6,1],[8,1]],[[1,4],[1,2],[1,4]]],[[5,1],[[4,1],[5,1],[6,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0068","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[5,[5,8,11],[[1,4],[1,2],[1,4]]],[6,[5,6,7],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[5, 8, 3, 9/2, 27/2, 27/2, 是], [6, 6, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[5, 8, 3, 9/2, 27/2, 27/2, 是], [6, 6, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153068,"parameters":{"laws":[[5,[5,8,11],[[1,4],[1,2],[1,4]]],[6,[5,6,7],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[5,1],[[11,1],[5,1],[8,1]],[[1,4],[1,4],[1,2]]],[[6,1],[[5,1],[6,1],[7,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0069","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[6,[6,7,8],[[1,4],[1,2],[1,4]]],[7,[6,7,8],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[6, 7, 1, 1/2, 3/2, 3/2, 是], [7, 7, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[6, 7, 1, 1/2, 3/2, 3/2, 是], [7, 7, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153069,"parameters":{"laws":[[6,[6,7,8],[[1,4],[1,2],[1,4]]],[7,[6,7,8],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[6,1],[[6,1],[7,1],[8,1]],[[1,4],[1,2],[1,4]]],[[7,1],[[6,1],[7,1],[8,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0070","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[7,[7,9,11],[[1,4],[1,2],[1,4]]],[8,[7,8,9],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[7, 9, 2, 2, 6, 6, 是], [8, 8, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[7, 9, 2, 2, 6, 6, 是], [8, 8, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153070,"parameters":{"laws":[[7,[7,9,11],[[1,4],[1,2],[1,4]]],[8,[7,8,9],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[7,1],[[11,1],[7,1],[9,1]],[[1,4],[1,4],[1,2]]],[[8,1],[[7,1],[8,1],[9,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0071","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[8,[8,11,14],[[1,4],[1,2],[1,4]]],[9,[8,9,10],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[8, 11, 3, 9/2, 27/2, 27/2, 是], [9, 9, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[8, 11, 3, 9/2, 27/2, 27/2, 是], [9, 9, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153071,"parameters":{"laws":[[8,[8,11,14],[[1,4],[1,2],[1,4]]],[9,[8,9,10],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[8,1],[[11,1],[14,1],[8,1]],[[1,2],[1,4],[1,4]]],[[9,1],[[10,1],[8,1],[9,1]],[[1,4],[1,4],[1,2]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0072","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[9,[9,10,11],[[1,4],[1,2],[1,4]]],[10,[9,10,11],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[10, 10, 0, 1/2, 1/2, 1/2, 是], [9, 10, 1, 1/2, 3/2, 3/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[10, 10, 0, 1/2, 1/2, 1/2, 是], [9, 10, 1, 1/2, 3/2, 3/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153072,"parameters":{"laws":[[9,[9,10,11],[[1,4],[1,2],[1,4]]],[10,[9,10,11],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[10,1],[[10,1],[11,1],[9,1]],[[1,2],[1,4],[1,4]]],[[9,1],[[10,1],[11,1],[9,1]],[[1,2],[1,4],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0073","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[10,[10,12,14],[[1,4],[1,2],[1,4]]],[11,[10,11,12],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[10, 12, 2, 2, 6, 6, 是], [11, 11, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[10, 12, 2, 2, 6, 6, 是], [11, 11, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153073,"parameters":{"laws":[[10,[10,12,14],[[1,4],[1,2],[1,4]]],[11,[10,11,12],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[10,1],[[10,1],[12,1],[14,1]],[[1,4],[1,2],[1,4]]],[[11,1],[[10,1],[11,1],[12,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0074","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[11,[11,14,17],[[1,4],[1,2],[1,4]]],[12,[11,12,13],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[11, 14, 3, 9/2, 27/2, 27/2, 是], [12, 12, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[11, 14, 3, 9/2, 27/2, 27/2, 是], [12, 12, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153074,"parameters":{"laws":[[11,[11,14,17],[[1,4],[1,2],[1,4]]],[12,[11,12,13],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[11,1],[[11,1],[14,1],[17,1]],[[1,4],[1,2],[1,4]]],[[12,1],[[11,1],[12,1],[13,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0075","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={\"laws\":[[12,[12,13,14],[[1,4],[1,2],[1,4]]],[13,[12,13,14],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[12, 13, 1, 1/2, 3/2, 3/2, 是], [13, 13, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)","explanation_zh":"Wave6第九層核對=([[12, 13, 1, 1/2, 3/2, 3/2, 是], [13, 13, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-multi-theta-table","seed":202611153075,"parameters":{"laws":[[12,[12,13,14],[[1,4],[1,2],[1,4]]],[13,[12,13,14],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-multi-theta-table","semantic_template":"qf-mod46-biasvar-multi-theta-table/{\"laws\":[[[12,1],[[12,1],[13,1],[14,1]],[[1,4],[1,2],[1,4]]],[[13,1],[[12,1],[13,1],[14,1]],[[1,4],[1,2],[1,4]]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0076","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-12,\"support\":[-13,-12,-11],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, -13, 1, 0, 是, 1/2, 1/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(-12, -13, 1, 0, 是, 1/2, 1/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153076,"parameters":{"theta":-12,"support":[-13,-12,-11],"probabilities":[[1,4],[1,2],[1,4]],"observed":-13,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-11,1],[-12,1],[-13,1]],\"theta\":[-12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0077","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-11,\"support\":[-11,-9,-7],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, -11, 0, 2, 否, 2, 6, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(-11, -11, 0, 2, 否, 2, 6, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153077,"parameters":{"theta":-11,"support":[-11,-9,-7],"probabilities":[[1,4],[1,2],[1,4]],"observed":-11,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-11,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-11,1],[-7,1],[-9,1]],\"theta\":[-11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0078","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-10,\"support\":[-13,-10,-7],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, -13, 9, 0, 是, 9/2, 9/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(-10, -13, 9, 0, 是, 9/2, 9/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153078,"parameters":{"theta":-10,"support":[-13,-10,-7],"probabilities":[[1,4],[1,2],[1,4]],"observed":-13,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-10,1],[-13,1],[-7,1]],\"theta\":[-10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0079","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-9,\"support\":[-9,-8,-7],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, -9, 0, 1, 否, 1/2, 3/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(-9, -9, 0, 1, 否, 1/2, 3/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153079,"parameters":{"theta":-9,"support":[-9,-8,-7],"probabilities":[[1,4],[1,2],[1,4]],"observed":-9,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-9,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-7,1],[-8,1],[-9,1]],\"theta\":[-9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0080","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-8,\"support\":[-10,-8,-6],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, -10, 4, 0, 是, 2, 2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(-8, -10, 4, 0, 是, 2, 2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153080,"parameters":{"theta":-8,"support":[-10,-8,-6],"probabilities":[[1,4],[1,2],[1,4]],"observed":-10,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-10,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-10,1],[-6,1],[-8,1]],\"theta\":[-8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0081","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-7,\"support\":[-7,-4,-1],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, -7, 0, 3, 否, 9/2, 27/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(-7, -7, 0, 3, 否, 9/2, 27/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153081,"parameters":{"theta":-7,"support":[-7,-4,-1],"probabilities":[[1,4],[1,2],[1,4]],"observed":-7,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-7,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[-4,1],[-7,1]],\"theta\":[-7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0082","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-6,\"support\":[-7,-6,-5],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, -7, 1, 0, 是, 1/2, 1/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(-6, -7, 1, 0, 是, 1/2, 1/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153082,"parameters":{"theta":-6,"support":[-7,-6,-5],"probabilities":[[1,4],[1,2],[1,4]],"observed":-7,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-7,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-5,1],[-6,1],[-7,1]],\"theta\":[-6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0083","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-5,\"support\":[-5,-3,-1],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, -5, 0, 2, 否, 2, 6, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(-5, -5, 0, 2, 否, 2, 6, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153083,"parameters":{"theta":-5,"support":[-5,-3,-1],"probabilities":[[1,4],[1,2],[1,4]],"observed":-5,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-5,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[-3,1],[-5,1]],\"theta\":[-5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0084","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-4,\"support\":[-7,-4,-1],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, -7, 9, 0, 是, 9/2, 9/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(-4, -7, 9, 0, 是, 9/2, 9/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153084,"parameters":{"theta":-4,"support":[-7,-4,-1],"probabilities":[[1,4],[1,2],[1,4]],"observed":-7,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-7,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[-4,1],[-7,1]],\"theta\":[-4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0085","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-3,\"support\":[-3,-2,-1],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, -3, 0, 1, 否, 1/2, 3/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(-3, -3, 0, 1, 否, 1/2, 3/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153085,"parameters":{"theta":-3,"support":[-3,-2,-1],"probabilities":[[1,4],[1,2],[1,4]],"observed":-3,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-3,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[-2,1],[-3,1]],\"theta\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0086","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-2,\"support\":[-4,-2,0],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, -4, 4, 0, 是, 2, 2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(-2, -4, 4, 0, 是, 2, 2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153086,"parameters":{"theta":-2,"support":[-4,-2,0],"probabilities":[[1,4],[1,2],[1,4]],"observed":-4,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-4,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-2,1],[-4,1],[0,1]],\"theta\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0087","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":-1,\"support\":[-1,2,5],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, -1, 0, 3, 否, 9/2, 27/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(-1, -1, 0, 3, 否, 9/2, 27/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153087,"parameters":{"theta":-1,"support":[-1,2,5],"probabilities":[[1,4],[1,2],[1,4]],"observed":-1,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-1,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[2,1],[5,1]],\"theta\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0088","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":0,\"support\":[-1,0,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, -1, 1, 0, 是, 1/2, 1/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(0, -1, 1, 0, 是, 1/2, 1/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153088,"parameters":{"theta":0,"support":[-1,0,1],"probabilities":[[1,4],[1,2],[1,4]],"observed":-1,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-1,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[0,1],[1,1]],\"theta\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0089","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":1,\"support\":[1,3,5],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 0, 2, 否, 2, 6, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(1, 1, 0, 2, 否, 2, 6, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153089,"parameters":{"theta":1,"support":[1,3,5],"probabilities":[[1,4],[1,2],[1,4]],"observed":1,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[1,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[1,1],[3,1],[5,1]],\"theta\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0090","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":2,\"support\":[-1,2,5],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, -1, 9, 0, 是, 9/2, 9/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(2, -1, 9, 0, 是, 9/2, 9/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153090,"parameters":{"theta":2,"support":[-1,2,5],"probabilities":[[1,4],[1,2],[1,4]],"observed":-1,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[-1,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[2,1],[5,1]],\"theta\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0091","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":3,\"support\":[3,4,5],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 0, 1, 否, 1/2, 3/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(3, 3, 0, 1, 否, 1/2, 3/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153091,"parameters":{"theta":3,"support":[3,4,5],"probabilities":[[1,4],[1,2],[1,4]],"observed":3,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[3,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[3,1],[4,1],[5,1]],\"theta\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0092","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":4,\"support\":[2,4,6],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 4, 0, 是, 2, 2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(4, 2, 4, 0, 是, 2, 2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153092,"parameters":{"theta":4,"support":[2,4,6],"probabilities":[[1,4],[1,2],[1,4]],"observed":2,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[2,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[4,1],[6,1]],\"theta\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0093","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":5,\"support\":[5,8,11],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 5, 0, 3, 否, 9/2, 27/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(5, 5, 0, 3, 否, 9/2, 27/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153093,"parameters":{"theta":5,"support":[5,8,11],"probabilities":[[1,4],[1,2],[1,4]],"observed":5,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[5,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[11,1],[5,1],[8,1]],\"theta\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0094","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":6,\"support\":[5,6,7],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 5, 1, 0, 是, 1/2, 1/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(6, 5, 1, 0, 是, 1/2, 1/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153094,"parameters":{"theta":6,"support":[5,6,7],"probabilities":[[1,4],[1,2],[1,4]],"observed":5,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[5,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[5,1],[6,1],[7,1]],\"theta\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0095","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":7,\"support\":[7,9,11],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 7, 0, 2, 否, 2, 6, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(7, 7, 0, 2, 否, 2, 6, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153095,"parameters":{"theta":7,"support":[7,9,11],"probabilities":[[1,4],[1,2],[1,4]],"observed":7,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[7,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[11,1],[7,1],[9,1]],\"theta\":[7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0096","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":8,\"support\":[5,8,11],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 5, 9, 0, 是, 9/2, 9/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(8, 5, 9, 0, 是, 9/2, 9/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153096,"parameters":{"theta":8,"support":[5,8,11],"probabilities":[[1,4],[1,2],[1,4]],"observed":5,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[5,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[11,1],[5,1],[8,1]],\"theta\":[8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0097","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":9,\"support\":[9,10,11],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 9, 0, 1, 否, 1/2, 3/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(9, 9, 0, 1, 否, 1/2, 3/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153097,"parameters":{"theta":9,"support":[9,10,11],"probabilities":[[1,4],[1,2],[1,4]],"observed":9,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[9,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[10,1],[11,1],[9,1]],\"theta\":[9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0098","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":10,\"support\":[8,10,12],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 8, 4, 0, 是, 2, 2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(10, 8, 4, 0, 是, 2, 2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153098,"parameters":{"theta":10,"support":[8,10,12],"probabilities":[[1,4],[1,2],[1,4]],"observed":8,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[8,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[10,1],[12,1],[8,1]],\"theta\":[10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0099","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":11,\"support\":[11,14,17],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 11, 0, 3, 否, 9/2, 27/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(11, 11, 0, 3, 否, 9/2, 27/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153099,"parameters":{"theta":11,"support":[11,14,17],"probabilities":[[1,4],[1,2],[1,4]],"observed":11,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[11,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[11,1],[14,1],[17,1]],\"theta\":[11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、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-20261115-r03-bias-variance-decomposition-0100","batch_id":"batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations","language":"zh-Hant","source_mko_ids":["mko-bias-variance-decomposition"],"title_zh":"偏誤－變異數分解","stem_zh":"逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-biased-unbiased-boundary；資料={\"theta\":12,\"support\":[11,12,13],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 11, 1, 0, 是, 1/2, 1/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)","explanation_zh":"Wave6第九層核對=(12, 11, 1, 0, 是, 1/2, 1/2, 是, 否, 否, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)；依完整明示資料獨立重算；cross term以E[T-E[T]]=0消去並直接核MSE=variance+bias^2；observed error不是MSE；unbiased只給MSE=variance，不推出zero MSE或consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["偏誤－變異數分解"]},"variant":{"family_id":"qf-mod46-biasvar-biased-unbiased-boundary","seed":202611153100,"parameters":{"theta":12,"support":[11,12,13],"probabilities":[[1,4],[1,2],[1,4]],"observed":11,"domain":"Exact_finite_bias_variance_decomposition","operation":"biasvar-biased-unbiased-boundary","semantic_template":"qf-mod46-biasvar-biased-unbiased-boundary/{\"observed\":[11,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[11,1],[12,1],[13,1]],\"theta\":[12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}