{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0001","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[0,1],\"coefficients\":[1,-2],\"shift\":2,\"n\":1,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 1/3, 0, 1/3)","explanation_zh":"Wave5第三層核對=(1, 0, 1/3, 0, 1/3)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032001,"parameters":{"limit":[0,1],"coefficients":[1,-2],"shift":2,"n":1,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[1,-2],\"limit\":[0,1],\"n\":1,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0002","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[1,2],\"coefficients\":[2,-3],\"shift\":2,\"n\":2,\"point\":1}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 5/4, 2, -3/4)","explanation_zh":"Wave5第三層核對=(2, 1, 5/4, 2, -3/4)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032002,"parameters":{"limit":[1,2],"coefficients":[2,-3],"shift":2,"n":2,"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[2,-3],\"limit\":[1,2],\"n\":2,\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0003","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[2,3],\"coefficients\":[3,-4],\"shift\":2,\"n\":3,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 13/5, 2, 3/5)","explanation_zh":"Wave5第三層核對=(3, 0, 13/5, 2, 3/5)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032003,"parameters":{"limit":[2,3],"coefficients":[3,-4],"shift":2,"n":3,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[3,-4],\"limit\":[2,3],\"n\":3,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0004","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[3,4],\"coefficients\":[4,-5],\"shift\":2,\"n\":4,\"point\":1}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(4, 1, 19/6, 4, -5/6)","explanation_zh":"Wave5第三層核對=(4, 1, 19/6, 4, -5/6)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032004,"parameters":{"limit":[3,4],"coefficients":[4,-5],"shift":2,"n":4,"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[4,-5],\"limit\":[3,4],\"n\":4,\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0005","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[4,5],\"coefficients\":[5,-6],\"shift\":2,\"n\":5,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 33/7, 4, 5/7)","explanation_zh":"Wave5第三層核對=(5, 0, 33/7, 4, 5/7)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032005,"parameters":{"limit":[4,5],"coefficients":[5,-6],"shift":2,"n":5,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[5,-6],\"limit\":[4,5],\"n\":5,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0006","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[5,6],\"coefficients\":[6,-7],\"shift\":2,\"n\":6,\"point\":1}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(6, 1, 41/8, 6, -7/8)","explanation_zh":"Wave5第三層核對=(6, 1, 41/8, 6, -7/8)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032006,"parameters":{"limit":[5,6],"coefficients":[6,-7],"shift":2,"n":6,"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[6,-7],\"limit\":[5,6],\"n\":6,\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0007","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[6,7],\"coefficients\":[7,-8],\"shift\":2,\"n\":7,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(7, 0, 61/9, 6, 7/9)","explanation_zh":"Wave5第三層核對=(7, 0, 61/9, 6, 7/9)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032007,"parameters":{"limit":[6,7],"coefficients":[7,-8],"shift":2,"n":7,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[7,-8],\"limit\":[6,7],\"n\":7,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0008","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[7,8],\"coefficients\":[8,-9],\"shift\":2,\"n\":8,\"point\":1}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(8, 1, 71/10, 8, -9/10)","explanation_zh":"Wave5第三層核對=(8, 1, 71/10, 8, -9/10)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032008,"parameters":{"limit":[7,8],"coefficients":[8,-9],"shift":2,"n":8,"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[8,-9],\"limit\":[7,8],\"n\":8,\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0009","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[8,9],\"coefficients\":[9,-10],\"shift\":2,\"n\":9,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(9, 0, 97/11, 8, 9/11)","explanation_zh":"Wave5第三層核對=(9, 0, 97/11, 8, 9/11)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032009,"parameters":{"limit":[8,9],"coefficients":[9,-10],"shift":2,"n":9,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[9,-10],\"limit\":[8,9],\"n\":9,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0010","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[9,10],\"coefficients\":[10,-11],\"shift\":2,\"n\":10,\"point\":1}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(10, 1, 109/12, 10, -11/12)","explanation_zh":"Wave5第三層核對=(10, 1, 109/12, 10, -11/12)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032010,"parameters":{"limit":[9,10],"coefficients":[10,-11],"shift":2,"n":10,"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[10,-11],\"limit\":[9,10],\"n\":10,\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0011","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[10,11],\"coefficients\":[11,-12],\"shift\":2,\"n\":11,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(11, 0, 141/13, 10, 11/13)","explanation_zh":"Wave5第三層核對=(11, 0, 141/13, 10, 11/13)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032011,"parameters":{"limit":[10,11],"coefficients":[11,-12],"shift":2,"n":11,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[11,-12],\"limit\":[10,11],\"n\":11,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0012","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[11,12],\"coefficients\":[12,-13],\"shift\":2,\"n\":12,\"point\":1}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(12, 1, 155/14, 12, -13/14)","explanation_zh":"Wave5第三層核對=(12, 1, 155/14, 12, -13/14)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032012,"parameters":{"limit":[11,12],"coefficients":[12,-13],"shift":2,"n":12,"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[12,-13],\"limit\":[11,12],\"n\":12,\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0013","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[12,13],\"coefficients\":[13,-14],\"shift\":2,\"n\":13,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(13, 0, 193/15, 12, 13/15)","explanation_zh":"Wave5第三層核對=(13, 0, 193/15, 12, 13/15)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032013,"parameters":{"limit":[12,13],"coefficients":[13,-14],"shift":2,"n":13,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[13,-14],\"limit\":[12,13],\"n\":13,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0014","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[13,14],\"coefficients\":[14,-15],\"shift\":2,\"n\":14,\"point\":1}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(14, 1, 209/16, 14, -15/16)","explanation_zh":"Wave5第三層核對=(14, 1, 209/16, 14, -15/16)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032014,"parameters":{"limit":[13,14],"coefficients":[14,-15],"shift":2,"n":14,"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[14,-15],\"limit\":[13,14],\"n\":14,\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0015","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[14,15],\"coefficients\":[15,-16],\"shift\":2,\"n\":15,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(15, 0, 253/17, 14, 15/17)","explanation_zh":"Wave5第三層核對=(15, 0, 253/17, 14, 15/17)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032015,"parameters":{"limit":[14,15],"coefficients":[15,-16],"shift":2,"n":15,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[15,-16],\"limit\":[14,15],\"n\":15,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0016","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[15,16],\"coefficients\":[16,-17],\"shift\":2,\"n\":16,\"point\":1}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(16, 1, 271/18, 16, -17/18)","explanation_zh":"Wave5第三層核對=(16, 1, 271/18, 16, -17/18)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032016,"parameters":{"limit":[15,16],"coefficients":[16,-17],"shift":2,"n":16,"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[16,-17],\"limit\":[15,16],\"n\":16,\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0017","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[16,17],\"coefficients\":[17,-18],\"shift\":2,\"n\":17,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(17, 0, 321/19, 16, 17/19)","explanation_zh":"Wave5第三層核對=(17, 0, 321/19, 16, 17/19)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032017,"parameters":{"limit":[16,17],"coefficients":[17,-18],"shift":2,"n":17,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[17,-18],\"limit\":[16,17],\"n\":17,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0018","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[17,18],\"coefficients\":[18,-19],\"shift\":2,\"n\":18,\"point\":1}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(18, 1, 341/20, 18, -19/20)","explanation_zh":"Wave5第三層核對=(18, 1, 341/20, 18, -19/20)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032018,"parameters":{"limit":[17,18],"coefficients":[18,-19],"shift":2,"n":18,"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[18,-19],\"limit\":[17,18],\"n\":18,\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0019","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[18,19],\"coefficients\":[19,-20],\"shift\":2,\"n\":19,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(19, 0, 397/21, 18, 19/21)","explanation_zh":"Wave5第三層核對=(19, 0, 397/21, 18, 19/21)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032019,"parameters":{"limit":[18,19],"coefficients":[19,-20],"shift":2,"n":19,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[19,-20],\"limit\":[18,19],\"n\":19,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0020","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[19,20],\"coefficients\":[20,-21],\"shift\":2,\"n\":20,\"point\":1}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(20, 1, 419/22, 20, -21/22)","explanation_zh":"Wave5第三層核對=(20, 1, 419/22, 20, -21/22)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032020,"parameters":{"limit":[19,20],"coefficients":[20,-21],"shift":2,"n":20,"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[20,-21],\"limit\":[19,20],\"n\":20,\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0021","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[20,21],\"coefficients\":[21,-22],\"shift\":2,\"n\":21,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(21, 0, 481/23, 20, 21/23)","explanation_zh":"Wave5第三層核對=(21, 0, 481/23, 20, 21/23)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032021,"parameters":{"limit":[20,21],"coefficients":[21,-22],"shift":2,"n":21,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[21,-22],\"limit\":[20,21],\"n\":21,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0022","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[21,22],\"coefficients\":[22,-23],\"shift\":2,\"n\":22,\"point\":1}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(22, 1, 505/24, 22, -23/24)","explanation_zh":"Wave5第三層核對=(22, 1, 505/24, 22, -23/24)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032022,"parameters":{"limit":[21,22],"coefficients":[22,-23],"shift":2,"n":22,"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[22,-23],\"limit\":[21,22],\"n\":22,\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0023","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[22,23],\"coefficients\":[23,-24],\"shift\":2,\"n\":23,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(23, 0, 573/25, 22, 23/25)","explanation_zh":"Wave5第三層核對=(23, 0, 573/25, 22, 23/25)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032023,"parameters":{"limit":[22,23],"coefficients":[23,-24],"shift":2,"n":23,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[23,-24],\"limit\":[22,23],\"n\":23,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0024","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[23,24],\"coefficients\":[24,-25],\"shift\":2,\"n\":24,\"point\":1}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(24, 1, 599/26, 24, -25/26)","explanation_zh":"Wave5第三層核對=(24, 1, 599/26, 24, -25/26)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032024,"parameters":{"limit":[23,24],"coefficients":[24,-25],"shift":2,"n":24,"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[24,-25],\"limit\":[23,24],\"n\":24,\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0025","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-evaluate；資料={\"limit\":[24,25],\"coefficients\":[25,-26],\"shift\":2,\"n\":25,\"point\":0}。請依 [n,point,f_n_x,limit_x,error] 欄序回答。","answer":{"kind":"expression","value":"(25, 0, 673/27, 24, 25/27)","explanation_zh":"Wave5第三層核對=(25, 0, 673/27, 24, 25/27)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-evaluate","seed":202611032025,"parameters":{"limit":[24,25],"coefficients":[25,-26],"shift":2,"n":25,"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-evaluate","semantic_template":"qf-mod34-pointwise-finite-domain-evaluate/{\"coefficients\":[25,-26],\"limit\":[24,25],\"n\":25,\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0026","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[0,1],\"coefficients\":[1,-2],\"shift\":2,\"epsilon\":[1,10],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/10, 9, 1/11, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/10, 9, 1/11, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032026,"parameters":{"limit":[0,1],"coefficients":[1,-2],"shift":2,"epsilon":[1,10],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[1,-2],\"epsilon\":[1,10],\"limit\":[0,1],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0027","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[1,2],\"coefficients\":[2,-3],\"shift\":2,\"epsilon\":[1,11],\"point\":1}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/11, 32, 3/34, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(1, 1/11, 32, 3/34, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032027,"parameters":{"limit":[1,2],"coefficients":[2,-3],"shift":2,"epsilon":[1,11],"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[2,-3],\"epsilon\":[1,11],\"limit\":[1,2],\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0028","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[2,3],\"coefficients\":[3,-4],\"shift\":2,\"epsilon\":[1,12],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/12, 35, 3/37, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/12, 35, 3/37, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032028,"parameters":{"limit":[2,3],"coefficients":[3,-4],"shift":2,"epsilon":[1,12],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[3,-4],\"epsilon\":[1,12],\"limit\":[2,3],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0029","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[3,4],\"coefficients\":[4,-5],\"shift\":2,\"epsilon\":[1,13],\"point\":1}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/13, 64, 5/66, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(1, 1/13, 64, 5/66, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032029,"parameters":{"limit":[3,4],"coefficients":[4,-5],"shift":2,"epsilon":[1,13],"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[4,-5],\"epsilon\":[1,13],\"limit\":[3,4],\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0030","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[4,5],\"coefficients\":[5,-6],\"shift\":2,\"epsilon\":[1,14],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/14, 69, 5/71, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/14, 69, 5/71, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032030,"parameters":{"limit":[4,5],"coefficients":[5,-6],"shift":2,"epsilon":[1,14],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[5,-6],\"epsilon\":[1,14],\"limit\":[4,5],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0031","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[5,6],\"coefficients\":[6,-7],\"shift\":2,\"epsilon\":[1,15],\"point\":1}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/15, 104, 7/106, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(1, 1/15, 104, 7/106, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032031,"parameters":{"limit":[5,6],"coefficients":[6,-7],"shift":2,"epsilon":[1,15],"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[6,-7],\"epsilon\":[1,15],\"limit\":[5,6],\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0032","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[6,7],\"coefficients\":[7,-8],\"shift\":2,\"epsilon\":[1,16],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/16, 111, 7/113, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/16, 111, 7/113, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032032,"parameters":{"limit":[6,7],"coefficients":[7,-8],"shift":2,"epsilon":[1,16],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[7,-8],\"epsilon\":[1,16],\"limit\":[6,7],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0033","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[7,8],\"coefficients\":[8,-9],\"shift\":2,\"epsilon\":[1,17],\"point\":1}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/17, 152, 9/154, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(1, 1/17, 152, 9/154, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032033,"parameters":{"limit":[7,8],"coefficients":[8,-9],"shift":2,"epsilon":[1,17],"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[8,-9],\"epsilon\":[1,17],\"limit\":[7,8],\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0034","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[8,9],\"coefficients\":[9,-10],\"shift\":2,\"epsilon\":[1,18],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/18, 161, 9/163, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/18, 161, 9/163, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032034,"parameters":{"limit":[8,9],"coefficients":[9,-10],"shift":2,"epsilon":[1,18],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[9,-10],\"epsilon\":[1,18],\"limit\":[8,9],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0035","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[9,10],\"coefficients\":[10,-11],\"shift\":2,\"epsilon\":[1,19],\"point\":1}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/19, 208, 11/210, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(1, 1/19, 208, 11/210, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032035,"parameters":{"limit":[9,10],"coefficients":[10,-11],"shift":2,"epsilon":[1,19],"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[10,-11],\"epsilon\":[1,19],\"limit\":[9,10],\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0036","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[10,11],\"coefficients\":[11,-12],\"shift\":2,\"epsilon\":[1,20],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/20, 219, 11/221, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/20, 219, 11/221, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032036,"parameters":{"limit":[10,11],"coefficients":[11,-12],"shift":2,"epsilon":[1,20],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[11,-12],\"epsilon\":[1,20],\"limit\":[10,11],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0037","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[11,12],\"coefficients\":[12,-13],\"shift\":2,\"epsilon\":[1,21],\"point\":1}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/21, 272, 13/274, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(1, 1/21, 272, 13/274, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032037,"parameters":{"limit":[11,12],"coefficients":[12,-13],"shift":2,"epsilon":[1,21],"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[12,-13],\"epsilon\":[1,21],\"limit\":[11,12],\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0038","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[12,13],\"coefficients\":[13,-14],\"shift\":2,\"epsilon\":[1,22],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/22, 285, 13/287, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/22, 285, 13/287, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032038,"parameters":{"limit":[12,13],"coefficients":[13,-14],"shift":2,"epsilon":[1,22],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[13,-14],\"epsilon\":[1,22],\"limit\":[12,13],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0039","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[13,14],\"coefficients\":[14,-15],\"shift\":2,\"epsilon\":[1,23],\"point\":1}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/23, 344, 15/346, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(1, 1/23, 344, 15/346, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032039,"parameters":{"limit":[13,14],"coefficients":[14,-15],"shift":2,"epsilon":[1,23],"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[14,-15],\"epsilon\":[1,23],\"limit\":[13,14],\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0040","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[14,15],\"coefficients\":[15,-16],\"shift\":2,\"epsilon\":[1,24],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/24, 359, 15/361, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/24, 359, 15/361, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032040,"parameters":{"limit":[14,15],"coefficients":[15,-16],"shift":2,"epsilon":[1,24],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[15,-16],\"epsilon\":[1,24],\"limit\":[14,15],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0041","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[15,16],\"coefficients\":[16,-17],\"shift\":2,\"epsilon\":[1,25],\"point\":1}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/25, 424, 17/426, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(1, 1/25, 424, 17/426, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032041,"parameters":{"limit":[15,16],"coefficients":[16,-17],"shift":2,"epsilon":[1,25],"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[16,-17],\"epsilon\":[1,25],\"limit\":[15,16],\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0042","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[16,17],\"coefficients\":[17,-18],\"shift\":2,\"epsilon\":[1,26],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/26, 441, 17/443, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/26, 441, 17/443, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032042,"parameters":{"limit":[16,17],"coefficients":[17,-18],"shift":2,"epsilon":[1,26],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[17,-18],\"epsilon\":[1,26],\"limit\":[16,17],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0043","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[17,18],\"coefficients\":[18,-19],\"shift\":2,\"epsilon\":[1,27],\"point\":1}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/27, 512, 19/514, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(1, 1/27, 512, 19/514, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032043,"parameters":{"limit":[17,18],"coefficients":[18,-19],"shift":2,"epsilon":[1,27],"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[18,-19],\"epsilon\":[1,27],\"limit\":[17,18],\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0044","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[18,19],\"coefficients\":[19,-20],\"shift\":2,\"epsilon\":[1,28],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/28, 531, 19/533, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/28, 531, 19/533, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032044,"parameters":{"limit":[18,19],"coefficients":[19,-20],"shift":2,"epsilon":[1,28],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[19,-20],\"epsilon\":[1,28],\"limit\":[18,19],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0045","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[19,20],\"coefficients\":[20,-21],\"shift\":2,\"epsilon\":[1,29],\"point\":1}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/29, 608, 21/610, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(1, 1/29, 608, 21/610, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032045,"parameters":{"limit":[19,20],"coefficients":[20,-21],"shift":2,"epsilon":[1,29],"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[20,-21],\"epsilon\":[1,29],\"limit\":[19,20],\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0046","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[20,21],\"coefficients\":[21,-22],\"shift\":2,\"epsilon\":[1,30],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/30, 629, 21/631, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/30, 629, 21/631, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032046,"parameters":{"limit":[20,21],"coefficients":[21,-22],"shift":2,"epsilon":[1,30],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[21,-22],\"epsilon\":[1,30],\"limit\":[20,21],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0047","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[21,22],\"coefficients\":[22,-23],\"shift\":2,\"epsilon\":[1,31],\"point\":1}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/31, 712, 23/714, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(1, 1/31, 712, 23/714, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032047,"parameters":{"limit":[21,22],"coefficients":[22,-23],"shift":2,"epsilon":[1,31],"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[22,-23],\"epsilon\":[1,31],\"limit\":[21,22],\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0048","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[22,23],\"coefficients\":[23,-24],\"shift\":2,\"epsilon\":[1,32],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/32, 735, 23/737, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/32, 735, 23/737, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032048,"parameters":{"limit":[22,23],"coefficients":[23,-24],"shift":2,"epsilon":[1,32],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[23,-24],\"epsilon\":[1,32],\"limit\":[22,23],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0049","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[23,24],\"coefficients\":[24,-25],\"shift\":2,\"epsilon\":[1,33],\"point\":1}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/33, 824, 25/826, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(1, 1/33, 824, 25/826, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032049,"parameters":{"limit":[23,24],"coefficients":[24,-25],"shift":2,"epsilon":[1,33],"point":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[24,-25],\"epsilon\":[1,33],\"limit\":[23,24],\"point\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0050","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-epsilon-witness；資料={\"limit\":[24,25],\"coefficients\":[25,-26],\"shift\":2,\"epsilon\":[1,34],\"point\":0}。請依 [point,epsilon,N,error_at_N,strict_tail,certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/34, 849, 25/851, 是, forall_n_ge_N)","explanation_zh":"Wave5第三層核對=(0, 1/34, 849, 25/851, 是, forall_n_ge_N)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-epsilon-witness","seed":202611032050,"parameters":{"limit":[24,25],"coefficients":[25,-26],"shift":2,"epsilon":[1,34],"point":0,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-epsilon-witness","semantic_template":"qf-mod34-pointwise-epsilon-witness/{\"coefficients\":[25,-26],\"epsilon\":[1,34],\"limit\":[24,25],\"point\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0051","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[0,1],\"coefficients\":[1,-2],\"shift\":2,\"epsilon\":[1,10]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/10, [[0, 9], [1, 19]], 19, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/10, [[0, 9], [1, 19]], 19, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032051,"parameters":{"limit":[0,1],"coefficients":[1,-2],"shift":2,"epsilon":[1,10],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[1,-2],\"epsilon\":[1,10],\"limit\":[0,1],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0052","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[1,2],\"coefficients\":[2,-3],\"shift\":2,\"epsilon\":[1,11]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/11, [[0, 21], [1, 32]], 32, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/11, [[0, 21], [1, 32]], 32, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032052,"parameters":{"limit":[1,2],"coefficients":[2,-3],"shift":2,"epsilon":[1,11],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[2,-3],\"epsilon\":[1,11],\"limit\":[1,2],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0053","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[2,3],\"coefficients\":[3,-4],\"shift\":2,\"epsilon\":[1,12]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/12, [[0, 35], [1, 47]], 47, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/12, [[0, 35], [1, 47]], 47, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032053,"parameters":{"limit":[2,3],"coefficients":[3,-4],"shift":2,"epsilon":[1,12],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[3,-4],\"epsilon\":[1,12],\"limit\":[2,3],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0054","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[3,4],\"coefficients\":[4,-5],\"shift\":2,\"epsilon\":[1,13]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/13, [[0, 51], [1, 64]], 64, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/13, [[0, 51], [1, 64]], 64, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032054,"parameters":{"limit":[3,4],"coefficients":[4,-5],"shift":2,"epsilon":[1,13],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[4,-5],\"epsilon\":[1,13],\"limit\":[3,4],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0055","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[4,5],\"coefficients\":[5,-6],\"shift\":2,\"epsilon\":[1,14]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/14, [[0, 69], [1, 83]], 83, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/14, [[0, 69], [1, 83]], 83, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032055,"parameters":{"limit":[4,5],"coefficients":[5,-6],"shift":2,"epsilon":[1,14],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[5,-6],\"epsilon\":[1,14],\"limit\":[4,5],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0056","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[5,6],\"coefficients\":[6,-7],\"shift\":2,\"epsilon\":[1,15]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/15, [[0, 89], [1, 104]], 104, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/15, [[0, 89], [1, 104]], 104, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032056,"parameters":{"limit":[5,6],"coefficients":[6,-7],"shift":2,"epsilon":[1,15],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[6,-7],\"epsilon\":[1,15],\"limit\":[5,6],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0057","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[6,7],\"coefficients\":[7,-8],\"shift\":2,\"epsilon\":[1,16]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/16, [[0, 111], [1, 127]], 127, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/16, [[0, 111], [1, 127]], 127, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032057,"parameters":{"limit":[6,7],"coefficients":[7,-8],"shift":2,"epsilon":[1,16],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[7,-8],\"epsilon\":[1,16],\"limit\":[6,7],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0058","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[7,8],\"coefficients\":[8,-9],\"shift\":2,\"epsilon\":[1,17]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/17, [[0, 135], [1, 152]], 152, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/17, [[0, 135], [1, 152]], 152, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032058,"parameters":{"limit":[7,8],"coefficients":[8,-9],"shift":2,"epsilon":[1,17],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[8,-9],\"epsilon\":[1,17],\"limit\":[7,8],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0059","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[8,9],\"coefficients\":[9,-10],\"shift\":2,\"epsilon\":[1,18]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/18, [[0, 161], [1, 179]], 179, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/18, [[0, 161], [1, 179]], 179, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032059,"parameters":{"limit":[8,9],"coefficients":[9,-10],"shift":2,"epsilon":[1,18],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[9,-10],\"epsilon\":[1,18],\"limit\":[8,9],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0060","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[9,10],\"coefficients\":[10,-11],\"shift\":2,\"epsilon\":[1,19]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/19, [[0, 189], [1, 208]], 208, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/19, [[0, 189], [1, 208]], 208, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032060,"parameters":{"limit":[9,10],"coefficients":[10,-11],"shift":2,"epsilon":[1,19],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[10,-11],\"epsilon\":[1,19],\"limit\":[9,10],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0061","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[10,11],\"coefficients\":[11,-12],\"shift\":2,\"epsilon\":[1,20]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/20, [[0, 219], [1, 239]], 239, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/20, [[0, 219], [1, 239]], 239, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032061,"parameters":{"limit":[10,11],"coefficients":[11,-12],"shift":2,"epsilon":[1,20],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[11,-12],\"epsilon\":[1,20],\"limit\":[10,11],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0062","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[11,12],\"coefficients\":[12,-13],\"shift\":2,\"epsilon\":[1,21]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/21, [[0, 251], [1, 272]], 272, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/21, [[0, 251], [1, 272]], 272, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032062,"parameters":{"limit":[11,12],"coefficients":[12,-13],"shift":2,"epsilon":[1,21],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[12,-13],\"epsilon\":[1,21],\"limit\":[11,12],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0063","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[12,13],\"coefficients\":[13,-14],\"shift\":2,\"epsilon\":[1,22]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/22, [[0, 285], [1, 307]], 307, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/22, [[0, 285], [1, 307]], 307, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032063,"parameters":{"limit":[12,13],"coefficients":[13,-14],"shift":2,"epsilon":[1,22],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[13,-14],\"epsilon\":[1,22],\"limit\":[12,13],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0064","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[13,14],\"coefficients\":[14,-15],\"shift\":2,\"epsilon\":[1,23]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/23, [[0, 321], [1, 344]], 344, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/23, [[0, 321], [1, 344]], 344, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032064,"parameters":{"limit":[13,14],"coefficients":[14,-15],"shift":2,"epsilon":[1,23],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[14,-15],\"epsilon\":[1,23],\"limit\":[13,14],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0065","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[14,15],\"coefficients\":[15,-16],\"shift\":2,\"epsilon\":[1,24]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/24, [[0, 359], [1, 383]], 383, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/24, [[0, 359], [1, 383]], 383, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032065,"parameters":{"limit":[14,15],"coefficients":[15,-16],"shift":2,"epsilon":[1,24],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[15,-16],\"epsilon\":[1,24],\"limit\":[14,15],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0066","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[15,16],\"coefficients\":[16,-17],\"shift\":2,\"epsilon\":[1,25]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/25, [[0, 399], [1, 424]], 424, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/25, [[0, 399], [1, 424]], 424, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032066,"parameters":{"limit":[15,16],"coefficients":[16,-17],"shift":2,"epsilon":[1,25],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[16,-17],\"epsilon\":[1,25],\"limit\":[15,16],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0067","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[16,17],\"coefficients\":[17,-18],\"shift\":2,\"epsilon\":[1,26]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/26, [[0, 441], [1, 467]], 467, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/26, [[0, 441], [1, 467]], 467, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032067,"parameters":{"limit":[16,17],"coefficients":[17,-18],"shift":2,"epsilon":[1,26],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[17,-18],\"epsilon\":[1,26],\"limit\":[16,17],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0068","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[17,18],\"coefficients\":[18,-19],\"shift\":2,\"epsilon\":[1,27]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/27, [[0, 485], [1, 512]], 512, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/27, [[0, 485], [1, 512]], 512, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032068,"parameters":{"limit":[17,18],"coefficients":[18,-19],"shift":2,"epsilon":[1,27],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[18,-19],\"epsilon\":[1,27],\"limit\":[17,18],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0069","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[18,19],\"coefficients\":[19,-20],\"shift\":2,\"epsilon\":[1,28]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/28, [[0, 531], [1, 559]], 559, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/28, [[0, 531], [1, 559]], 559, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032069,"parameters":{"limit":[18,19],"coefficients":[19,-20],"shift":2,"epsilon":[1,28],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[19,-20],\"epsilon\":[1,28],\"limit\":[18,19],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0070","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[19,20],\"coefficients\":[20,-21],\"shift\":2,\"epsilon\":[1,29]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/29, [[0, 579], [1, 608]], 608, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/29, [[0, 579], [1, 608]], 608, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032070,"parameters":{"limit":[19,20],"coefficients":[20,-21],"shift":2,"epsilon":[1,29],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[20,-21],\"epsilon\":[1,29],\"limit\":[19,20],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0071","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[20,21],\"coefficients\":[21,-22],\"shift\":2,\"epsilon\":[1,30]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/30, [[0, 629], [1, 659]], 659, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/30, [[0, 629], [1, 659]], 659, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032071,"parameters":{"limit":[20,21],"coefficients":[21,-22],"shift":2,"epsilon":[1,30],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[21,-22],\"epsilon\":[1,30],\"limit\":[20,21],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0072","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[21,22],\"coefficients\":[22,-23],\"shift\":2,\"epsilon\":[1,31]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/31, [[0, 681], [1, 712]], 712, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/31, [[0, 681], [1, 712]], 712, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032072,"parameters":{"limit":[21,22],"coefficients":[22,-23],"shift":2,"epsilon":[1,31],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[22,-23],\"epsilon\":[1,31],\"limit\":[21,22],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0073","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[22,23],\"coefficients\":[23,-24],\"shift\":2,\"epsilon\":[1,32]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/32, [[0, 735], [1, 767]], 767, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/32, [[0, 735], [1, 767]], 767, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032073,"parameters":{"limit":[22,23],"coefficients":[23,-24],"shift":2,"epsilon":[1,32],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[23,-24],\"epsilon\":[1,32],\"limit\":[22,23],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0074","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[23,24],\"coefficients\":[24,-25],\"shift\":2,\"epsilon\":[1,33]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/33, [[0, 791], [1, 824]], 824, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/33, [[0, 791], [1, 824]], 824, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032074,"parameters":{"limit":[23,24],"coefficients":[24,-25],"shift":2,"epsilon":[1,33],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[24,-25],\"epsilon\":[1,33],\"limit\":[23,24],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0075","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-domain-uniform-max；資料={\"limit\":[24,25],\"coefficients\":[25,-26],\"shift\":2,\"epsilon\":[1,34]}。請依 [epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/34, [[0, 849], [1, 883]], 883, 是, finite_domain_maximum_only_not_infinite_domain_theorem)","explanation_zh":"Wave5第三層核對=(1/34, [[0, 849], [1, 883]], 883, 是, finite_domain_maximum_only_not_infinite_domain_theorem)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-domain-uniform-max","seed":202611032075,"parameters":{"limit":[24,25],"coefficients":[25,-26],"shift":2,"epsilon":[1,34],"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-domain-uniform-max","semantic_template":"qf-mod34-pointwise-finite-domain-uniform-max/{\"coefficients\":[25,-26],\"epsilon\":[1,34],\"limit\":[24,25],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0076","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[0,1],\"coefficients\":[1,-2],\"shift\":2,\"prefix_length\":1}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1, [[0, [1/2]], [1, [0]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(1, [[0, [1/2]], [1, [0]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032076,"parameters":{"limit":[0,1],"coefficients":[1,-2],"shift":2,"prefix_length":1,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[1,-2],\"limit\":[0,1],\"prefix_length\":1,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0077","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[1,2],\"coefficients\":[2,-3],\"shift\":2,\"prefix_length\":2}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2, [[0, [2, 5/3]], [1, [1/2, 1]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(2, [[0, [2, 5/3]], [1, [1/2, 1]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032077,"parameters":{"limit":[1,2],"coefficients":[2,-3],"shift":2,"prefix_length":2,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[2,-3],\"limit\":[1,2],\"prefix_length\":2,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0078","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[2,3],\"coefficients\":[3,-4],\"shift\":2,\"prefix_length\":3}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(3, [[0, [7/2, 3, 11/4]], [1, [1, 5/3, 2]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(3, [[0, [7/2, 3, 11/4]], [1, [1, 5/3, 2]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032078,"parameters":{"limit":[2,3],"coefficients":[3,-4],"shift":2,"prefix_length":3,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[3,-4],\"limit\":[2,3],\"prefix_length\":3,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0079","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[3,4],\"coefficients\":[4,-5],\"shift\":2,\"prefix_length\":4}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(4, [[0, [5, 13/3, 4, 19/5]], [1, [3/2, 7/3, 11/4, 3]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(4, [[0, [5, 13/3, 4, 19/5]], [1, [3/2, 7/3, 11/4, 3]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032079,"parameters":{"limit":[3,4],"coefficients":[4,-5],"shift":2,"prefix_length":4,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[4,-5],\"limit\":[3,4],\"prefix_length\":4,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0080","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[4,5],\"coefficients\":[5,-6],\"shift\":2,\"prefix_length\":5}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(5, [[0, [13/2, 17/3, 21/4, 5, 29/6]], [1, [2, 3, 7/2, 19/5, 4]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(5, [[0, [13/2, 17/3, 21/4, 5, 29/6]], [1, [2, 3, 7/2, 19/5, 4]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032080,"parameters":{"limit":[4,5],"coefficients":[5,-6],"shift":2,"prefix_length":5,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[5,-6],\"limit\":[4,5],\"prefix_length\":5,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0081","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[5,6],\"coefficients\":[6,-7],\"shift\":2,\"prefix_length\":6}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(6, [[0, [8, 7, 13/2, 31/5, 6, 41/7]], [1, [5/2, 11/3, 17/4, 23/5, 29/6, 5]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(6, [[0, [8, 7, 13/2, 31/5, 6, 41/7]], [1, [5/2, 11/3, 17/4, 23/5, 29/6, 5]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032081,"parameters":{"limit":[5,6],"coefficients":[6,-7],"shift":2,"prefix_length":6,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[6,-7],\"limit\":[5,6],\"prefix_length\":6,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0082","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[6,7],\"coefficients\":[7,-8],\"shift\":2,\"prefix_length\":7}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(7, [[0, [19/2, 25/3, 31/4, 37/5, 43/6, 7, 55/8]], [1, [3, 13/3, 5, 27/5, 17/3, 41/7, 6]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(7, [[0, [19/2, 25/3, 31/4, 37/5, 43/6, 7, 55/8]], [1, [3, 13/3, 5, 27/5, 17/3, 41/7, 6]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032082,"parameters":{"limit":[6,7],"coefficients":[7,-8],"shift":2,"prefix_length":7,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[7,-8],\"limit\":[6,7],\"prefix_length\":7,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0083","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[7,8],\"coefficients\":[8,-9],\"shift\":2,\"prefix_length\":8}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(8, [[0, [11, 29/3, 9, 43/5, 25/3, 57/7, 8, 71/9]], [1, [7/2, 5, 23/4, 31/5, 13/2, 47/7, 55/8, 7]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(8, [[0, [11, 29/3, 9, 43/5, 25/3, 57/7, 8, 71/9]], [1, [7/2, 5, 23/4, 31/5, 13/2, 47/7, 55/8, 7]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032083,"parameters":{"limit":[7,8],"coefficients":[8,-9],"shift":2,"prefix_length":8,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[8,-9],\"limit\":[7,8],\"prefix_length\":8,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0084","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[8,9],\"coefficients\":[9,-10],\"shift\":2,\"prefix_length\":9}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(9, [[0, [25/2, 11, 41/4, 49/5, 19/2, 65/7, 73/8, 9, 89/10]], [1, [4, 17/3, 13/2, 7, 22/3, 53/7, 31/4, 71/9, 8]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(9, [[0, [25/2, 11, 41/4, 49/5, 19/2, 65/7, 73/8, 9, 89/10]], [1, [4, 17/3, 13/2, 7, 22/3, 53/7, 31/4, 71/9, 8]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032084,"parameters":{"limit":[8,9],"coefficients":[9,-10],"shift":2,"prefix_length":9,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[9,-10],\"limit\":[8,9],\"prefix_length\":9,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0085","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[9,10],\"coefficients\":[10,-11],\"shift\":2,\"prefix_length\":10}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(10, [[0, [14, 37/3, 23/2, 11, 32/3, 73/7, 41/4, 91/9, 10, 109/11]], [1, [9/2, 19/3, 29/4, 39/5, 49/6, 59/7, 69/8, 79/9, 89/10, 9]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(10, [[0, [14, 37/3, 23/2, 11, 32/3, 73/7, 41/4, 91/9, 10, 109/11]], [1, [9/2, 19/3, 29/4, 39/5, 49/6, 59/7, 69/8, 79/9, 89/10, 9]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032085,"parameters":{"limit":[9,10],"coefficients":[10,-11],"shift":2,"prefix_length":10,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[10,-11],\"limit\":[9,10],\"prefix_length\":10,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0086","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[10,11],\"coefficients\":[11,-12],\"shift\":2,\"prefix_length\":11}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(11, [[0, [31/2, 41/3, 51/4, 61/5, 71/6, 81/7, 91/8, 101/9, 111/10, 11, 131/12]], [1, [5, 7, 8, 43/5, 9, 65/7, 19/2, 29/3, 49/5, 109/11, 10]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(11, [[0, [31/2, 41/3, 51/4, 61/5, 71/6, 81/7, 91/8, 101/9, 111/10, 11, 131/12]], [1, [5, 7, 8, 43/5, 9, 65/7, 19/2, 29/3, 49/5, 109/11, 10]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032086,"parameters":{"limit":[10,11],"coefficients":[11,-12],"shift":2,"prefix_length":11,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[11,-12],\"limit\":[10,11],\"prefix_length\":11,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0087","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[11,12],\"coefficients\":[12,-13],\"shift\":2,\"prefix_length\":12}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(12, [[0, [17, 15, 14, 67/5, 13, 89/7, 25/2, 37/3, 61/5, 133/11, 12, 155/13]], [1, [11/2, 23/3, 35/4, 47/5, 59/6, 71/7, 83/8, 95/9, 107/10, 119/11, 131/12, 11]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(12, [[0, [17, 15, 14, 67/5, 13, 89/7, 25/2, 37/3, 61/5, 133/11, 12, 155/13]], [1, [11/2, 23/3, 35/4, 47/5, 59/6, 71/7, 83/8, 95/9, 107/10, 119/11, 131/12, 11]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032087,"parameters":{"limit":[11,12],"coefficients":[12,-13],"shift":2,"prefix_length":12,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[12,-13],\"limit\":[11,12],\"prefix_length\":12,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0088","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[12,13],\"coefficients\":[13,-14],\"shift\":2,\"prefix_length\":13}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(13, [[0, [37/2, 49/3, 61/4, 73/5, 85/6, 97/7, 109/8, 121/9, 133/10, 145/11, 157/12, 13, 181/14]], [1, [6, 25/3, 19/2, 51/5, 32/3, 11, 45/4, 103/9, 58/5, 129/11, 71/6, 155/13, 12]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(13, [[0, [37/2, 49/3, 61/4, 73/5, 85/6, 97/7, 109/8, 121/9, 133/10, 145/11, 157/12, 13, 181/14]], [1, [6, 25/3, 19/2, 51/5, 32/3, 11, 45/4, 103/9, 58/5, 129/11, 71/6, 155/13, 12]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032088,"parameters":{"limit":[12,13],"coefficients":[13,-14],"shift":2,"prefix_length":13,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[13,-14],\"limit\":[12,13],\"prefix_length\":13,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0089","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[13,14],\"coefficients\":[14,-15],\"shift\":2,\"prefix_length\":14}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(14, [[0, [20, 53/3, 33/2, 79/5, 46/3, 15, 59/4, 131/9, 72/5, 157/11, 85/6, 183/13, 14, 209/15]], [1, [13/2, 9, 41/4, 11, 23/2, 83/7, 97/8, 37/3, 25/2, 139/11, 51/4, 167/13, 181/14, 13]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(14, [[0, [20, 53/3, 33/2, 79/5, 46/3, 15, 59/4, 131/9, 72/5, 157/11, 85/6, 183/13, 14, 209/15]], [1, [13/2, 9, 41/4, 11, 23/2, 83/7, 97/8, 37/3, 25/2, 139/11, 51/4, 167/13, 181/14, 13]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032089,"parameters":{"limit":[13,14],"coefficients":[14,-15],"shift":2,"prefix_length":14,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[14,-15],\"limit\":[13,14],\"prefix_length\":14,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0090","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[14,15],\"coefficients\":[15,-16],\"shift\":2,\"prefix_length\":15}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(15, [[0, [43/2, 19, 71/4, 17, 33/2, 113/7, 127/8, 47/3, 31/2, 169/11, 61/4, 197/13, 211/14, 15, 239/16]], [1, [7, 29/3, 11, 59/5, 37/3, 89/7, 13, 119/9, 67/5, 149/11, 41/3, 179/13, 97/7, 209/15, 14]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(15, [[0, [43/2, 19, 71/4, 17, 33/2, 113/7, 127/8, 47/3, 31/2, 169/11, 61/4, 197/13, 211/14, 15, 239/16]], [1, [7, 29/3, 11, 59/5, 37/3, 89/7, 13, 119/9, 67/5, 149/11, 41/3, 179/13, 97/7, 209/15, 14]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032090,"parameters":{"limit":[14,15],"coefficients":[15,-16],"shift":2,"prefix_length":15,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[15,-16],\"limit\":[14,15],\"prefix_length\":15,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0091","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[15,16],\"coefficients\":[16,-17],\"shift\":2,\"prefix_length\":16}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(16, [[0, [23, 61/3, 19, 91/5, 53/3, 121/7, 17, 151/9, 83/5, 181/11, 49/3, 211/13, 113/7, 241/15, 16, 271/17]], [1, [15/2, 31/3, 47/4, 63/5, 79/6, 95/7, 111/8, 127/9, 143/10, 159/11, 175/12, 191/13, 207/14, 223/15, 239/16, 15]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(16, [[0, [23, 61/3, 19, 91/5, 53/3, 121/7, 17, 151/9, 83/5, 181/11, 49/3, 211/13, 113/7, 241/15, 16, 271/17]], [1, [15/2, 31/3, 47/4, 63/5, 79/6, 95/7, 111/8, 127/9, 143/10, 159/11, 175/12, 191/13, 207/14, 223/15, 239/16, 15]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032091,"parameters":{"limit":[15,16],"coefficients":[16,-17],"shift":2,"prefix_length":16,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[16,-17],\"limit\":[15,16],\"prefix_length\":16,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0092","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[16,17],\"coefficients\":[17,-18],\"shift\":2,\"prefix_length\":17}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(17, [[0, [49/2, 65/3, 81/4, 97/5, 113/6, 129/7, 145/8, 161/9, 177/10, 193/11, 209/12, 225/13, 241/14, 257/15, 273/16, 17, 305/18]], [1, [8, 11, 25/2, 67/5, 14, 101/7, 59/4, 15, 76/5, 169/11, 31/2, 203/13, 110/7, 79/5, 127/8, 271/17, 16]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(17, [[0, [49/2, 65/3, 81/4, 97/5, 113/6, 129/7, 145/8, 161/9, 177/10, 193/11, 209/12, 225/13, 241/14, 257/15, 273/16, 17, 305/18]], [1, [8, 11, 25/2, 67/5, 14, 101/7, 59/4, 15, 76/5, 169/11, 31/2, 203/13, 110/7, 79/5, 127/8, 271/17, 16]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032092,"parameters":{"limit":[16,17],"coefficients":[17,-18],"shift":2,"prefix_length":17,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[17,-18],\"limit\":[16,17],\"prefix_length\":17,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0093","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[17,18],\"coefficients\":[18,-19],\"shift\":2,\"prefix_length\":18}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(18, [[0, [26, 23, 43/2, 103/5, 20, 137/7, 77/4, 19, 94/5, 205/11, 37/2, 239/13, 128/7, 91/5, 145/8, 307/17, 18, 341/19]], [1, [17/2, 35/3, 53/4, 71/5, 89/6, 107/7, 125/8, 143/9, 161/10, 179/11, 197/12, 215/13, 233/14, 251/15, 269/16, 287/17, 305/18, 17]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(18, [[0, [26, 23, 43/2, 103/5, 20, 137/7, 77/4, 19, 94/5, 205/11, 37/2, 239/13, 128/7, 91/5, 145/8, 307/17, 18, 341/19]], [1, [17/2, 35/3, 53/4, 71/5, 89/6, 107/7, 125/8, 143/9, 161/10, 179/11, 197/12, 215/13, 233/14, 251/15, 269/16, 287/17, 305/18, 17]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032093,"parameters":{"limit":[17,18],"coefficients":[18,-19],"shift":2,"prefix_length":18,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[18,-19],\"limit\":[17,18],\"prefix_length\":18,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0094","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[18,19],\"coefficients\":[19,-20],\"shift\":2,\"prefix_length\":19}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(19, [[0, [55/2, 73/3, 91/4, 109/5, 127/6, 145/7, 163/8, 181/9, 199/10, 217/11, 235/12, 253/13, 271/14, 289/15, 307/16, 325/17, 343/18, 19, 379/20]], [1, [9, 37/3, 14, 15, 47/3, 113/7, 33/2, 151/9, 17, 189/11, 52/3, 227/13, 123/7, 53/3, 71/4, 303/17, 161/9, 341/19, 18]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(19, [[0, [55/2, 73/3, 91/4, 109/5, 127/6, 145/7, 163/8, 181/9, 199/10, 217/11, 235/12, 253/13, 271/14, 289/15, 307/16, 325/17, 343/18, 19, 379/20]], [1, [9, 37/3, 14, 15, 47/3, 113/7, 33/2, 151/9, 17, 189/11, 52/3, 227/13, 123/7, 53/3, 71/4, 303/17, 161/9, 341/19, 18]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032094,"parameters":{"limit":[18,19],"coefficients":[19,-20],"shift":2,"prefix_length":19,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[19,-20],\"limit\":[18,19],\"prefix_length\":19,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0095","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[19,20],\"coefficients\":[20,-21],\"shift\":2,\"prefix_length\":20}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(20, [[0, [29, 77/3, 24, 23, 67/3, 153/7, 43/2, 191/9, 21, 229/11, 62/3, 267/13, 143/7, 61/3, 81/4, 343/17, 181/9, 381/19, 20, 419/21]], [1, [19/2, 13, 59/4, 79/5, 33/2, 17, 139/8, 53/3, 179/10, 199/11, 73/4, 239/13, 37/2, 93/5, 299/16, 319/17, 113/6, 359/19, 379/20, 19]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(20, [[0, [29, 77/3, 24, 23, 67/3, 153/7, 43/2, 191/9, 21, 229/11, 62/3, 267/13, 143/7, 61/3, 81/4, 343/17, 181/9, 381/19, 20, 419/21]], [1, [19/2, 13, 59/4, 79/5, 33/2, 17, 139/8, 53/3, 179/10, 199/11, 73/4, 239/13, 37/2, 93/5, 299/16, 319/17, 113/6, 359/19, 379/20, 19]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032095,"parameters":{"limit":[19,20],"coefficients":[20,-21],"shift":2,"prefix_length":20,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[20,-21],\"limit\":[19,20],\"prefix_length\":20,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0096","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[20,21],\"coefficients\":[21,-22],\"shift\":2,\"prefix_length\":21}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(21, [[0, [61/2, 27, 101/4, 121/5, 47/2, 23, 181/8, 67/3, 221/10, 241/11, 87/4, 281/13, 43/2, 107/5, 341/16, 361/17, 127/6, 401/19, 421/20, 21, 461/22]], [1, [10, 41/3, 31/2, 83/5, 52/3, 125/7, 73/4, 167/9, 94/5, 19, 115/6, 251/13, 136/7, 293/15, 157/8, 335/17, 178/9, 377/19, 199/10, 419/21, 20]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(21, [[0, [61/2, 27, 101/4, 121/5, 47/2, 23, 181/8, 67/3, 221/10, 241/11, 87/4, 281/13, 43/2, 107/5, 341/16, 361/17, 127/6, 401/19, 421/20, 21, 461/22]], [1, [10, 41/3, 31/2, 83/5, 52/3, 125/7, 73/4, 167/9, 94/5, 19, 115/6, 251/13, 136/7, 293/15, 157/8, 335/17, 178/9, 377/19, 199/10, 419/21, 20]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032096,"parameters":{"limit":[20,21],"coefficients":[21,-22],"shift":2,"prefix_length":21,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[21,-22],\"limit\":[20,21],\"prefix_length\":21,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0097","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[21,22],\"coefficients\":[22,-23],\"shift\":2,\"prefix_length\":22}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(22, [[0, [32, 85/3, 53/2, 127/5, 74/3, 169/7, 95/4, 211/9, 116/5, 23, 137/6, 295/13, 158/7, 337/15, 179/8, 379/17, 200/9, 421/19, 221/10, 463/21, 22, 505/23]], [1, [21/2, 43/3, 65/4, 87/5, 109/6, 131/7, 153/8, 175/9, 197/10, 219/11, 241/12, 263/13, 285/14, 307/15, 329/16, 351/17, 373/18, 395/19, 417/20, 439/21, 461/22, 21]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(22, [[0, [32, 85/3, 53/2, 127/5, 74/3, 169/7, 95/4, 211/9, 116/5, 23, 137/6, 295/13, 158/7, 337/15, 179/8, 379/17, 200/9, 421/19, 221/10, 463/21, 22, 505/23]], [1, [21/2, 43/3, 65/4, 87/5, 109/6, 131/7, 153/8, 175/9, 197/10, 219/11, 241/12, 263/13, 285/14, 307/15, 329/16, 351/17, 373/18, 395/19, 417/20, 439/21, 461/22, 21]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032097,"parameters":{"limit":[21,22],"coefficients":[22,-23],"shift":2,"prefix_length":22,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[22,-23],\"limit\":[21,22],\"prefix_length\":22,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0098","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[22,23],\"coefficients\":[23,-24],\"shift\":2,\"prefix_length\":23}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(23, [[0, [67/2, 89/3, 111/4, 133/5, 155/6, 177/7, 199/8, 221/9, 243/10, 265/11, 287/12, 309/13, 331/14, 353/15, 375/16, 397/17, 419/18, 441/19, 463/20, 485/21, 507/22, 23, 551/24]], [1, [11, 15, 17, 91/5, 19, 137/7, 20, 61/3, 103/5, 229/11, 21, 275/13, 149/7, 107/5, 43/2, 367/17, 65/3, 413/19, 109/5, 153/7, 241/11, 505/23, 22]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(23, [[0, [67/2, 89/3, 111/4, 133/5, 155/6, 177/7, 199/8, 221/9, 243/10, 265/11, 287/12, 309/13, 331/14, 353/15, 375/16, 397/17, 419/18, 441/19, 463/20, 485/21, 507/22, 23, 551/24]], [1, [11, 15, 17, 91/5, 19, 137/7, 20, 61/3, 103/5, 229/11, 21, 275/13, 149/7, 107/5, 43/2, 367/17, 65/3, 413/19, 109/5, 153/7, 241/11, 505/23, 22]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032098,"parameters":{"limit":[22,23],"coefficients":[23,-24],"shift":2,"prefix_length":23,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[23,-24],\"limit\":[22,23],\"prefix_length\":23,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0099","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[23,24],\"coefficients\":[24,-25],\"shift\":2,\"prefix_length\":24}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(24, [[0, [35, 31, 29, 139/5, 27, 185/7, 26, 77/3, 127/5, 277/11, 25, 323/13, 173/7, 123/5, 49/2, 415/17, 73/3, 461/19, 121/5, 169/7, 265/11, 553/23, 24, 599/25]], [1, [23/2, 47/3, 71/4, 19, 119/6, 143/7, 167/8, 191/9, 43/2, 239/11, 263/12, 287/13, 311/14, 67/3, 359/16, 383/17, 407/18, 431/19, 91/4, 479/21, 503/22, 527/23, 551/24, 23]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(24, [[0, [35, 31, 29, 139/5, 27, 185/7, 26, 77/3, 127/5, 277/11, 25, 323/13, 173/7, 123/5, 49/2, 415/17, 73/3, 461/19, 121/5, 169/7, 265/11, 553/23, 24, 599/25]], [1, [23/2, 47/3, 71/4, 19, 119/6, 143/7, 167/8, 191/9, 43/2, 239/11, 263/12, 287/13, 311/14, 67/3, 359/16, 383/17, 407/18, 431/19, 91/4, 479/21, 503/22, 527/23, 551/24, 23]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032099,"parameters":{"limit":[23,24],"coefficients":[24,-25],"shift":2,"prefix_length":24,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[24,-25],\"limit\":[23,24],\"prefix_length\":24,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0100","batch_id":"batch-20261014-advance-20261103-r02-pointwise-convergence-of-functions-foundations","language":"zh-Hant","source_mko_ids":["mko-pointwise-convergence-of-functions"],"title_zh":"函數列逐點收斂","stem_zh":"同一有限domain上的symbolic rational function tail。操作=pointwise-finite-prefix-boundary；資料={\"limit\":[24,25],\"coefficients\":[25,-26],\"shift\":2,\"prefix_length\":25}。請依 [prefix_length,value_rows,sufficient,boundary] 欄序回答。","answer":{"kind":"expression","value":"(25, [[0, [73/2, 97/3, 121/4, 29, 169/6, 193/7, 217/8, 241/9, 53/2, 289/11, 313/12, 337/13, 361/14, 77/3, 409/16, 433/17, 457/18, 481/19, 101/4, 529/21, 553/22, 577/23, 601/24, 25, 649/26]], [1, [12, 49/3, 37/2, 99/5, 62/3, 149/7, 87/4, 199/9, 112/5, 249/11, 137/6, 23, 162/7, 349/15, 187/8, 399/17, 212/9, 449/19, 237/10, 499/21, 262/11, 549/23, 287/12, 599/25, 24]]], 否, finite_prefix_not_pointwise_convergence_certificate)","explanation_zh":"Wave5第三層核對=(25, [[0, [73/2, 97/3, 121/4, 29, 169/6, 193/7, 217/8, 241/9, 53/2, 289/11, 313/12, 337/13, 361/14, 77/3, 409/16, 433/17, 457/18, 481/19, 101/4, 529/21, 553/22, 577/23, 601/24, 25, 649/26]], [1, [12, 49/3, 37/2, 99/5, 62/3, 149/7, 87/4, 199/9, 112/5, 249/11, 137/6, 23, 162/7, 349/15, 187/8, 399/17, 212/9, 449/19, 237/10, 499/21, 262/11, 549/23, 287/12, 599/25, 24]]], 否, finite_prefix_not_pointwise_convergence_certificate)；依完整明示資料重算；逐點量詞保留N依x與epsilon且tail對所有n大於等於N；有限domain最大witness不外推無限domain；有限prefix不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["函數列逐點收斂"]},"variant":{"family_id":"qf-mod34-pointwise-finite-prefix-boundary","seed":202611032100,"parameters":{"limit":[24,25],"coefficients":[25,-26],"shift":2,"prefix_length":25,"domain":"Finite_domain_symbolic_pointwise_tail","operation":"pointwise-finite-prefix-boundary","semantic_template":"qf-mod34-pointwise-finite-prefix-boundary/{\"coefficients\":[25,-26],\"limit\":[24,25],\"prefix_length\":25,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}