{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0001","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[1,2],\"coefficient\":1,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 1/2, 1, 2, [0, 1/6, 1/4], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 1/2, 1, 2, [0, 1/6, 1/4], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082001,"parameters":{"limit":[1,2],"coefficient":1,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":1,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0002","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[2,3],\"coefficient\":2,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 2/3, 2, 2, [-1/3, 0, 1/6], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 2/3, 2, 2, [-1/3, 0, 1/6], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082002,"parameters":{"limit":[2,3],"coefficient":2,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":2,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0003","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[3,4],\"coefficient\":3,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 3/4, 3, 2, [-3/4, -1/4, 0], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 3/4, 3, 2, [-3/4, -1/4, 0], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082003,"parameters":{"limit":[3,4],"coefficient":3,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":3,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0004","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[4,5],\"coefficient\":4,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 4/5, 4, 2, [-6/5, -8/15, -1/5], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 4/5, 4, 2, [-6/5, -8/15, -1/5], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082004,"parameters":{"limit":[4,5],"coefficient":4,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":4,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0005","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[5,6],\"coefficient\":5,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 5/6, 5, 2, [-5/3, -5/6, -5/12], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 5/6, 5, 2, [-5/3, -5/6, -5/12], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082005,"parameters":{"limit":[5,6],"coefficient":5,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":5,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0006","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[6,7],\"coefficient\":6,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 6/7, 6, 2, [-15/7, -8/7, -9/14], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 6/7, 6, 2, [-15/7, -8/7, -9/14], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082006,"parameters":{"limit":[6,7],"coefficient":6,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":6,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0007","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[7,8],\"coefficient\":7,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 7/8, 7, 2, [-21/8, -35/24, -7/8], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 7/8, 7, 2, [-21/8, -35/24, -7/8], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082007,"parameters":{"limit":[7,8],"coefficient":7,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":7,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0008","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[8,9],\"coefficient\":8,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 8/9, 8, 2, [-28/9, -16/9, -10/9], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 8/9, 8, 2, [-28/9, -16/9, -10/9], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082008,"parameters":{"limit":[8,9],"coefficient":8,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":8,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0009","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[9,10],\"coefficient\":9,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 9/10, 9, 2, [-18/5, -21/10, -27/20], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 9/10, 9, 2, [-18/5, -21/10, -27/20], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082009,"parameters":{"limit":[9,10],"coefficient":9,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":9,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0010","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[10,11],\"coefficient\":10,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 10/11, 10, 2, [-45/11, -80/33, -35/22], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 10/11, 10, 2, [-45/11, -80/33, -35/22], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082010,"parameters":{"limit":[10,11],"coefficient":10,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":10,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0011","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[11,12],\"coefficient\":11,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 11/12, 11, 2, [-55/12, -11/4, -11/6], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 11/12, 11, 2, [-55/12, -11/4, -11/6], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082011,"parameters":{"limit":[11,12],"coefficient":11,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":11,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0012","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[12,13],\"coefficient\":12,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 12/13, 12, 2, [-66/13, -40/13, -27/13], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 12/13, 12, 2, [-66/13, -40/13, -27/13], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082012,"parameters":{"limit":[12,13],"coefficient":12,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":12,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0013","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[13,14],\"coefficient\":13,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 13/14, 13, 2, [-39/7, -143/42, -65/28], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 13/14, 13, 2, [-39/7, -143/42, -65/28], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082013,"parameters":{"limit":[13,14],"coefficient":13,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":13,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0014","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[14,15],\"coefficient\":14,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 14/15, 14, 2, [-91/15, -56/15, -77/30], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 14/15, 14, 2, [-91/15, -56/15, -77/30], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082014,"parameters":{"limit":[14,15],"coefficient":14,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":14,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0015","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[15,16],\"coefficient\":15,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 15/16, 15, 2, [-105/16, -65/16, -45/16], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 15/16, 15, 2, [-105/16, -65/16, -45/16], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082015,"parameters":{"limit":[15,16],"coefficient":15,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":15,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0016","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[16,17],\"coefficient\":16,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 16/17, 16, 2, [-120/17, -224/51, -52/17], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 16/17, 16, 2, [-120/17, -224/51, -52/17], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082016,"parameters":{"limit":[16,17],"coefficient":16,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":16,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0017","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[17,18],\"coefficient\":17,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 17/18, 17, 2, [-68/9, -85/18, -119/36], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 17/18, 17, 2, [-68/9, -85/18, -119/36], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082017,"parameters":{"limit":[17,18],"coefficient":17,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":17,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0018","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[18,19],\"coefficient\":18,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 18/19, 18, 2, [-153/19, -96/19, -135/38], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 18/19, 18, 2, [-153/19, -96/19, -135/38], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082018,"parameters":{"limit":[18,19],"coefficient":18,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":18,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0019","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[19,20],\"coefficient\":19,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 19/20, 19, 2, [-171/20, -323/60, -19/5], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 19/20, 19, 2, [-171/20, -323/60, -19/5], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082019,"parameters":{"limit":[19,20],"coefficient":19,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":19,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0020","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[20,21],\"coefficient\":20,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 20/21, 20, 2, [-190/21, -40/7, -85/21], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 20/21, 20, 2, [-190/21, -40/7, -85/21], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082020,"parameters":{"limit":[20,21],"coefficient":20,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":20,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0021","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[21,22],\"coefficient\":21,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 21/22, 21, 2, [-105/11, -133/22, -189/44], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 21/22, 21, 2, [-105/11, -133/22, -189/44], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082021,"parameters":{"limit":[21,22],"coefficient":21,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":21,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0022","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[22,23],\"coefficient\":22,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 22/23, 22, 2, [-231/23, -440/69, -209/46], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 22/23, 22, 2, [-231/23, -440/69, -209/46], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082022,"parameters":{"limit":[22,23],"coefficient":22,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":22,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0023","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[23,24],\"coefficient\":23,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 23/24, 23, 2, [-253/24, -161/24, -115/24], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 23/24, 23, 2, [-253/24, -161/24, -115/24], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082023,"parameters":{"limit":[23,24],"coefficient":23,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":23,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0024","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[24,25],\"coefficient\":24,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 24/25, 24, 2, [-276/25, -176/25, -126/25], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 24/25, 24, 2, [-276/25, -176/25, -126/25], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082024,"parameters":{"limit":[24,25],"coefficient":24,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":24,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0025","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-increasing-limit-certificate；資料={\"limit\":[25,26],\"coefficient\":25,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 25/26, 25, 2, [-150/13, -575/78, -275/52], 是, nondecreasing_bounded_above_supremum_limit)","explanation_zh":"Wave6第二層核對=(increasing, 25/26, 25, 2, [-150/13, -575/78, -275/52], 是, nondecreasing_bounded_above_supremum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082025,"parameters":{"limit":[25,26],"coefficient":25,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-increasing-limit-certificate","semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":25,\"limit\":[25,26],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0026","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[1,2],\"coefficient\":1,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 1/2, 1, 2, [1, 5/6, 3/4], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 1/2, 1, 2, [1, 5/6, 3/4], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082026,"parameters":{"limit":[1,2],"coefficient":1,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":1,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0027","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[2,3],\"coefficient\":2,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 2/3, 2, 2, [5/3, 4/3, 7/6], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 2/3, 2, 2, [5/3, 4/3, 7/6], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082027,"parameters":{"limit":[2,3],"coefficient":2,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":2,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0028","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[3,4],\"coefficient\":3,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 3/4, 3, 2, [9/4, 7/4, 3/2], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 3/4, 3, 2, [9/4, 7/4, 3/2], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082028,"parameters":{"limit":[3,4],"coefficient":3,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":3,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0029","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[4,5],\"coefficient\":4,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 4/5, 4, 2, [14/5, 32/15, 9/5], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 4/5, 4, 2, [14/5, 32/15, 9/5], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082029,"parameters":{"limit":[4,5],"coefficient":4,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":4,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0030","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[5,6],\"coefficient\":5,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 5/6, 5, 2, [10/3, 5/2, 25/12], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 5/6, 5, 2, [10/3, 5/2, 25/12], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082030,"parameters":{"limit":[5,6],"coefficient":5,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":5,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0031","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[6,7],\"coefficient\":6,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 6/7, 6, 2, [27/7, 20/7, 33/14], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 6/7, 6, 2, [27/7, 20/7, 33/14], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082031,"parameters":{"limit":[6,7],"coefficient":6,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":6,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0032","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[7,8],\"coefficient\":7,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 7/8, 7, 2, [35/8, 77/24, 21/8], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 7/8, 7, 2, [35/8, 77/24, 21/8], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082032,"parameters":{"limit":[7,8],"coefficient":7,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":7,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0033","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[8,9],\"coefficient\":8,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 8/9, 8, 2, [44/9, 32/9, 26/9], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 8/9, 8, 2, [44/9, 32/9, 26/9], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082033,"parameters":{"limit":[8,9],"coefficient":8,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":8,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0034","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[9,10],\"coefficient\":9,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 9/10, 9, 2, [27/5, 39/10, 63/20], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 9/10, 9, 2, [27/5, 39/10, 63/20], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082034,"parameters":{"limit":[9,10],"coefficient":9,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":9,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0035","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[10,11],\"coefficient\":10,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 10/11, 10, 2, [65/11, 140/33, 75/22], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 10/11, 10, 2, [65/11, 140/33, 75/22], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082035,"parameters":{"limit":[10,11],"coefficient":10,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":10,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0036","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[11,12],\"coefficient\":11,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 11/12, 11, 2, [77/12, 55/12, 11/3], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 11/12, 11, 2, [77/12, 55/12, 11/3], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082036,"parameters":{"limit":[11,12],"coefficient":11,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":11,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0037","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[12,13],\"coefficient\":12,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 12/13, 12, 2, [90/13, 64/13, 51/13], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 12/13, 12, 2, [90/13, 64/13, 51/13], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082037,"parameters":{"limit":[12,13],"coefficient":12,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":12,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0038","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[13,14],\"coefficient\":13,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 13/14, 13, 2, [52/7, 221/42, 117/28], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 13/14, 13, 2, [52/7, 221/42, 117/28], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082038,"parameters":{"limit":[13,14],"coefficient":13,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":13,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0039","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[14,15],\"coefficient\":14,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 14/15, 14, 2, [119/15, 28/5, 133/30], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 14/15, 14, 2, [119/15, 28/5, 133/30], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082039,"parameters":{"limit":[14,15],"coefficient":14,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":14,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0040","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[15,16],\"coefficient\":15,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 15/16, 15, 2, [135/16, 95/16, 75/16], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 15/16, 15, 2, [135/16, 95/16, 75/16], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082040,"parameters":{"limit":[15,16],"coefficient":15,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":15,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0041","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[16,17],\"coefficient\":16,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 16/17, 16, 2, [152/17, 320/51, 84/17], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 16/17, 16, 2, [152/17, 320/51, 84/17], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082041,"parameters":{"limit":[16,17],"coefficient":16,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":16,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0042","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[17,18],\"coefficient\":17,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 17/18, 17, 2, [85/9, 119/18, 187/36], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 17/18, 17, 2, [85/9, 119/18, 187/36], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082042,"parameters":{"limit":[17,18],"coefficient":17,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":17,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0043","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[18,19],\"coefficient\":18,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 18/19, 18, 2, [189/19, 132/19, 207/38], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 18/19, 18, 2, [189/19, 132/19, 207/38], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082043,"parameters":{"limit":[18,19],"coefficient":18,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":18,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0044","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[19,20],\"coefficient\":19,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 19/20, 19, 2, [209/20, 437/60, 57/10], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 19/20, 19, 2, [209/20, 437/60, 57/10], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082044,"parameters":{"limit":[19,20],"coefficient":19,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":19,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0045","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[20,21],\"coefficient\":20,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 20/21, 20, 2, [230/21, 160/21, 125/21], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 20/21, 20, 2, [230/21, 160/21, 125/21], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082045,"parameters":{"limit":[20,21],"coefficient":20,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":20,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0046","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[21,22],\"coefficient\":21,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 21/22, 21, 2, [126/11, 175/22, 273/44], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 21/22, 21, 2, [126/11, 175/22, 273/44], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082046,"parameters":{"limit":[21,22],"coefficient":21,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":21,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0047","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[22,23],\"coefficient\":22,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 22/23, 22, 2, [275/23, 572/69, 297/46], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 22/23, 22, 2, [275/23, 572/69, 297/46], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082047,"parameters":{"limit":[22,23],"coefficient":22,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":22,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0048","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[23,24],\"coefficient\":23,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 23/24, 23, 2, [299/24, 69/8, 161/24], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 23/24, 23, 2, [299/24, 69/8, 161/24], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082048,"parameters":{"limit":[23,24],"coefficient":23,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":23,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0049","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[24,25],\"coefficient\":24,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 24/25, 24, 2, [324/25, 224/25, 174/25], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 24/25, 24, 2, [324/25, 224/25, 174/25], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082049,"parameters":{"limit":[24,25],"coefficient":24,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":24,\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0050","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-decreasing-limit-certificate；資料={\"limit\":[25,26],\"coefficient\":25,\"shift\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 25/26, 25, 2, [175/13, 725/78, 375/52], 是, nonincreasing_bounded_below_infimum_limit)","explanation_zh":"Wave6第二層核對=(decreasing, 25/26, 25, 2, [175/13, 725/78, 375/52], 是, nonincreasing_bounded_below_infimum_limit)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082050,"parameters":{"limit":[25,26],"coefficient":25,"shift":2,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-decreasing-limit-certificate","semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":25,\"limit\":[25,26],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0051","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[1,2],\"coefficient\":1,\"shift\":2,\"epsilon\":[1,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 1/2, 1/10, 9, 9/22, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 1/2, 1/10, 9, 9/22, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082051,"parameters":{"limit":[1,2],"coefficient":1,"shift":2,"direction":"increasing","epsilon":[1,10],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":1,\"direction\":\"increasing\",\"epsilon\":[1,10],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0052","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"decreasing\",\"limit\":[2,3],\"coefficient\":2,\"shift\":2,\"epsilon\":[1,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 2/3, 1/11, 21, 52/69, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(decreasing, 2/3, 1/11, 21, 52/69, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082052,"parameters":{"limit":[2,3],"coefficient":2,"shift":2,"direction":"decreasing","epsilon":[1,11],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":2,\"direction\":\"decreasing\",\"epsilon\":[1,11],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0053","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[3,4],\"coefficient\":3,\"shift\":2,\"epsilon\":[1,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 3/4, 1/12, 35, 99/148, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 3/4, 1/12, 35, 99/148, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082053,"parameters":{"limit":[3,4],"coefficient":3,"shift":2,"direction":"increasing","epsilon":[1,12],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":3,\"direction\":\"increasing\",\"epsilon\":[1,12],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0054","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"decreasing\",\"limit\":[4,5],\"coefficient\":4,\"shift\":2,\"epsilon\":[1,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 4/5, 1/13, 51, 232/265, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(decreasing, 4/5, 1/13, 51, 232/265, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082054,"parameters":{"limit":[4,5],"coefficient":4,"shift":2,"direction":"decreasing","epsilon":[1,13],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":4,\"direction\":\"decreasing\",\"epsilon\":[1,13],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0055","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[5,6],\"coefficient\":5,\"shift\":2,\"epsilon\":[1,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 5/6, 1/14, 69, 325/426, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 5/6, 1/14, 69, 325/426, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082055,"parameters":{"limit":[5,6],"coefficient":5,"shift":2,"direction":"increasing","epsilon":[1,14],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":5,\"direction\":\"increasing\",\"epsilon\":[1,14],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0056","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"decreasing\",\"limit\":[6,7],\"coefficient\":6,\"shift\":2,\"epsilon\":[1,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 6/7, 1/15, 89, 12/13, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(decreasing, 6/7, 1/15, 89, 12/13, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082056,"parameters":{"limit":[6,7],"coefficient":6,"shift":2,"direction":"decreasing","epsilon":[1,15],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":6,\"direction\":\"decreasing\",\"epsilon\":[1,15],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0057","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[7,8],\"coefficient\":7,\"shift\":2,\"epsilon\":[1,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 7/8, 1/16, 111, 735/904, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 7/8, 1/16, 111, 735/904, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082057,"parameters":{"limit":[7,8],"coefficient":7,"shift":2,"direction":"increasing","epsilon":[1,16],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":7,\"direction\":\"increasing\",\"epsilon\":[1,16],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0058","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"decreasing\",\"limit\":[8,9],\"coefficient\":8,\"shift\":2,\"epsilon\":[1,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 8/9, 1/17, 135, 1168/1233, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(decreasing, 8/9, 1/17, 135, 1168/1233, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082058,"parameters":{"limit":[8,9],"coefficient":8,"shift":2,"direction":"decreasing","epsilon":[1,17],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":8,\"direction\":\"decreasing\",\"epsilon\":[1,17],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0059","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[9,10],\"coefficient\":9,\"shift\":2,\"epsilon\":[1,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 9/10, 1/18, 161, 1377/1630, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 9/10, 1/18, 161, 1377/1630, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082059,"parameters":{"limit":[9,10],"coefficient":9,"shift":2,"direction":"increasing","epsilon":[1,18],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":9,\"direction\":\"increasing\",\"epsilon\":[1,18],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0060","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"decreasing\",\"limit\":[10,11],\"coefficient\":10,\"shift\":2,\"epsilon\":[1,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 10/11, 1/19, 189, 2020/2101, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(decreasing, 10/11, 1/19, 189, 2020/2101, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082060,"parameters":{"limit":[10,11],"coefficient":10,"shift":2,"direction":"decreasing","epsilon":[1,19],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":10,\"direction\":\"decreasing\",\"epsilon\":[1,19],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0061","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[11,12],\"coefficient\":11,\"shift\":2,\"epsilon\":[1,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 11/12, 1/20, 219, 2299/2652, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 11/12, 1/20, 219, 2299/2652, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082061,"parameters":{"limit":[11,12],"coefficient":11,"shift":2,"direction":"increasing","epsilon":[1,20],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":11,\"direction\":\"increasing\",\"epsilon\":[1,20],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0062","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"decreasing\",\"limit\":[12,13],\"coefficient\":12,\"shift\":2,\"epsilon\":[1,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 12/13, 1/21, 251, 3192/3289, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(decreasing, 12/13, 1/21, 251, 3192/3289, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082062,"parameters":{"limit":[12,13],"coefficient":12,"shift":2,"direction":"decreasing","epsilon":[1,21],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":12,\"direction\":\"decreasing\",\"epsilon\":[1,21],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0063","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[13,14],\"coefficient\":13,\"shift\":2,\"epsilon\":[1,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 13/14, 1/22, 285, 507/574, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 13/14, 1/22, 285, 507/574, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082063,"parameters":{"limit":[13,14],"coefficient":13,"shift":2,"direction":"increasing","epsilon":[1,22],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":13,\"direction\":\"increasing\",\"epsilon\":[1,22],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0064","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"decreasing\",\"limit\":[14,15],\"coefficient\":14,\"shift\":2,\"epsilon\":[1,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 14/15, 1/23, 321, 4732/4845, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(decreasing, 14/15, 1/23, 321, 4732/4845, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082064,"parameters":{"limit":[14,15],"coefficient":14,"shift":2,"direction":"decreasing","epsilon":[1,23],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":14,\"direction\":\"decreasing\",\"epsilon\":[1,23],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0065","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[15,16],\"coefficient\":15,\"shift\":2,\"epsilon\":[1,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 15/16, 1/24, 359, 5175/5776, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 15/16, 1/24, 359, 5175/5776, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082065,"parameters":{"limit":[15,16],"coefficient":15,"shift":2,"direction":"increasing","epsilon":[1,24],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":15,\"direction\":\"increasing\",\"epsilon\":[1,24],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0066","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"decreasing\",\"limit\":[16,17],\"coefficient\":16,\"shift\":2,\"epsilon\":[1,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 16/17, 1/25, 399, 6688/6817, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(decreasing, 16/17, 1/25, 399, 6688/6817, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082066,"parameters":{"limit":[16,17],"coefficient":16,"shift":2,"direction":"decreasing","epsilon":[1,25],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":16,\"direction\":\"decreasing\",\"epsilon\":[1,25],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0067","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[17,18],\"coefficient\":17,\"shift\":2,\"epsilon\":[1,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 17/18, 1/26, 441, 7225/7974, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 17/18, 1/26, 441, 7225/7974, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082067,"parameters":{"limit":[17,18],"coefficient":17,"shift":2,"direction":"increasing","epsilon":[1,26],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":17,\"direction\":\"increasing\",\"epsilon\":[1,26],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0068","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"decreasing\",\"limit\":[18,19],\"coefficient\":18,\"shift\":2,\"epsilon\":[1,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 18/19, 1/27, 485, 9108/9253, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(decreasing, 18/19, 1/27, 485, 9108/9253, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082068,"parameters":{"limit":[18,19],"coefficient":18,"shift":2,"direction":"decreasing","epsilon":[1,27],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":18,\"direction\":\"decreasing\",\"epsilon\":[1,27],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0069","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[19,20],\"coefficient\":19,\"shift\":2,\"epsilon\":[1,28]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 19/20, 1/28, 531, 9747/10660, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 19/20, 1/28, 531, 9747/10660, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082069,"parameters":{"limit":[19,20],"coefficient":19,"shift":2,"direction":"increasing","epsilon":[1,28],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":19,\"direction\":\"increasing\",\"epsilon\":[1,28],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0070","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"decreasing\",\"limit\":[20,21],\"coefficient\":20,\"shift\":2,\"epsilon\":[1,29]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 20/21, 1/29, 579, 1720/1743, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(decreasing, 20/21, 1/29, 579, 1720/1743, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082070,"parameters":{"limit":[20,21],"coefficient":20,"shift":2,"direction":"decreasing","epsilon":[1,29],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":20,\"direction\":\"decreasing\",\"epsilon\":[1,29],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0071","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[21,22],\"coefficient\":21,\"shift\":2,\"epsilon\":[1,30]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 21/22, 1/30, 629, 12789/13882, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 21/22, 1/30, 629, 12789/13882, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082071,"parameters":{"limit":[21,22],"coefficient":21,"shift":2,"direction":"increasing","epsilon":[1,30],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":21,\"direction\":\"increasing\",\"epsilon\":[1,30],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0072","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"decreasing\",\"limit\":[22,23],\"coefficient\":22,\"shift\":2,\"epsilon\":[1,31]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 22/23, 1/31, 681, 15532/15709, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(decreasing, 22/23, 1/31, 681, 15532/15709, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082072,"parameters":{"limit":[22,23],"coefficient":22,"shift":2,"direction":"decreasing","epsilon":[1,31],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":22,\"direction\":\"decreasing\",\"epsilon\":[1,31],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0073","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[23,24],\"coefficient\":23,\"shift\":2,\"epsilon\":[1,32]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 23/24, 1/32, 735, 16399/17688, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 23/24, 1/32, 735, 16399/17688, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082073,"parameters":{"limit":[23,24],"coefficient":23,"shift":2,"direction":"increasing","epsilon":[1,32],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":23,\"direction\":\"increasing\",\"epsilon\":[1,32],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0074","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"decreasing\",\"limit\":[24,25],\"coefficient\":24,\"shift\":2,\"epsilon\":[1,33]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decreasing, 24/25, 1/33, 791, 19632/19825, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(decreasing, 24/25, 1/33, 791, 19632/19825, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082074,"parameters":{"limit":[24,25],"coefficient":24,"shift":2,"direction":"decreasing","epsilon":[1,33],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":24,\"direction\":\"decreasing\",\"epsilon\":[1,33],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0075","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-epsilon-witness；資料={\"direction\":\"increasing\",\"limit\":[25,26],\"coefficient\":25,\"shift\":2,\"epsilon\":[1,34]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(increasing, 25/26, 1/34, 849, 20625/22126, 是, forall_n_ge_N_by_monotone_symbolic_tail)","explanation_zh":"Wave6第二層核對=(increasing, 25/26, 1/34, 849, 20625/22126, 是, forall_n_ge_N_by_monotone_symbolic_tail)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082075,"parameters":{"limit":[25,26],"coefficient":25,"shift":2,"direction":"increasing","epsilon":[1,34],"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-epsilon-witness","semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":25,\"direction\":\"increasing\",\"epsilon\":[1,34],\"limit\":[25,26],\"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":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0076","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"monotone_unbounded\",\"prefix_length\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2], 是, 否, 否, monotone_without_bound_not_covered)","explanation_zh":"Wave6第二層核對=(monotone_unbounded, [0, 1, 2], 是, 否, 否, monotone_without_bound_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082076,"parameters":{"mode":"monotone_unbounded","prefix_length":3,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0077","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"bounded_nonmonotone\",\"prefix_length\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1], 否, 是, 否, bounded_without_monotonicity_not_covered)","explanation_zh":"Wave6第二層核對=(bounded_nonmonotone, [1, -1, 1, -1], 否, 是, 否, bounded_without_monotonicity_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082077,"parameters":{"mode":"bounded_nonmonotone","prefix_length":4,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0078","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"finite_prefix_only\",\"prefix_length\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_prefix_only, 5, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)","explanation_zh":"Wave6第二層核對=(finite_prefix_only, 5, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082078,"parameters":{"mode":"finite_prefix_only","prefix_length":5,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0079","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"monotone_unbounded\",\"prefix_length\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5], 是, 否, 否, monotone_without_bound_not_covered)","explanation_zh":"Wave6第二層核對=(monotone_unbounded, [0, 1, 2, 3, 4, 5], 是, 否, 否, monotone_without_bound_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082079,"parameters":{"mode":"monotone_unbounded","prefix_length":6,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0080","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"bounded_nonmonotone\",\"prefix_length\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1], 否, 是, 否, bounded_without_monotonicity_not_covered)","explanation_zh":"Wave6第二層核對=(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1], 否, 是, 否, bounded_without_monotonicity_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082080,"parameters":{"mode":"bounded_nonmonotone","prefix_length":7,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0081","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"finite_prefix_only\",\"prefix_length\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_prefix_only, 8, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)","explanation_zh":"Wave6第二層核對=(finite_prefix_only, 8, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082081,"parameters":{"mode":"finite_prefix_only","prefix_length":8,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0082","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"monotone_unbounded\",\"prefix_length\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8], 是, 否, 否, monotone_without_bound_not_covered)","explanation_zh":"Wave6第二層核對=(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8], 是, 否, 否, monotone_without_bound_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082082,"parameters":{"mode":"monotone_unbounded","prefix_length":9,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0083","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"bounded_nonmonotone\",\"prefix_length\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1], 否, 是, 否, bounded_without_monotonicity_not_covered)","explanation_zh":"Wave6第二層核對=(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1], 否, 是, 否, bounded_without_monotonicity_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082083,"parameters":{"mode":"bounded_nonmonotone","prefix_length":10,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0084","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"finite_prefix_only\",\"prefix_length\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_prefix_only, 11, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)","explanation_zh":"Wave6第二層核對=(finite_prefix_only, 11, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082084,"parameters":{"mode":"finite_prefix_only","prefix_length":11,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0085","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"monotone_unbounded\",\"prefix_length\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, 否, 否, monotone_without_bound_not_covered)","explanation_zh":"Wave6第二層核對=(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], 是, 否, 否, monotone_without_bound_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082085,"parameters":{"mode":"monotone_unbounded","prefix_length":12,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0086","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"bounded_nonmonotone\",\"prefix_length\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], 否, 是, 否, bounded_without_monotonicity_not_covered)","explanation_zh":"Wave6第二層核對=(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], 否, 是, 否, bounded_without_monotonicity_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082086,"parameters":{"mode":"bounded_nonmonotone","prefix_length":13,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0087","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"finite_prefix_only\",\"prefix_length\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_prefix_only, 14, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)","explanation_zh":"Wave6第二層核對=(finite_prefix_only, 14, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082087,"parameters":{"mode":"finite_prefix_only","prefix_length":14,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0088","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"monotone_unbounded\",\"prefix_length\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], 是, 否, 否, monotone_without_bound_not_covered)","explanation_zh":"Wave6第二層核對=(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], 是, 否, 否, monotone_without_bound_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082088,"parameters":{"mode":"monotone_unbounded","prefix_length":15,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0089","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"bounded_nonmonotone\",\"prefix_length\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1], 否, 是, 否, bounded_without_monotonicity_not_covered)","explanation_zh":"Wave6第二層核對=(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1], 否, 是, 否, bounded_without_monotonicity_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082089,"parameters":{"mode":"bounded_nonmonotone","prefix_length":16,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0090","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"finite_prefix_only\",\"prefix_length\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_prefix_only, 17, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)","explanation_zh":"Wave6第二層核對=(finite_prefix_only, 17, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082090,"parameters":{"mode":"finite_prefix_only","prefix_length":17,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0091","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"monotone_unbounded\",\"prefix_length\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], 是, 否, 否, monotone_without_bound_not_covered)","explanation_zh":"Wave6第二層核對=(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], 是, 否, 否, monotone_without_bound_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082091,"parameters":{"mode":"monotone_unbounded","prefix_length":18,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0092","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"bounded_nonmonotone\",\"prefix_length\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], 否, 是, 否, bounded_without_monotonicity_not_covered)","explanation_zh":"Wave6第二層核對=(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], 否, 是, 否, bounded_without_monotonicity_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082092,"parameters":{"mode":"bounded_nonmonotone","prefix_length":19,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0093","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"finite_prefix_only\",\"prefix_length\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_prefix_only, 20, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)","explanation_zh":"Wave6第二層核對=(finite_prefix_only, 20, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082093,"parameters":{"mode":"finite_prefix_only","prefix_length":20,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0094","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"monotone_unbounded\",\"prefix_length\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], 是, 否, 否, monotone_without_bound_not_covered)","explanation_zh":"Wave6第二層核對=(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], 是, 否, 否, monotone_without_bound_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082094,"parameters":{"mode":"monotone_unbounded","prefix_length":21,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0095","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"bounded_nonmonotone\",\"prefix_length\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1], 否, 是, 否, bounded_without_monotonicity_not_covered)","explanation_zh":"Wave6第二層核對=(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1], 否, 是, 否, bounded_without_monotonicity_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082095,"parameters":{"mode":"bounded_nonmonotone","prefix_length":22,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0096","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"finite_prefix_only\",\"prefix_length\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_prefix_only, 23, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)","explanation_zh":"Wave6第二層核對=(finite_prefix_only, 23, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082096,"parameters":{"mode":"finite_prefix_only","prefix_length":23,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0097","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"monotone_unbounded\",\"prefix_length\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], 是, 否, 否, monotone_without_bound_not_covered)","explanation_zh":"Wave6第二層核對=(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], 是, 否, 否, monotone_without_bound_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082097,"parameters":{"mode":"monotone_unbounded","prefix_length":24,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0098","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"bounded_nonmonotone\",\"prefix_length\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], 否, 是, 否, bounded_without_monotonicity_not_covered)","explanation_zh":"Wave6第二層核對=(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], 否, 是, 否, bounded_without_monotonicity_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082098,"parameters":{"mode":"bounded_nonmonotone","prefix_length":25,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0099","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"finite_prefix_only\",\"prefix_length\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_prefix_only, 26, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)","explanation_zh":"Wave6第二層核對=(finite_prefix_only, 26, 否, 否, 否, finite_prefix_not_universal_monotone_bounded_certificate)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082099,"parameters":{"mode":"finite_prefix_only","prefix_length":26,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或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-20261108-r02-monotone-sequence-convergence-theorem-0100","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-monotone-sequence-convergence-theorem"],"title_zh":"單調有界數列收斂定理","stem_zh":"實數列的symbolic全稱單調與相應有界證書。操作=monoseq-hypothesis-boundary；資料={\"mode\":\"monotone_unbounded\",\"prefix_length\":27}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], 是, 否, 否, monotone_without_bound_not_covered)","explanation_zh":"Wave6第二層核對=(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], 是, 否, 否, monotone_without_bound_not_covered)；依完整明示資料獨立重算；nondecreasing加上界時極限為sup，nonincreasing加下界時為inf；sup不必是maximum；epsilon證明以s-epsilon不是上界取得index再由monotonicity控制tail；finite prefix不足且convergence不推monotone。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["單調有界數列收斂定理"]},"variant":{"family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082100,"parameters":{"mode":"monotone_unbounded","prefix_length":27,"domain":"Symbolic_monotone_bounded_real_sequence","operation":"monoseq-hypothesis-boundary","semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":27}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}