{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0001","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":-3,\"base_intercept\":1,\"error_scale\":1,\"n\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 1, 1, 1, f_n(x)=-3x+1+1x/2, 1/2, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(-3, 1, 1, 1, f_n(x)=-3x+1+1x/2, 1/2, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173001,"parameters":{"base_slope":-3,"base_intercept":1,"error_scale":1,"n":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[1,1],base_slope:[-3,1],error_scale:[1,1],n:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0002","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":-2,\"base_intercept\":2,\"error_scale\":2,\"n\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 2, 2, 2, f_n(x)=-2x+2+2x/3, 2/3, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(-2, 2, 2, 2, f_n(x)=-2x+2+2x/3, 2/3, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173002,"parameters":{"base_slope":-2,"base_intercept":2,"error_scale":2,"n":2,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[2,1],base_slope:[-2,1],error_scale:[2,1],n:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0003","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":-1,\"base_intercept\":3,\"error_scale\":3,\"n\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 3, 3, 3, f_n(x)=-1x+3+3x/4, 3/4, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(-1, 3, 3, 3, f_n(x)=-1x+3+3x/4, 3/4, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173003,"parameters":{"base_slope":-1,"base_intercept":3,"error_scale":3,"n":3,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[3,1],base_slope:[-1,1],error_scale:[3,1],n:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0004","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":0,\"base_intercept\":4,\"error_scale\":4,\"n\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 4, 4, 4, f_n(x)=0x+4+4x/5, 4/5, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(0, 4, 4, 4, f_n(x)=0x+4+4x/5, 4/5, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173004,"parameters":{"base_slope":0,"base_intercept":4,"error_scale":4,"n":4,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[4,1],base_slope:[0,1],error_scale:[4,1],n:4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0005","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":1,\"base_intercept\":5,\"error_scale\":5,\"n\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 5, 5, 5, f_n(x)=1x+5+5x/6, 5/6, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(1, 5, 5, 5, f_n(x)=1x+5+5x/6, 5/6, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173005,"parameters":{"base_slope":1,"base_intercept":5,"error_scale":5,"n":5,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[5,1],base_slope:[1,1],error_scale:[5,1],n:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0006","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":2,\"base_intercept\":6,\"error_scale\":6,\"n\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 6, 6, f_n(x)=2x+6+6x/7, 6/7, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(2, 6, 6, 6, f_n(x)=2x+6+6x/7, 6/7, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173006,"parameters":{"base_slope":2,"base_intercept":6,"error_scale":6,"n":6,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[6,1],base_slope:[2,1],error_scale:[6,1],n:6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0007","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":3,\"base_intercept\":7,\"error_scale\":7,\"n\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 7, 7, 7, f_n(x)=3x+7+7x/8, 7/8, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(3, 7, 7, 7, f_n(x)=3x+7+7x/8, 7/8, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173007,"parameters":{"base_slope":3,"base_intercept":7,"error_scale":7,"n":7,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[7,1],base_slope:[3,1],error_scale:[7,1],n:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0008","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":-3,\"base_intercept\":8,\"error_scale\":8,\"n\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 8, 8, 8, f_n(x)=-3x+8+8x/9, 8/9, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(-3, 8, 8, 8, f_n(x)=-3x+8+8x/9, 8/9, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173008,"parameters":{"base_slope":-3,"base_intercept":8,"error_scale":8,"n":8,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[8,1],base_slope:[-3,1],error_scale:[8,1],n:8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0009","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":-2,\"base_intercept\":9,\"error_scale\":9,\"n\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 9, 9, 9, f_n(x)=-2x+9+9x/10, 9/10, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(-2, 9, 9, 9, f_n(x)=-2x+9+9x/10, 9/10, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173009,"parameters":{"base_slope":-2,"base_intercept":9,"error_scale":9,"n":9,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[9,1],base_slope:[-2,1],error_scale:[9,1],n:9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0010","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":-1,\"base_intercept\":10,\"error_scale\":10,\"n\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 10, 10, 10, f_n(x)=-1x+10+10x/11, 10/11, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(-1, 10, 10, 10, f_n(x)=-1x+10+10x/11, 10/11, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173010,"parameters":{"base_slope":-1,"base_intercept":10,"error_scale":10,"n":10,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[10,1],base_slope:[-1,1],error_scale:[10,1],n:10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0011","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":0,\"base_intercept\":11,\"error_scale\":11,\"n\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 11, 11, 11, f_n(x)=0x+11+11x/12, 11/12, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(0, 11, 11, 11, f_n(x)=0x+11+11x/12, 11/12, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173011,"parameters":{"base_slope":0,"base_intercept":11,"error_scale":11,"n":11,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[11,1],base_slope:[0,1],error_scale:[11,1],n:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0012","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":1,\"base_intercept\":12,\"error_scale\":12,\"n\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 12, 12, 12, f_n(x)=1x+12+12x/13, 12/13, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(1, 12, 12, 12, f_n(x)=1x+12+12x/13, 12/13, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173012,"parameters":{"base_slope":1,"base_intercept":12,"error_scale":12,"n":12,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[12,1],base_slope:[1,1],error_scale:[12,1],n:12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0013","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":2,\"base_intercept\":13,\"error_scale\":13,\"n\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 13, 13, 13, f_n(x)=2x+13+13x/14, 13/14, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(2, 13, 13, 13, f_n(x)=2x+13+13x/14, 13/14, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173013,"parameters":{"base_slope":2,"base_intercept":13,"error_scale":13,"n":13,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[13,1],base_slope:[2,1],error_scale:[13,1],n:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0014","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":3,\"base_intercept\":14,\"error_scale\":14,\"n\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 14, 14, 14, f_n(x)=3x+14+14x/15, 14/15, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(3, 14, 14, 14, f_n(x)=3x+14+14x/15, 14/15, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173014,"parameters":{"base_slope":3,"base_intercept":14,"error_scale":14,"n":14,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[14,1],base_slope:[3,1],error_scale:[14,1],n:14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0015","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":-3,\"base_intercept\":15,\"error_scale\":15,\"n\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 15, 15, 15, f_n(x)=-3x+15+15x/16, 15/16, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(-3, 15, 15, 15, f_n(x)=-3x+15+15x/16, 15/16, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173015,"parameters":{"base_slope":-3,"base_intercept":15,"error_scale":15,"n":15,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[15,1],base_slope:[-3,1],error_scale:[15,1],n:15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0016","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":-2,\"base_intercept\":16,\"error_scale\":16,\"n\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 16, 16, 16, f_n(x)=-2x+16+16x/17, 16/17, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(-2, 16, 16, 16, f_n(x)=-2x+16+16x/17, 16/17, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173016,"parameters":{"base_slope":-2,"base_intercept":16,"error_scale":16,"n":16,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[16,1],base_slope:[-2,1],error_scale:[16,1],n:16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0017","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":-1,\"base_intercept\":17,\"error_scale\":17,\"n\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 17, 17, 17, f_n(x)=-1x+17+17x/18, 17/18, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(-1, 17, 17, 17, f_n(x)=-1x+17+17x/18, 17/18, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173017,"parameters":{"base_slope":-1,"base_intercept":17,"error_scale":17,"n":17,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[17,1],base_slope:[-1,1],error_scale:[17,1],n:17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0018","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":0,\"base_intercept\":18,\"error_scale\":18,\"n\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 18, 18, 18, f_n(x)=0x+18+18x/19, 18/19, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(0, 18, 18, 18, f_n(x)=0x+18+18x/19, 18/19, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173018,"parameters":{"base_slope":0,"base_intercept":18,"error_scale":18,"n":18,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[18,1],base_slope:[0,1],error_scale:[18,1],n:18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0019","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":1,\"base_intercept\":19,\"error_scale\":19,\"n\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 19, 19, 19, f_n(x)=1x+19+19x/20, 19/20, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(1, 19, 19, 19, f_n(x)=1x+19+19x/20, 19/20, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173019,"parameters":{"base_slope":1,"base_intercept":19,"error_scale":19,"n":19,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[19,1],base_slope:[1,1],error_scale:[19,1],n:19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0020","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":2,\"base_intercept\":20,\"error_scale\":20,\"n\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 20, 20, 20, f_n(x)=2x+20+20x/21, 20/21, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(2, 20, 20, 20, f_n(x)=2x+20+20x/21, 20/21, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173020,"parameters":{"base_slope":2,"base_intercept":20,"error_scale":20,"n":20,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[20,1],base_slope:[2,1],error_scale:[20,1],n:20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0021","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":3,\"base_intercept\":21,\"error_scale\":21,\"n\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 21, 21, 21, f_n(x)=3x+21+21x/22, 21/22, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(3, 21, 21, 21, f_n(x)=3x+21+21x/22, 21/22, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173021,"parameters":{"base_slope":3,"base_intercept":21,"error_scale":21,"n":21,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[21,1],base_slope:[3,1],error_scale:[21,1],n:21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0022","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":-3,\"base_intercept\":22,\"error_scale\":22,\"n\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 22, 22, 22, f_n(x)=-3x+22+22x/23, 22/23, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(-3, 22, 22, 22, f_n(x)=-3x+22+22x/23, 22/23, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173022,"parameters":{"base_slope":-3,"base_intercept":22,"error_scale":22,"n":22,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[22,1],base_slope:[-3,1],error_scale:[22,1],n:22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0023","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":-2,\"base_intercept\":23,\"error_scale\":23,\"n\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 23, 23, 23, f_n(x)=-2x+23+23x/24, 23/24, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(-2, 23, 23, 23, f_n(x)=-2x+23+23x/24, 23/24, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173023,"parameters":{"base_slope":-2,"base_intercept":23,"error_scale":23,"n":23,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[23,1],base_slope:[-2,1],error_scale:[23,1],n:23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0024","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":-1,\"base_intercept\":24,\"error_scale\":24,\"n\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 24, 24, 24, f_n(x)=-1x+24+24x/25, 24/25, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(-1, 24, 24, 24, f_n(x)=-1x+24+24x/25, 24/25, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173024,"parameters":{"base_slope":-1,"base_intercept":24,"error_scale":24,"n":24,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[24,1],base_slope:[-1,1],error_scale:[24,1],n:24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0025","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-sup-certificate；資料={\"base_slope\":0,\"base_intercept\":25,\"error_scale\":25,\"n\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 25, 25, 25, f_n(x)=0x+25+25x/26, 25/26, sup_on_[0,1]_attained_at_x=1, 是)","explanation_zh":"Wave6第十一層核對=(0, 25, 25, 25, f_n(x)=0x+25+25x/26, 25/26, sup_on_[0,1]_attained_at_x=1, 是)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-sup-certificate","seed":202611173025,"parameters":{"base_slope":0,"base_intercept":25,"error_scale":25,"n":25,"domain":"Exact_uniform_limit_continuity","operation":"uniform-sup-certificate","semantic_template":"uniform-sup-certificate|{base_intercept:[25,1],base_slope:[0,1],error_scale:[25,1],n:25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0026","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":1,\"epsilon\":[1,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1/7, 7, 1/8, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(1, 1/7, 7, 1/8, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173026,"parameters":{"error_scale":1,"epsilon":[1,7],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,7],error_scale:[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0027","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":2,\"epsilon\":[2,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1/4, 8, 2/9, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(2, 1/4, 8, 2/9, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173027,"parameters":{"error_scale":2,"epsilon":[2,8],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,4],error_scale:[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0028","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":3,\"epsilon\":[3,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 1/3, 9, 3/10, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(3, 1/3, 9, 3/10, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173028,"parameters":{"error_scale":3,"epsilon":[3,9],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,3],error_scale:[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0029","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":4,\"epsilon\":[4,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 2/5, 10, 4/11, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(4, 2/5, 10, 4/11, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173029,"parameters":{"error_scale":4,"epsilon":[4,10],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[2,5],error_scale:[4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0030","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":5,\"epsilon\":[5,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 5/11, 11, 5/12, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(5, 5/11, 11, 5/12, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173030,"parameters":{"error_scale":5,"epsilon":[5,11],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[5,11],error_scale:[5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0031","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":6,\"epsilon\":[1,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 1/7, 42, 6/43, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(6, 1/7, 42, 6/43, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173031,"parameters":{"error_scale":6,"epsilon":[1,7],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,7],error_scale:[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0032","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":7,\"epsilon\":[2,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 1/4, 28, 7/29, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(7, 1/4, 28, 7/29, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173032,"parameters":{"error_scale":7,"epsilon":[2,8],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,4],error_scale:[7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0033","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":8,\"epsilon\":[3,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 1/3, 24, 8/25, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(8, 1/3, 24, 8/25, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173033,"parameters":{"error_scale":8,"epsilon":[3,9],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,3],error_scale:[8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0034","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":9,\"epsilon\":[4,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 2/5, 22, 9/23, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(9, 2/5, 22, 9/23, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173034,"parameters":{"error_scale":9,"epsilon":[4,10],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[2,5],error_scale:[9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0035","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":10,\"epsilon\":[5,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 5/11, 22, 10/23, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(10, 5/11, 22, 10/23, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173035,"parameters":{"error_scale":10,"epsilon":[5,11],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[5,11],error_scale:[10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0036","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":11,\"epsilon\":[1,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 1/7, 77, 11/78, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(11, 1/7, 77, 11/78, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173036,"parameters":{"error_scale":11,"epsilon":[1,7],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,7],error_scale:[11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0037","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":12,\"epsilon\":[2,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 1/4, 48, 12/49, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(12, 1/4, 48, 12/49, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173037,"parameters":{"error_scale":12,"epsilon":[2,8],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,4],error_scale:[12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0038","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":13,\"epsilon\":[3,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 1/3, 39, 13/40, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(13, 1/3, 39, 13/40, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173038,"parameters":{"error_scale":13,"epsilon":[3,9],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,3],error_scale:[13,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0039","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":14,\"epsilon\":[4,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 2/5, 35, 7/18, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(14, 2/5, 35, 7/18, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173039,"parameters":{"error_scale":14,"epsilon":[4,10],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[2,5],error_scale:[14,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0040","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":15,\"epsilon\":[5,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 5/11, 33, 15/34, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(15, 5/11, 33, 15/34, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173040,"parameters":{"error_scale":15,"epsilon":[5,11],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[5,11],error_scale:[15,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0041","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":16,\"epsilon\":[1,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 1/7, 112, 16/113, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(16, 1/7, 112, 16/113, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173041,"parameters":{"error_scale":16,"epsilon":[1,7],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,7],error_scale:[16,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0042","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":17,\"epsilon\":[2,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 1/4, 68, 17/69, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(17, 1/4, 68, 17/69, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173042,"parameters":{"error_scale":17,"epsilon":[2,8],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,4],error_scale:[17,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0043","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":18,\"epsilon\":[3,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 1/3, 54, 18/55, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(18, 1/3, 54, 18/55, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173043,"parameters":{"error_scale":18,"epsilon":[3,9],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,3],error_scale:[18,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0044","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":19,\"epsilon\":[4,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 2/5, 47, 19/48, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(19, 2/5, 47, 19/48, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173044,"parameters":{"error_scale":19,"epsilon":[4,10],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[2,5],error_scale:[19,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0045","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":20,\"epsilon\":[5,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 5/11, 44, 4/9, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(20, 5/11, 44, 4/9, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173045,"parameters":{"error_scale":20,"epsilon":[5,11],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[5,11],error_scale:[20,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0046","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":21,\"epsilon\":[1,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 1/7, 147, 21/148, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(21, 1/7, 147, 21/148, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173046,"parameters":{"error_scale":21,"epsilon":[1,7],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,7],error_scale:[21,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0047","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":22,\"epsilon\":[2,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 1/4, 88, 22/89, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(22, 1/4, 88, 22/89, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173047,"parameters":{"error_scale":22,"epsilon":[2,8],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,4],error_scale:[22,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0048","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":23,\"epsilon\":[3,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 1/3, 69, 23/70, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(23, 1/3, 69, 23/70, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173048,"parameters":{"error_scale":23,"epsilon":[3,9],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[1,3],error_scale:[23,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0049","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":24,\"epsilon\":[4,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 2/5, 60, 24/61, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(24, 2/5, 60, 24/61, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173049,"parameters":{"error_scale":24,"epsilon":[4,10],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[2,5],error_scale:[24,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0050","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-epsilon-index；資料={\"error_scale\":25,\"epsilon\":[5,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, 5/11, 55, 25/56, 是, same_N_controls_every_x_in_[0,1])","explanation_zh":"Wave6第十一層核對=(25, 5/11, 55, 25/56, 是, same_N_controls_every_x_in_[0,1])；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-epsilon-index","seed":202611173050,"parameters":{"error_scale":25,"epsilon":[5,11],"domain":"Exact_uniform_limit_continuity","operation":"uniform-epsilon-index","semantic_template":"uniform-epsilon-index|{epsilon:[5,11],error_scale:[25,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0051","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":1,\"lipschitz\":1,\"error_bound\":[1,4],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1/4, 1, 1/3, 5/6, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(1, 1, 1/4, 1, 1/3, 5/6, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173051,"parameters":{"n":1,"lipschitz":1,"error_bound":[1,4],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,4],lipschitz:[1,1],n:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0052","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":2,\"lipschitz\":2,\"error_bound\":[1,5],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, 1/5, 1, 1/6, 11/15, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(2, 2, 1/5, 1, 1/6, 11/15, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173052,"parameters":{"n":2,"lipschitz":2,"error_bound":[1,5],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,5],lipschitz:[2,1],n:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0053","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":3,\"lipschitz\":3,\"error_bound\":[1,6],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 3, 1/6, 1, 1/9, 2/3, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(3, 3, 1/6, 1, 1/9, 2/3, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173053,"parameters":{"n":3,"lipschitz":3,"error_bound":[1,6],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,6],lipschitz:[3,1],n:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0054","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":4,\"lipschitz\":4,\"error_bound\":[1,7],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 4, 1/7, 1, 1/12, 13/21, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(4, 4, 1/7, 1, 1/12, 13/21, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173054,"parameters":{"n":4,"lipschitz":4,"error_bound":[1,7],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,7],lipschitz:[4,1],n:4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0055","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":5,\"lipschitz\":5,\"error_bound\":[1,8],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 5, 1/8, 1, 1/15, 7/12, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(5, 5, 1/8, 1, 1/15, 7/12, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173055,"parameters":{"n":5,"lipschitz":5,"error_bound":[1,8],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,8],lipschitz:[5,1],n:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0056","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":6,\"lipschitz\":1,\"error_bound\":[1,9],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 1, 1/9, 1, 1/3, 5/9, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(6, 1, 1/9, 1, 1/3, 5/9, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173056,"parameters":{"n":6,"lipschitz":1,"error_bound":[1,9],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,9],lipschitz:[1,1],n:6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0057","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":7,\"lipschitz\":2,\"error_bound\":[1,10],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 2, 1/10, 1, 1/6, 8/15, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(7, 2, 1/10, 1, 1/6, 8/15, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173057,"parameters":{"n":7,"lipschitz":2,"error_bound":[1,10],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,10],lipschitz:[2,1],n:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0058","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":8,\"lipschitz\":3,\"error_bound\":[1,11],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 3, 1/11, 1, 1/9, 17/33, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(8, 3, 1/11, 1, 1/9, 17/33, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173058,"parameters":{"n":8,"lipschitz":3,"error_bound":[1,11],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,11],lipschitz:[3,1],n:8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0059","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":9,\"lipschitz\":4,\"error_bound\":[1,12],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 4, 1/12, 1, 1/12, 1/2, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(9, 4, 1/12, 1, 1/12, 1/2, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173059,"parameters":{"n":9,"lipschitz":4,"error_bound":[1,12],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,12],lipschitz:[4,1],n:9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0060","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":10,\"lipschitz\":5,\"error_bound\":[1,13],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 5, 1/13, 1, 1/15, 19/39, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(10, 5, 1/13, 1, 1/15, 19/39, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173060,"parameters":{"n":10,"lipschitz":5,"error_bound":[1,13],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,13],lipschitz:[5,1],n:10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0061","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":11,\"lipschitz\":1,\"error_bound\":[1,14],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 1, 1/14, 1, 1/3, 10/21, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(11, 1, 1/14, 1, 1/3, 10/21, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173061,"parameters":{"n":11,"lipschitz":1,"error_bound":[1,14],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,14],lipschitz:[1,1],n:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0062","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":12,\"lipschitz\":2,\"error_bound\":[1,15],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 2, 1/15, 1, 1/6, 7/15, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(12, 2, 1/15, 1, 1/6, 7/15, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173062,"parameters":{"n":12,"lipschitz":2,"error_bound":[1,15],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,15],lipschitz:[2,1],n:12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0063","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":13,\"lipschitz\":3,\"error_bound\":[1,16],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 3, 1/16, 1, 1/9, 11/24, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(13, 3, 1/16, 1, 1/9, 11/24, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173063,"parameters":{"n":13,"lipschitz":3,"error_bound":[1,16],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,16],lipschitz:[3,1],n:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0064","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":14,\"lipschitz\":4,\"error_bound\":[1,17],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 4, 1/17, 1, 1/12, 23/51, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(14, 4, 1/17, 1, 1/12, 23/51, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173064,"parameters":{"n":14,"lipschitz":4,"error_bound":[1,17],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,17],lipschitz:[4,1],n:14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0065","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":15,\"lipschitz\":5,\"error_bound\":[1,18],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 5, 1/18, 1, 1/15, 4/9, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(15, 5, 1/18, 1, 1/15, 4/9, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173065,"parameters":{"n":15,"lipschitz":5,"error_bound":[1,18],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,18],lipschitz:[5,1],n:15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0066","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":16,\"lipschitz\":1,\"error_bound\":[1,19],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 1, 1/19, 1, 1/3, 25/57, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(16, 1, 1/19, 1, 1/3, 25/57, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173066,"parameters":{"n":16,"lipschitz":1,"error_bound":[1,19],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,19],lipschitz:[1,1],n:16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0067","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":17,\"lipschitz\":2,\"error_bound\":[1,20],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 2, 1/20, 1, 1/6, 13/30, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(17, 2, 1/20, 1, 1/6, 13/30, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173067,"parameters":{"n":17,"lipschitz":2,"error_bound":[1,20],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,20],lipschitz:[2,1],n:17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0068","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":18,\"lipschitz\":3,\"error_bound\":[1,21],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 3, 1/21, 1, 1/9, 3/7, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(18, 3, 1/21, 1, 1/9, 3/7, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173068,"parameters":{"n":18,"lipschitz":3,"error_bound":[1,21],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,21],lipschitz:[3,1],n:18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0069","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":19,\"lipschitz\":4,\"error_bound\":[1,22],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 4, 1/22, 1, 1/12, 14/33, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(19, 4, 1/22, 1, 1/12, 14/33, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173069,"parameters":{"n":19,"lipschitz":4,"error_bound":[1,22],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,22],lipschitz:[4,1],n:19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0070","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":20,\"lipschitz\":5,\"error_bound\":[1,23],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 5, 1/23, 1, 1/15, 29/69, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(20, 5, 1/23, 1, 1/15, 29/69, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173070,"parameters":{"n":20,"lipschitz":5,"error_bound":[1,23],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,23],lipschitz:[5,1],n:20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0071","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":21,\"lipschitz\":1,\"error_bound\":[1,24],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 1, 1/24, 1, 1/3, 5/12, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(21, 1, 1/24, 1, 1/3, 5/12, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173071,"parameters":{"n":21,"lipschitz":1,"error_bound":[1,24],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,24],lipschitz:[1,1],n:21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0072","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":22,\"lipschitz\":2,\"error_bound\":[1,25],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 2, 1/25, 1, 1/6, 31/75, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(22, 2, 1/25, 1, 1/6, 31/75, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173072,"parameters":{"n":22,"lipschitz":2,"error_bound":[1,25],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,25],lipschitz:[2,1],n:22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0073","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":23,\"lipschitz\":3,\"error_bound\":[1,26],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 3, 1/26, 1, 1/9, 16/39, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(23, 3, 1/26, 1, 1/9, 16/39, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173073,"parameters":{"n":23,"lipschitz":3,"error_bound":[1,26],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,26],lipschitz:[3,1],n:23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0074","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":24,\"lipschitz\":4,\"error_bound\":[1,27],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 4, 1/27, 1, 1/12, 11/27, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(24, 4, 1/27, 1, 1/12, 11/27, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173074,"parameters":{"n":24,"lipschitz":4,"error_bound":[1,27],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,27],lipschitz:[4,1],n:24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0075","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-continuity-transfer；資料={\"n\":25,\"lipschitz\":5,\"error_bound\":[1,28],\"epsilon\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, 5, 1/28, 1, 1/15, 17/42, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation_zh":"Wave6第十一層核對=(25, 5, 1/28, 1, 1/15, 17/42, 是, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173075,"parameters":{"n":25,"lipschitz":5,"error_bound":[1,28],"epsilon":1,"domain":"Exact_uniform_limit_continuity","operation":"uniform-continuity-transfer","semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,28],lipschitz:[5,1],n:25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0076","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"pointwise_only\",\"scale\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(pointwise_only, 2, f_n(x)=x^(n+2)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)","explanation_zh":"Wave6第十一層核對=(pointwise_only, 2, f_n(x)=x^(n+2)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173076,"parameters":{"mode":"pointwise_only","scale":2,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0077","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"discontinuous_terms\",\"scale\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_terms, 3, f_n(x)=1_if_x>=1/4_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)","explanation_zh":"Wave6第十一層核對=(discontinuous_terms, 3, f_n(x)=1_if_x>=1/4_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173077,"parameters":{"mode":"discontinuous_terms","scale":3,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0078","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"finite_grid\",\"scale\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid, 4, grid={k/5:0<=k<=5}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, 否)","explanation_zh":"Wave6第十一層核對=(finite_grid, 4, grid={k/5:0<=k<=5}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173078,"parameters":{"mode":"finite_grid","scale":4,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"finite_grid\",scale:4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0079","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"moving_spike\",\"scale\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(moving_spike, 5, f_n(x)=max(0,1-(n+5)*abs(x-1/(n+5)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, 否)","explanation_zh":"Wave6第十一層核對=(moving_spike, 5, f_n(x)=max(0,1-(n+5)*abs(x-1/(n+5)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173079,"parameters":{"mode":"moving_spike","scale":5,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"moving_spike\",scale:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0080","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"pointwise_only\",\"scale\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(pointwise_only, 6, f_n(x)=x^(n+6)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)","explanation_zh":"Wave6第十一層核對=(pointwise_only, 6, f_n(x)=x^(n+6)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173080,"parameters":{"mode":"pointwise_only","scale":6,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0081","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"discontinuous_terms\",\"scale\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_terms, 7, f_n(x)=1_if_x>=1/8_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)","explanation_zh":"Wave6第十一層核對=(discontinuous_terms, 7, f_n(x)=1_if_x>=1/8_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173081,"parameters":{"mode":"discontinuous_terms","scale":7,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0082","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"finite_grid\",\"scale\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid, 8, grid={k/9:0<=k<=9}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, 否)","explanation_zh":"Wave6第十一層核對=(finite_grid, 8, grid={k/9:0<=k<=9}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173082,"parameters":{"mode":"finite_grid","scale":8,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"finite_grid\",scale:8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0083","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"moving_spike\",\"scale\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(moving_spike, 9, f_n(x)=max(0,1-(n+9)*abs(x-1/(n+9)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, 否)","explanation_zh":"Wave6第十一層核對=(moving_spike, 9, f_n(x)=max(0,1-(n+9)*abs(x-1/(n+9)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173083,"parameters":{"mode":"moving_spike","scale":9,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"moving_spike\",scale:9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0084","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"pointwise_only\",\"scale\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(pointwise_only, 10, f_n(x)=x^(n+10)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)","explanation_zh":"Wave6第十一層核對=(pointwise_only, 10, f_n(x)=x^(n+10)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173084,"parameters":{"mode":"pointwise_only","scale":10,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0085","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"discontinuous_terms\",\"scale\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_terms, 11, f_n(x)=1_if_x>=1/12_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)","explanation_zh":"Wave6第十一層核對=(discontinuous_terms, 11, f_n(x)=1_if_x>=1/12_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173085,"parameters":{"mode":"discontinuous_terms","scale":11,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0086","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"finite_grid\",\"scale\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid, 12, grid={k/13:0<=k<=13}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, 否)","explanation_zh":"Wave6第十一層核對=(finite_grid, 12, grid={k/13:0<=k<=13}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173086,"parameters":{"mode":"finite_grid","scale":12,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"finite_grid\",scale:12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0087","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"moving_spike\",\"scale\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(moving_spike, 13, f_n(x)=max(0,1-(n+13)*abs(x-1/(n+13)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, 否)","explanation_zh":"Wave6第十一層核對=(moving_spike, 13, f_n(x)=max(0,1-(n+13)*abs(x-1/(n+13)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173087,"parameters":{"mode":"moving_spike","scale":13,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"moving_spike\",scale:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0088","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"pointwise_only\",\"scale\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(pointwise_only, 14, f_n(x)=x^(n+14)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)","explanation_zh":"Wave6第十一層核對=(pointwise_only, 14, f_n(x)=x^(n+14)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173088,"parameters":{"mode":"pointwise_only","scale":14,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0089","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"discontinuous_terms\",\"scale\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_terms, 15, f_n(x)=1_if_x>=1/16_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)","explanation_zh":"Wave6第十一層核對=(discontinuous_terms, 15, f_n(x)=1_if_x>=1/16_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173089,"parameters":{"mode":"discontinuous_terms","scale":15,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0090","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"finite_grid\",\"scale\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid, 16, grid={k/17:0<=k<=17}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, 否)","explanation_zh":"Wave6第十一層核對=(finite_grid, 16, grid={k/17:0<=k<=17}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173090,"parameters":{"mode":"finite_grid","scale":16,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"finite_grid\",scale:16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0091","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"moving_spike\",\"scale\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(moving_spike, 17, f_n(x)=max(0,1-(n+17)*abs(x-1/(n+17)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, 否)","explanation_zh":"Wave6第十一層核對=(moving_spike, 17, f_n(x)=max(0,1-(n+17)*abs(x-1/(n+17)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173091,"parameters":{"mode":"moving_spike","scale":17,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"moving_spike\",scale:17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0092","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"pointwise_only\",\"scale\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(pointwise_only, 18, f_n(x)=x^(n+18)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)","explanation_zh":"Wave6第十一層核對=(pointwise_only, 18, f_n(x)=x^(n+18)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173092,"parameters":{"mode":"pointwise_only","scale":18,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0093","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"discontinuous_terms\",\"scale\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_terms, 19, f_n(x)=1_if_x>=1/20_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)","explanation_zh":"Wave6第十一層核對=(discontinuous_terms, 19, f_n(x)=1_if_x>=1/20_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173093,"parameters":{"mode":"discontinuous_terms","scale":19,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0094","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"finite_grid\",\"scale\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid, 20, grid={k/21:0<=k<=21}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, 否)","explanation_zh":"Wave6第十一層核對=(finite_grid, 20, grid={k/21:0<=k<=21}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173094,"parameters":{"mode":"finite_grid","scale":20,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"finite_grid\",scale:20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0095","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"moving_spike\",\"scale\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(moving_spike, 21, f_n(x)=max(0,1-(n+21)*abs(x-1/(n+21)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, 否)","explanation_zh":"Wave6第十一層核對=(moving_spike, 21, f_n(x)=max(0,1-(n+21)*abs(x-1/(n+21)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173095,"parameters":{"mode":"moving_spike","scale":21,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"moving_spike\",scale:21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0096","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"pointwise_only\",\"scale\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(pointwise_only, 22, f_n(x)=x^(n+22)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)","explanation_zh":"Wave6第十一層核對=(pointwise_only, 22, f_n(x)=x^(n+22)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173096,"parameters":{"mode":"pointwise_only","scale":22,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0097","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"discontinuous_terms\",\"scale\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_terms, 23, f_n(x)=1_if_x>=1/24_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)","explanation_zh":"Wave6第十一層核對=(discontinuous_terms, 23, f_n(x)=1_if_x>=1/24_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173097,"parameters":{"mode":"discontinuous_terms","scale":23,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0098","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"finite_grid\",\"scale\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid, 24, grid={k/25:0<=k<=25}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, 否)","explanation_zh":"Wave6第十一層核對=(finite_grid, 24, grid={k/25:0<=k<=25}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173098,"parameters":{"mode":"finite_grid","scale":24,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"finite_grid\",scale:24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0099","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"moving_spike\",\"scale\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(moving_spike, 25, f_n(x)=max(0,1-(n+25)*abs(x-1/(n+25)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, 否)","explanation_zh":"Wave6第十一層核對=(moving_spike, 25, f_n(x)=max(0,1-(n+25)*abs(x-1/(n+25)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173099,"parameters":{"mode":"moving_spike","scale":25,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"moving_spike\",scale:25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0100","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-uniform-limit-continuity-theorem"],"title_zh":"一致極限保持連續","stem_zh":"共同domain、逐項continuity與同一N控制全部x的uniform責任。操作=uniform-limit-boundary；資料={\"mode\":\"pointwise_only\",\"scale\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(pointwise_only, 26, f_n(x)=x^(n+26)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)","explanation_zh":"Wave6第十一層核對=(pointwise_only, 26, f_n(x)=x^(n+26)_on_[0,1], pointwise_limit_discontinuous_at_1, 否)；依完整明示資料獨立重算；sup error與epsilon/3 transfer使用exact bounds；pointwise convergence、finite grid或不連續approximants不足以推出結論。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["一致極限保持連續"]},"variant":{"family_id":"qf-mod48-uniform-limit-boundary","seed":202611173100,"parameters":{"mode":"pointwise_only","scale":26,"domain":"Exact_uniform_limit_continuity","operation":"uniform-limit-boundary","semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般series、uniform-limit或topology定理證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}