{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0001","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":-4,\"intercept\":1,\"slope\":-3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-3, 1, -4, 1/3, 13, 13/3, 13, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(-3, 1, -4, 1/3, 13, 13/3, 13, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195001,"parameters":{"slope":-3,"intercept":1,"approximation":-4,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[-4,1],\"intercept\":[1,1],\"slope\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0002","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":-3,\"intercept\":2,\"slope\":-2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-2, 2, -3, 1, 8, 4, 8, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(-2, 2, -3, 1, 8, 4, 8, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195002,"parameters":{"slope":-2,"intercept":2,"approximation":-3,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[-3,1],\"intercept\":[2,1],\"slope\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0003","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":-2,\"intercept\":3,\"slope\":-1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-1, 3, -2, 3, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(-1, 3, -2, 3, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195003,"parameters":{"slope":-1,"intercept":3,"approximation":-2,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[-2,1],\"intercept\":[3,1],\"slope\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0004","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":-1,\"intercept\":4,\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 4, -1, -2, 2, 1, 2, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(2, 4, -1, -2, 2, 1, 2, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195004,"parameters":{"slope":2,"intercept":4,"approximation":-1,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[-1,1],\"intercept\":[4,1],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0005","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":0,\"intercept\":5,\"slope\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(1, 5, 0, -5, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(1, 5, 0, -5, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195005,"parameters":{"slope":1,"intercept":5,"approximation":0,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[0,1],\"intercept\":[5,1],\"slope\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0006","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":1,\"intercept\":6,\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 6, 1, -3, 8, 4, 8, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(2, 6, 1, -3, 8, 4, 8, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195006,"parameters":{"slope":2,"intercept":6,"approximation":1,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[1,1],\"intercept\":[6,1],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0007","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":2,\"intercept\":7,\"slope\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 7, 2, -7/3, 13, 13/3, 13, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(3, 7, 2, -7/3, 13, 13/3, 13, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195007,"parameters":{"slope":3,"intercept":7,"approximation":2,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[2,1],\"intercept\":[7,1],\"slope\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0008","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":3,\"intercept\":8,\"slope\":-3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-3, 8, 3, 8/3, 1, 1/3, 1, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(-3, 8, 3, 8/3, 1, 1/3, 1, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195008,"parameters":{"slope":-3,"intercept":8,"approximation":3,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[3,1],\"intercept\":[8,1],\"slope\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0009","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":4,\"intercept\":9,\"slope\":-2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-2, 9, 4, 9/2, 1, 1/2, 1, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(-2, 9, 4, 9/2, 1, 1/2, 1, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195009,"parameters":{"slope":-2,"intercept":9,"approximation":4,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[4,1],\"intercept\":[9,1],\"slope\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0010","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":5,\"intercept\":10,\"slope\":-1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-1, 10, 5, 10, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(-1, 10, 5, 10, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195010,"parameters":{"slope":-1,"intercept":10,"approximation":5,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[5,1],\"intercept\":[10,1],\"slope\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0011","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":6,\"intercept\":11,\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 11, 6, -11/2, 23, 23/2, 23, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(2, 11, 6, -11/2, 23, 23/2, 23, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195011,"parameters":{"slope":2,"intercept":11,"approximation":6,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[6,1],\"intercept\":[11,1],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0012","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":7,\"intercept\":12,\"slope\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(1, 12, 7, -12, 19, 19, 19, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(1, 12, 7, -12, 19, 19, 19, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195012,"parameters":{"slope":1,"intercept":12,"approximation":7,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[7,1],\"intercept\":[12,1],\"slope\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0013","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":8,\"intercept\":13,\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 13, 8, -13/2, 29, 29/2, 29, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(2, 13, 8, -13/2, 29, 29/2, 29, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195013,"parameters":{"slope":2,"intercept":13,"approximation":8,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[8,1],\"intercept\":[13,1],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0014","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":9,\"intercept\":14,\"slope\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 14, 9, -14/3, 41, 41/3, 41, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(3, 14, 9, -14/3, 41, 41/3, 41, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195014,"parameters":{"slope":3,"intercept":14,"approximation":9,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[9,1],\"intercept\":[14,1],\"slope\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0015","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":10,\"intercept\":15,\"slope\":-3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-3, 15, 10, 5, 15, 5, 15, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(-3, 15, 10, 5, 15, 5, 15, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195015,"parameters":{"slope":-3,"intercept":15,"approximation":10,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[10,1],\"intercept\":[15,1],\"slope\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0016","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":11,\"intercept\":16,\"slope\":-2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-2, 16, 11, 8, 6, 3, 6, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(-2, 16, 11, 8, 6, 3, 6, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195016,"parameters":{"slope":-2,"intercept":16,"approximation":11,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[11,1],\"intercept\":[16,1],\"slope\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0017","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":12,\"intercept\":17,\"slope\":-1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-1, 17, 12, 17, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(-1, 17, 12, 17, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195017,"parameters":{"slope":-1,"intercept":17,"approximation":12,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[12,1],\"intercept\":[17,1],\"slope\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0018","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":13,\"intercept\":18,\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 18, 13, -9, 44, 22, 44, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(2, 18, 13, -9, 44, 22, 44, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195018,"parameters":{"slope":2,"intercept":18,"approximation":13,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[13,1],\"intercept\":[18,1],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0019","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":14,\"intercept\":19,\"slope\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(1, 19, 14, -19, 33, 33, 33, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(1, 19, 14, -19, 33, 33, 33, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195019,"parameters":{"slope":1,"intercept":19,"approximation":14,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[14,1],\"intercept\":[19,1],\"slope\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0020","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":15,\"intercept\":20,\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 20, 15, -10, 50, 25, 50, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(2, 20, 15, -10, 50, 25, 50, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195020,"parameters":{"slope":2,"intercept":20,"approximation":15,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[15,1],\"intercept\":[20,1],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0021","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":16,\"intercept\":21,\"slope\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 21, 16, -7, 69, 23, 69, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(3, 21, 16, -7, 69, 23, 69, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195021,"parameters":{"slope":3,"intercept":21,"approximation":16,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[16,1],\"intercept\":[21,1],\"slope\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0022","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":17,\"intercept\":22,\"slope\":-3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-3, 22, 17, 22/3, 29, 29/3, 29, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(-3, 22, 17, 22/3, 29, 29/3, 29, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195022,"parameters":{"slope":-3,"intercept":22,"approximation":17,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[17,1],\"intercept\":[22,1],\"slope\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0023","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":18,\"intercept\":23,\"slope\":-2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-2, 23, 18, 23/2, 13, 13/2, 13, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(-2, 23, 18, 23/2, 13, 13/2, 13, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195023,"parameters":{"slope":-2,"intercept":23,"approximation":18,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[18,1],\"intercept\":[23,1],\"slope\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0024","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":19,\"intercept\":24,\"slope\":-1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-1, 24, 19, 24, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(-1, 24, 19, 24, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195024,"parameters":{"slope":-1,"intercept":24,"approximation":19,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[19,1],\"intercept\":[24,1],\"slope\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0025","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={\"approximation\":20,\"intercept\":25,\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 25, 20, -25/2, 65, 65/2, 65, 是, residual_and_forward_error_are_distinct_quantities)","explanation_zh":"Wave6第十三層核對=(2, 25, 20, -25/2, 65, 65/2, 65, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195025,"parameters":{"slope":2,"intercept":25,"approximation":20,"domain":"Exact_numerical_error_density","operation":"numerr-affine-residual-forward","semantic_template":"numerr-affine-residual-forward|{\"approximation\":[20,1],\"intercept\":[25,1],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0026","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[1,3],\"slope\":-3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-3, 1/3, 1, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(-3, 1/3, 1, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195026,"parameters":{"slope":-3,"input_error_bound":[1,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[1,3],\"slope\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0027","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[2,3],\"slope\":-2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-2, 2/3, 4/3, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(-2, 2/3, 4/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195027,"parameters":{"slope":-2,"input_error_bound":[2,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[2,3],\"slope\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0028","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[3,3],\"slope\":-1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-1, 1, 1, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(-1, 1, 1, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195028,"parameters":{"slope":-1,"input_error_bound":[3,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[1,1],\"slope\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0029","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[4,3],\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 4/3, 8/3, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(2, 4/3, 8/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195029,"parameters":{"slope":2,"input_error_bound":[4,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[4,3],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0030","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[5,3],\"slope\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(1, 5/3, 5/3, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(1, 5/3, 5/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195030,"parameters":{"slope":1,"input_error_bound":[5,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[5,3],\"slope\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0031","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[6,3],\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 2, 4, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(2, 2, 4, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195031,"parameters":{"slope":2,"input_error_bound":[6,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[2,1],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0032","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[7,3],\"slope\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 7/3, 7, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(3, 7/3, 7, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195032,"parameters":{"slope":3,"input_error_bound":[7,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[7,3],\"slope\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0033","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[8,3],\"slope\":-3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-3, 8/3, 8, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(-3, 8/3, 8, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195033,"parameters":{"slope":-3,"input_error_bound":[8,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[8,3],\"slope\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0034","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[9,3],\"slope\":-2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-2, 3, 6, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(-2, 3, 6, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195034,"parameters":{"slope":-2,"input_error_bound":[9,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[3,1],\"slope\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0035","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[10,3],\"slope\":-1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-1, 10/3, 10/3, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(-1, 10/3, 10/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195035,"parameters":{"slope":-1,"input_error_bound":[10,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[10,3],\"slope\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0036","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[11,3],\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 11/3, 22/3, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(2, 11/3, 22/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195036,"parameters":{"slope":2,"input_error_bound":[11,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[11,3],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0037","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[12,3],\"slope\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(1, 4, 4, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(1, 4, 4, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195037,"parameters":{"slope":1,"input_error_bound":[12,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[4,1],\"slope\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0038","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[13,3],\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 13/3, 26/3, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(2, 13/3, 26/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195038,"parameters":{"slope":2,"input_error_bound":[13,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[13,3],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0039","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[14,3],\"slope\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 14/3, 14, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(3, 14/3, 14, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195039,"parameters":{"slope":3,"input_error_bound":[14,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[14,3],\"slope\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0040","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[15,3],\"slope\":-3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-3, 5, 15, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(-3, 5, 15, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195040,"parameters":{"slope":-3,"input_error_bound":[15,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[5,1],\"slope\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0041","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[16,3],\"slope\":-2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-2, 16/3, 32/3, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(-2, 16/3, 32/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195041,"parameters":{"slope":-2,"input_error_bound":[16,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[16,3],\"slope\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0042","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[17,3],\"slope\":-1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-1, 17/3, 17/3, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(-1, 17/3, 17/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195042,"parameters":{"slope":-1,"input_error_bound":[17,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[17,3],\"slope\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0043","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[18,3],\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 6, 12, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(2, 6, 12, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195043,"parameters":{"slope":2,"input_error_bound":[18,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[6,1],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0044","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[19,3],\"slope\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(1, 19/3, 19/3, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(1, 19/3, 19/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195044,"parameters":{"slope":1,"input_error_bound":[19,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[19,3],\"slope\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0045","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[20,3],\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 20/3, 40/3, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(2, 20/3, 40/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195045,"parameters":{"slope":2,"input_error_bound":[20,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[20,3],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0046","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[21,3],\"slope\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 7, 21, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(3, 7, 21, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195046,"parameters":{"slope":3,"input_error_bound":[21,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[7,1],\"slope\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0047","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[22,3],\"slope\":-3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-3, 22/3, 22, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(-3, 22/3, 22, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195047,"parameters":{"slope":-3,"input_error_bound":[22,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[22,3],\"slope\":[-3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0048","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[23,3],\"slope\":-2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-2, 23/3, 46/3, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(-2, 23/3, 46/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195048,"parameters":{"slope":-2,"input_error_bound":[23,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[23,3],\"slope\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0049","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[24,3],\"slope\":-1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(-1, 8, 8, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(-1, 8, 8, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195049,"parameters":{"slope":-1,"input_error_bound":[24,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[8,1],\"slope\":[-1,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0050","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={\"input_error_bound\":[25,3],\"slope\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 25/3, 50/3, affine_output_absolute_error_bound)","explanation_zh":"Wave6第十三層核對=(2, 25/3, 50/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-affine-propagation","seed":202611195050,"parameters":{"slope":2,"input_error_bound":[25,3],"domain":"Exact_numerical_error_density","operation":"numerr-affine-propagation","semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[25,3],\"slope\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0051","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":1,\"slope_a\":1,\"slope_b\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(1, 1, 1, 2, 1/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(1, 1, 1, 2, 1/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195051,"parameters":{"residual":1,"slope_a":1,"slope_b":2,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[1,1],\"slope_a\":[1,1],\"slope_b\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0052","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":2,\"slope_a\":1,\"slope_b\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 1, 2, 3, 2/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(2, 1, 2, 3, 2/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195052,"parameters":{"residual":2,"slope_a":1,"slope_b":3,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[2,1],\"slope_a\":[1,1],\"slope_b\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0053","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":3,\"slope_a\":1,\"slope_b\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 1, 3, 4, 3/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(3, 1, 3, 4, 3/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195053,"parameters":{"residual":3,"slope_a":1,"slope_b":4,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[3,1],\"slope_a\":[1,1],\"slope_b\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0054","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":4,\"slope_a\":1,\"slope_b\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(4, 1, 4, 5, 4/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(4, 1, 4, 5, 4/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195054,"parameters":{"residual":4,"slope_a":1,"slope_b":5,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[4,1],\"slope_a\":[1,1],\"slope_b\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0055","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":5,\"slope_a\":1,\"slope_b\":6}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(5, 1, 5, 6, 5/6, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(5, 1, 5, 6, 5/6, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195055,"parameters":{"residual":5,"slope_a":1,"slope_b":6,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[5,1],\"slope_a\":[1,1],\"slope_b\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0056","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":6,\"slope_a\":1,\"slope_b\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(6, 1, 6, 2, 3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(6, 1, 6, 2, 3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195056,"parameters":{"residual":6,"slope_a":1,"slope_b":2,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[6,1],\"slope_a\":[1,1],\"slope_b\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0057","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":7,\"slope_a\":1,\"slope_b\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(7, 1, 7, 3, 7/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(7, 1, 7, 3, 7/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195057,"parameters":{"residual":7,"slope_a":1,"slope_b":3,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[7,1],\"slope_a\":[1,1],\"slope_b\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0058","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":8,\"slope_a\":1,\"slope_b\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(8, 1, 8, 4, 2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(8, 1, 8, 4, 2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195058,"parameters":{"residual":8,"slope_a":1,"slope_b":4,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[8,1],\"slope_a\":[1,1],\"slope_b\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0059","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":9,\"slope_a\":1,\"slope_b\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(9, 1, 9, 5, 9/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(9, 1, 9, 5, 9/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195059,"parameters":{"residual":9,"slope_a":1,"slope_b":5,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[9,1],\"slope_a\":[1,1],\"slope_b\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0060","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":10,\"slope_a\":1,\"slope_b\":6}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(10, 1, 10, 6, 5/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(10, 1, 10, 6, 5/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195060,"parameters":{"residual":10,"slope_a":1,"slope_b":6,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[10,1],\"slope_a\":[1,1],\"slope_b\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0061","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":11,\"slope_a\":1,\"slope_b\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(11, 1, 11, 2, 11/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(11, 1, 11, 2, 11/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195061,"parameters":{"residual":11,"slope_a":1,"slope_b":2,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[11,1],\"slope_a\":[1,1],\"slope_b\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0062","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":12,\"slope_a\":1,\"slope_b\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(12, 1, 12, 3, 4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(12, 1, 12, 3, 4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195062,"parameters":{"residual":12,"slope_a":1,"slope_b":3,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[12,1],\"slope_a\":[1,1],\"slope_b\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0063","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":13,\"slope_a\":1,\"slope_b\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(13, 1, 13, 4, 13/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(13, 1, 13, 4, 13/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195063,"parameters":{"residual":13,"slope_a":1,"slope_b":4,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[13,1],\"slope_a\":[1,1],\"slope_b\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0064","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":14,\"slope_a\":1,\"slope_b\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(14, 1, 14, 5, 14/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(14, 1, 14, 5, 14/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195064,"parameters":{"residual":14,"slope_a":1,"slope_b":5,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[14,1],\"slope_a\":[1,1],\"slope_b\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0065","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":15,\"slope_a\":1,\"slope_b\":6}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(15, 1, 15, 6, 5/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(15, 1, 15, 6, 5/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195065,"parameters":{"residual":15,"slope_a":1,"slope_b":6,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[15,1],\"slope_a\":[1,1],\"slope_b\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0066","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":16,\"slope_a\":1,\"slope_b\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(16, 1, 16, 2, 8, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(16, 1, 16, 2, 8, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195066,"parameters":{"residual":16,"slope_a":1,"slope_b":2,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[16,1],\"slope_a\":[1,1],\"slope_b\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0067","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":17,\"slope_a\":1,\"slope_b\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(17, 1, 17, 3, 17/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(17, 1, 17, 3, 17/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195067,"parameters":{"residual":17,"slope_a":1,"slope_b":3,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[17,1],\"slope_a\":[1,1],\"slope_b\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0068","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":18,\"slope_a\":1,\"slope_b\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(18, 1, 18, 4, 9/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(18, 1, 18, 4, 9/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195068,"parameters":{"residual":18,"slope_a":1,"slope_b":4,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[18,1],\"slope_a\":[1,1],\"slope_b\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0069","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":19,\"slope_a\":1,\"slope_b\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(19, 1, 19, 5, 19/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(19, 1, 19, 5, 19/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195069,"parameters":{"residual":19,"slope_a":1,"slope_b":5,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[19,1],\"slope_a\":[1,1],\"slope_b\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0070","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":20,\"slope_a\":1,\"slope_b\":6}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(20, 1, 20, 6, 10/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(20, 1, 20, 6, 10/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195070,"parameters":{"residual":20,"slope_a":1,"slope_b":6,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[20,1],\"slope_a\":[1,1],\"slope_b\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0071","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":21,\"slope_a\":1,\"slope_b\":2}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(21, 1, 21, 2, 21/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(21, 1, 21, 2, 21/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195071,"parameters":{"residual":21,"slope_a":1,"slope_b":2,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[21,1],\"slope_a\":[1,1],\"slope_b\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0072","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":22,\"slope_a\":1,\"slope_b\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(22, 1, 22, 3, 22/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(22, 1, 22, 3, 22/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195072,"parameters":{"residual":22,"slope_a":1,"slope_b":3,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[22,1],\"slope_a\":[1,1],\"slope_b\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0073","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":23,\"slope_a\":1,\"slope_b\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(23, 1, 23, 4, 23/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(23, 1, 23, 4, 23/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195073,"parameters":{"residual":23,"slope_a":1,"slope_b":4,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[23,1],\"slope_a\":[1,1],\"slope_b\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0074","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":24,\"slope_a\":1,\"slope_b\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(24, 1, 24, 5, 24/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(24, 1, 24, 5, 24/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195074,"parameters":{"residual":24,"slope_a":1,"slope_b":5,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[24,1],\"slope_a\":[1,1],\"slope_b\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0075","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={\"residual\":25,\"slope_a\":1,\"slope_b\":6}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(25, 1, 25, 6, 25/6, 是, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation_zh":"Wave6第十三層核對=(25, 1, 25, 6, 25/6, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195075,"parameters":{"residual":25,"slope_a":1,"slope_b":6,"domain":"Exact_numerical_error_density","operation":"numerr-conditioning-witness","semantic_template":"numerr-conditioning-witness|{\"residual\":[25,1],\"slope_a\":[1,1],\"slope_b\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0076","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"zero_reference\",\"model\":{\"approximation\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_reference, 1, reference=0, relative_error=undefined, 否)","explanation_zh":"Wave6第十三層核對=(zero_reference, 1, reference=0, relative_error=undefined, 否)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195076,"parameters":{"mode":"zero_reference","model":{"approximation":1},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[1,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0077","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[2,2],\"tolerance\":[2,2]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(inclusive_tolerance, 1, 1, 是, error<=tolerance_is_true_at_equality)","explanation_zh":"Wave6第十三層核對=(inclusive_tolerance, 1, 1, 是, error<=tolerance_is_true_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195077,"parameters":{"mode":"inclusive_tolerance","model":{"error":[2,2],"tolerance":[2,2]},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[1,1],\"tolerance\":[1,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0078","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"strict_tolerance\",\"model\":{\"error\":[3,2],\"tolerance\":[3,2]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(strict_tolerance, 3/2, 3/2, 否, error<tolerance_is_false_at_equality)","explanation_zh":"Wave6第十三層核對=(strict_tolerance, 3/2, 3/2, 否, error<tolerance_is_false_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195078,"parameters":{"mode":"strict_tolerance","model":{"error":[3,2],"tolerance":[3,2]},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"strict_tolerance\",\"model\":{\"error\":[3,2],\"tolerance\":[3,2]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0079","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":4,\"reference\":-4}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(signed_vs_absolute, 4, -4, 8, 8, 否, signed_error_and_absolute_error_preserve_different_information)","explanation_zh":"Wave6第十三層核對=(signed_vs_absolute, 4, -4, 8, 8, 否, signed_error_and_absolute_error_preserve_different_information)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195079,"parameters":{"mode":"signed_vs_absolute","model":{"approximation":4,"reference":-4},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":[4,1],\"reference\":[-4,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0080","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"zero_reference\",\"model\":{\"approximation\":5}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_reference, 5, reference=0, relative_error=undefined, 否)","explanation_zh":"Wave6第十三層核對=(zero_reference, 5, reference=0, relative_error=undefined, 否)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195080,"parameters":{"mode":"zero_reference","model":{"approximation":5},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[5,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0081","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[6,2],\"tolerance\":[6,2]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(inclusive_tolerance, 3, 3, 是, error<=tolerance_is_true_at_equality)","explanation_zh":"Wave6第十三層核對=(inclusive_tolerance, 3, 3, 是, error<=tolerance_is_true_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195081,"parameters":{"mode":"inclusive_tolerance","model":{"error":[6,2],"tolerance":[6,2]},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[3,1],\"tolerance\":[3,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0082","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"strict_tolerance\",\"model\":{\"error\":[7,2],\"tolerance\":[7,2]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(strict_tolerance, 7/2, 7/2, 否, error<tolerance_is_false_at_equality)","explanation_zh":"Wave6第十三層核對=(strict_tolerance, 7/2, 7/2, 否, error<tolerance_is_false_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195082,"parameters":{"mode":"strict_tolerance","model":{"error":[7,2],"tolerance":[7,2]},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"strict_tolerance\",\"model\":{\"error\":[7,2],\"tolerance\":[7,2]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0083","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":8,\"reference\":-8}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(signed_vs_absolute, 8, -8, 16, 16, 否, signed_error_and_absolute_error_preserve_different_information)","explanation_zh":"Wave6第十三層核對=(signed_vs_absolute, 8, -8, 16, 16, 否, signed_error_and_absolute_error_preserve_different_information)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195083,"parameters":{"mode":"signed_vs_absolute","model":{"approximation":8,"reference":-8},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":[8,1],\"reference\":[-8,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0084","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"zero_reference\",\"model\":{\"approximation\":9}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_reference, 9, reference=0, relative_error=undefined, 否)","explanation_zh":"Wave6第十三層核對=(zero_reference, 9, reference=0, relative_error=undefined, 否)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195084,"parameters":{"mode":"zero_reference","model":{"approximation":9},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[9,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0085","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[10,2],\"tolerance\":[10,2]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(inclusive_tolerance, 5, 5, 是, error<=tolerance_is_true_at_equality)","explanation_zh":"Wave6第十三層核對=(inclusive_tolerance, 5, 5, 是, error<=tolerance_is_true_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195085,"parameters":{"mode":"inclusive_tolerance","model":{"error":[10,2],"tolerance":[10,2]},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[5,1],\"tolerance\":[5,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0086","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"strict_tolerance\",\"model\":{\"error\":[11,2],\"tolerance\":[11,2]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(strict_tolerance, 11/2, 11/2, 否, error<tolerance_is_false_at_equality)","explanation_zh":"Wave6第十三層核對=(strict_tolerance, 11/2, 11/2, 否, error<tolerance_is_false_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195086,"parameters":{"mode":"strict_tolerance","model":{"error":[11,2],"tolerance":[11,2]},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"strict_tolerance\",\"model\":{\"error\":[11,2],\"tolerance\":[11,2]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0087","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":12,\"reference\":-12}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(signed_vs_absolute, 12, -12, 24, 24, 否, signed_error_and_absolute_error_preserve_different_information)","explanation_zh":"Wave6第十三層核對=(signed_vs_absolute, 12, -12, 24, 24, 否, signed_error_and_absolute_error_preserve_different_information)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195087,"parameters":{"mode":"signed_vs_absolute","model":{"approximation":12,"reference":-12},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":[12,1],\"reference\":[-12,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0088","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"zero_reference\",\"model\":{\"approximation\":13}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_reference, 13, reference=0, relative_error=undefined, 否)","explanation_zh":"Wave6第十三層核對=(zero_reference, 13, reference=0, relative_error=undefined, 否)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195088,"parameters":{"mode":"zero_reference","model":{"approximation":13},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[13,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0089","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[14,2],\"tolerance\":[14,2]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(inclusive_tolerance, 7, 7, 是, error<=tolerance_is_true_at_equality)","explanation_zh":"Wave6第十三層核對=(inclusive_tolerance, 7, 7, 是, error<=tolerance_is_true_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195089,"parameters":{"mode":"inclusive_tolerance","model":{"error":[14,2],"tolerance":[14,2]},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[7,1],\"tolerance\":[7,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0090","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"strict_tolerance\",\"model\":{\"error\":[15,2],\"tolerance\":[15,2]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(strict_tolerance, 15/2, 15/2, 否, error<tolerance_is_false_at_equality)","explanation_zh":"Wave6第十三層核對=(strict_tolerance, 15/2, 15/2, 否, error<tolerance_is_false_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195090,"parameters":{"mode":"strict_tolerance","model":{"error":[15,2],"tolerance":[15,2]},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"strict_tolerance\",\"model\":{\"error\":[15,2],\"tolerance\":[15,2]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0091","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":16,\"reference\":-16}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(signed_vs_absolute, 16, -16, 32, 32, 否, signed_error_and_absolute_error_preserve_different_information)","explanation_zh":"Wave6第十三層核對=(signed_vs_absolute, 16, -16, 32, 32, 否, signed_error_and_absolute_error_preserve_different_information)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195091,"parameters":{"mode":"signed_vs_absolute","model":{"approximation":16,"reference":-16},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":[16,1],\"reference\":[-16,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0092","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"zero_reference\",\"model\":{\"approximation\":17}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_reference, 17, reference=0, relative_error=undefined, 否)","explanation_zh":"Wave6第十三層核對=(zero_reference, 17, reference=0, relative_error=undefined, 否)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195092,"parameters":{"mode":"zero_reference","model":{"approximation":17},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[17,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0093","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[18,2],\"tolerance\":[18,2]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(inclusive_tolerance, 9, 9, 是, error<=tolerance_is_true_at_equality)","explanation_zh":"Wave6第十三層核對=(inclusive_tolerance, 9, 9, 是, error<=tolerance_is_true_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195093,"parameters":{"mode":"inclusive_tolerance","model":{"error":[18,2],"tolerance":[18,2]},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[9,1],\"tolerance\":[9,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0094","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"strict_tolerance\",\"model\":{\"error\":[19,2],\"tolerance\":[19,2]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(strict_tolerance, 19/2, 19/2, 否, error<tolerance_is_false_at_equality)","explanation_zh":"Wave6第十三層核對=(strict_tolerance, 19/2, 19/2, 否, error<tolerance_is_false_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195094,"parameters":{"mode":"strict_tolerance","model":{"error":[19,2],"tolerance":[19,2]},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"strict_tolerance\",\"model\":{\"error\":[19,2],\"tolerance\":[19,2]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0095","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":20,\"reference\":-20}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(signed_vs_absolute, 20, -20, 40, 40, 否, signed_error_and_absolute_error_preserve_different_information)","explanation_zh":"Wave6第十三層核對=(signed_vs_absolute, 20, -20, 40, 40, 否, signed_error_and_absolute_error_preserve_different_information)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195095,"parameters":{"mode":"signed_vs_absolute","model":{"approximation":20,"reference":-20},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":[20,1],\"reference\":[-20,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0096","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"zero_reference\",\"model\":{\"approximation\":21}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_reference, 21, reference=0, relative_error=undefined, 否)","explanation_zh":"Wave6第十三層核對=(zero_reference, 21, reference=0, relative_error=undefined, 否)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195096,"parameters":{"mode":"zero_reference","model":{"approximation":21},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[21,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0097","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[22,2],\"tolerance\":[22,2]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(inclusive_tolerance, 11, 11, 是, error<=tolerance_is_true_at_equality)","explanation_zh":"Wave6第十三層核對=(inclusive_tolerance, 11, 11, 是, error<=tolerance_is_true_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195097,"parameters":{"mode":"inclusive_tolerance","model":{"error":[22,2],"tolerance":[22,2]},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[11,1],\"tolerance\":[11,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0098","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"strict_tolerance\",\"model\":{\"error\":[23,2],\"tolerance\":[23,2]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(strict_tolerance, 23/2, 23/2, 否, error<tolerance_is_false_at_equality)","explanation_zh":"Wave6第十三層核對=(strict_tolerance, 23/2, 23/2, 否, error<tolerance_is_false_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195098,"parameters":{"mode":"strict_tolerance","model":{"error":[23,2],"tolerance":[23,2]},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"strict_tolerance\",\"model\":{\"error\":[23,2],\"tolerance\":[23,2]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0099","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":24,\"reference\":-24}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(signed_vs_absolute, 24, -24, 48, 48, 否, signed_error_and_absolute_error_preserve_different_information)","explanation_zh":"Wave6第十三層核對=(signed_vs_absolute, 24, -24, 48, 48, 否, signed_error_and_absolute_error_preserve_different_information)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195099,"parameters":{"mode":"signed_vs_absolute","model":{"approximation":24,"reference":-24},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":[24,1],\"reference\":[-24,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r05-0100","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","language":"zh-Hant","source_mko_ids":["mko-numerical-approximation-error"],"title_zh":"數值近似誤差","stem_zh":"exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={\"mode\":\"zero_reference\",\"model\":{\"approximation\":25}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_reference, 25, reference=0, relative_error=undefined, 否)","explanation_zh":"Wave6第十三層核對=(zero_reference, 25, reference=0, relative_error=undefined, 否)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["數值近似誤差"]},"variant":{"family_id":"qf-mod50-numerr-boundary","seed":202611195100,"parameters":{"mode":"zero_reference","model":{"approximation":25},"domain":"Exact_numerical_error_density","operation":"numerr-boundary","semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[25,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}