{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0001","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[1,-2,[3,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, -2, 3/2], 2, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([1, -2, 3/2], 2, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092001,"parameters":{"period_values":[1,-2,[3,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[1,-2,[3,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0002","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[2,-3,[4,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, -3, 2], 3, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([2, -3, 2], 3, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092002,"parameters":{"period_values":[2,-3,[4,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[2,-3,[4,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0003","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[3,-4,[5,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, -4, 5/2], 4, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([3, -4, 5/2], 4, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092003,"parameters":{"period_values":[3,-4,[5,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[3,-4,[5,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0004","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[4,-5,[6,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, -5, 3], 5, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([4, -5, 3], 5, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092004,"parameters":{"period_values":[4,-5,[6,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[4,-5,[6,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0005","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[5,-6,[7,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, -6, 7/2], 6, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([5, -6, 7/2], 6, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092005,"parameters":{"period_values":[5,-6,[7,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[5,-6,[7,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0006","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[6,-7,[8,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, -7, 4], 7, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([6, -7, 4], 7, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092006,"parameters":{"period_values":[6,-7,[8,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[6,-7,[8,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0007","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[7,-8,[9,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, -8, 9/2], 8, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([7, -8, 9/2], 8, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092007,"parameters":{"period_values":[7,-8,[9,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[7,-8,[9,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0008","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[8,-9,[10,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, -9, 5], 9, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([8, -9, 5], 9, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092008,"parameters":{"period_values":[8,-9,[10,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[8,-9,[10,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0009","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[9,-10,[11,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, -10, 11/2], 10, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([9, -10, 11/2], 10, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092009,"parameters":{"period_values":[9,-10,[11,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[9,-10,[11,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0010","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[10,-11,[12,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, -11, 6], 11, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([10, -11, 6], 11, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092010,"parameters":{"period_values":[10,-11,[12,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[10,-11,[12,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0011","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[11,-12,[13,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, -12, 13/2], 12, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([11, -12, 13/2], 12, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092011,"parameters":{"period_values":[11,-12,[13,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[11,-12,[13,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0012","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[12,-13,[14,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, -13, 7], 13, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([12, -13, 7], 13, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092012,"parameters":{"period_values":[12,-13,[14,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[12,-13,[14,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0013","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[13,-14,[15,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13, -14, 15/2], 14, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([13, -14, 15/2], 14, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092013,"parameters":{"period_values":[13,-14,[15,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[13,-14,[15,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0014","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[14,-15,[16,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14, -15, 8], 15, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([14, -15, 8], 15, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092014,"parameters":{"period_values":[14,-15,[16,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[14,-15,[16,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0015","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[15,-16,[17,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([15, -16, 17/2], 16, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([15, -16, 17/2], 16, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092015,"parameters":{"period_values":[15,-16,[17,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[15,-16,[17,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0016","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[16,-17,[18,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([16, -17, 9], 17, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([16, -17, 9], 17, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092016,"parameters":{"period_values":[16,-17,[18,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[16,-17,[18,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0017","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[17,-18,[19,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17, -18, 19/2], 18, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([17, -18, 19/2], 18, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092017,"parameters":{"period_values":[17,-18,[19,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[17,-18,[19,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0018","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[18,-19,[20,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([18, -19, 10], 19, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([18, -19, 10], 19, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092018,"parameters":{"period_values":[18,-19,[20,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[18,-19,[20,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0019","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[19,-20,[21,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([19, -20, 21/2], 20, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([19, -20, 21/2], 20, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092019,"parameters":{"period_values":[19,-20,[21,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[19,-20,[21,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0020","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[20,-21,[22,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([20, -21, 11], 21, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([20, -21, 11], 21, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092020,"parameters":{"period_values":[20,-21,[22,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[20,-21,[22,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0021","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[21,-22,[23,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([21, -22, 23/2], 22, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([21, -22, 23/2], 22, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092021,"parameters":{"period_values":[21,-22,[23,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[21,-22,[23,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0022","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[22,-23,[24,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([22, -23, 12], 23, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([22, -23, 12], 23, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092022,"parameters":{"period_values":[22,-23,[24,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[22,-23,[24,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0023","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[23,-24,[25,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([23, -24, 25/2], 24, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([23, -24, 25/2], 24, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092023,"parameters":{"period_values":[23,-24,[25,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[23,-24,[25,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0024","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[24,-25,[26,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([24, -25, 13], 25, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([24, -25, 13], 25, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092024,"parameters":{"period_values":[24,-25,[26,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[24,-25,[26,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0025","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-boundedness；資料={\"period_values\":[25,-26,[27,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([25, -26, 27/2], 26, 是, whole_infinite_periodic_sequence_bounded)","explanation_zh":"Wave6第三層核對=([25, -26, 27/2], 26, 是, whole_infinite_periodic_sequence_bounded)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-boundedness","seed":202611092025,"parameters":{"period_values":[25,-26,[27,2]],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-boundedness","semantic_template":"qf-mod40-bw-periodic-boundedness/{\"period_values\":[25,-26,[27,2]]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0026","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[1,-2,[3,2]],\"residue\":0,\"j\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 1, 3, 1, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(0, 1, 3, 1, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092026,"parameters":{"period_values":[1,-2,[3,2]],"residue":0,"j":1,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":1,\"period_values\":[1,-2,[3,2]],\"residue\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0027","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[2,-3,[4,2]],\"residue\":1,\"j\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 2, 7, -3, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(1, 2, 7, -3, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092027,"parameters":{"period_values":[2,-3,[4,2]],"residue":1,"j":2,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":2,\"period_values\":[2,-3,[4,2]],\"residue\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0028","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[3,-4,[5,2]],\"residue\":2,\"j\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 11, 5/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(2, 3, 11, 5/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092028,"parameters":{"period_values":[3,-4,[5,2]],"residue":2,"j":3,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":3,\"period_values\":[3,-4,[5,2]],\"residue\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0029","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[4,-5,[6,2]],\"residue\":0,\"j\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 4, 12, 4, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(0, 4, 12, 4, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092029,"parameters":{"period_values":[4,-5,[6,2]],"residue":0,"j":4,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":4,\"period_values\":[4,-5,[6,2]],\"residue\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0030","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[5,-6,[7,2]],\"residue\":1,\"j\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 5, 16, -6, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(1, 5, 16, -6, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092030,"parameters":{"period_values":[5,-6,[7,2]],"residue":1,"j":5,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":5,\"period_values\":[5,-6,[7,2]],\"residue\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0031","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[6,-7,[8,2]],\"residue\":2,\"j\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 6, 20, 4, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(2, 6, 20, 4, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092031,"parameters":{"period_values":[6,-7,[8,2]],"residue":2,"j":6,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":6,\"period_values\":[6,-7,[8,2]],\"residue\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0032","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[7,-8,[9,2]],\"residue\":0,\"j\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 7, 21, 7, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(0, 7, 21, 7, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092032,"parameters":{"period_values":[7,-8,[9,2]],"residue":0,"j":7,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":7,\"period_values\":[7,-8,[9,2]],\"residue\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0033","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[8,-9,[10,2]],\"residue\":1,\"j\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 8, 25, -9, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(1, 8, 25, -9, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092033,"parameters":{"period_values":[8,-9,[10,2]],"residue":1,"j":8,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":8,\"period_values\":[8,-9,[10,2]],\"residue\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0034","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[9,-10,[11,2]],\"residue\":2,\"j\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 9, 29, 11/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(2, 9, 29, 11/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092034,"parameters":{"period_values":[9,-10,[11,2]],"residue":2,"j":9,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":9,\"period_values\":[9,-10,[11,2]],\"residue\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0035","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[10,-11,[12,2]],\"residue\":0,\"j\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 10, 30, 10, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(0, 10, 30, 10, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092035,"parameters":{"period_values":[10,-11,[12,2]],"residue":0,"j":10,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":10,\"period_values\":[10,-11,[12,2]],\"residue\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0036","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[11,-12,[13,2]],\"residue\":1,\"j\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 11, 34, -12, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(1, 11, 34, -12, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092036,"parameters":{"period_values":[11,-12,[13,2]],"residue":1,"j":11,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":11,\"period_values\":[11,-12,[13,2]],\"residue\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0037","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[12,-13,[14,2]],\"residue\":2,\"j\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 12, 38, 7, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(2, 12, 38, 7, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092037,"parameters":{"period_values":[12,-13,[14,2]],"residue":2,"j":12,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":12,\"period_values\":[12,-13,[14,2]],\"residue\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0038","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[13,-14,[15,2]],\"residue\":0,\"j\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 13, 39, 13, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(0, 13, 39, 13, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092038,"parameters":{"period_values":[13,-14,[15,2]],"residue":0,"j":13,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":13,\"period_values\":[13,-14,[15,2]],\"residue\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0039","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[14,-15,[16,2]],\"residue\":1,\"j\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 14, 43, -15, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(1, 14, 43, -15, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092039,"parameters":{"period_values":[14,-15,[16,2]],"residue":1,"j":14,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":14,\"period_values\":[14,-15,[16,2]],\"residue\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0040","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[15,-16,[17,2]],\"residue\":2,\"j\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 15, 47, 17/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(2, 15, 47, 17/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092040,"parameters":{"period_values":[15,-16,[17,2]],"residue":2,"j":15,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":15,\"period_values\":[15,-16,[17,2]],\"residue\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0041","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[16,-17,[18,2]],\"residue\":0,\"j\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 16, 48, 16, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(0, 16, 48, 16, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092041,"parameters":{"period_values":[16,-17,[18,2]],"residue":0,"j":16,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":16,\"period_values\":[16,-17,[18,2]],\"residue\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0042","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[17,-18,[19,2]],\"residue\":1,\"j\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 17, 52, -18, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(1, 17, 52, -18, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092042,"parameters":{"period_values":[17,-18,[19,2]],"residue":1,"j":17,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":17,\"period_values\":[17,-18,[19,2]],\"residue\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0043","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[18,-19,[20,2]],\"residue\":2,\"j\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 18, 56, 10, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(2, 18, 56, 10, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092043,"parameters":{"period_values":[18,-19,[20,2]],"residue":2,"j":18,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":18,\"period_values\":[18,-19,[20,2]],\"residue\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0044","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[19,-20,[21,2]],\"residue\":0,\"j\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 19, 57, 19, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(0, 19, 57, 19, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092044,"parameters":{"period_values":[19,-20,[21,2]],"residue":0,"j":19,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":19,\"period_values\":[19,-20,[21,2]],\"residue\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0045","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[20,-21,[22,2]],\"residue\":1,\"j\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 20, 61, -21, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(1, 20, 61, -21, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092045,"parameters":{"period_values":[20,-21,[22,2]],"residue":1,"j":20,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":20,\"period_values\":[20,-21,[22,2]],\"residue\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0046","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[21,-22,[23,2]],\"residue\":2,\"j\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 21, 65, 23/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(2, 21, 65, 23/2, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092046,"parameters":{"period_values":[21,-22,[23,2]],"residue":2,"j":21,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":21,\"period_values\":[21,-22,[23,2]],\"residue\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0047","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[22,-23,[24,2]],\"residue\":0,\"j\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 22, 66, 22, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(0, 22, 66, 22, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092047,"parameters":{"period_values":[22,-23,[24,2]],"residue":0,"j":22,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":22,\"period_values\":[22,-23,[24,2]],\"residue\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0048","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[23,-24,[25,2]],\"residue\":1,\"j\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 23, 70, -24, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(1, 23, 70, -24, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092048,"parameters":{"period_values":[23,-24,[25,2]],"residue":1,"j":23,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":23,\"period_values\":[23,-24,[25,2]],\"residue\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0049","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[24,-25,[26,2]],\"residue\":2,\"j\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 24, 74, 13, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(2, 24, 74, 13, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092049,"parameters":{"period_values":[24,-25,[26,2]],"residue":2,"j":24,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":24,\"period_values\":[24,-25,[26,2]],\"residue\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0050","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-periodic-subsequence；資料={\"period_values\":[25,-26,[27,2]],\"residue\":0,\"j\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 25, 75, 25, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)","explanation_zh":"Wave6第三層核對=(0, 25, 75, 25, 是, k_j_residue_plus_period_times_j_strictly_increasing_constant_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-periodic-subsequence","seed":202611092050,"parameters":{"period_values":[25,-26,[27,2]],"residue":0,"j":25,"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-periodic-subsequence","semantic_template":"qf-mod40-bw-periodic-subsequence/{\"j\":25,\"period_values\":[25,-26,[27,2]],\"residue\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0051","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[1,2],\"coefficient\":1,\"shift\":2,\"parity\":0,\"epsilon\":[1,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 1/2, 1/10, 5, 7/12, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 1/2, 1/10, 5, 7/12, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092051,"parameters":{"limit":[1,2],"coefficient":1,"shift":2,"parity":0,"epsilon":[1,10],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":1,\"epsilon\":[1,10],\"limit\":[1,2],\"parity\":0,\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0052","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[2,3],\"coefficient\":2,\"shift\":3,\"parity\":1,\"epsilon\":[1,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, k(j)=2j+1, 2/3, 1/11, 10, 7/12, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(1, k(j)=2j+1, 2/3, 1/11, 10, 7/12, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092052,"parameters":{"limit":[2,3],"coefficient":2,"shift":3,"parity":1,"epsilon":[1,11],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":2,\"epsilon\":[1,11],\"limit\":[2,3],\"parity\":1,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0053","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[3,4],\"coefficient\":3,\"shift\":4,\"parity\":0,\"epsilon\":[1,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 3/4, 1/12, 17, 63/76, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 3/4, 1/12, 17, 63/76, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092053,"parameters":{"limit":[3,4],"coefficient":3,"shift":4,"parity":0,"epsilon":[1,12],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":3,\"epsilon\":[1,12],\"limit\":[3,4],\"parity\":0,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0054","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[4,5],\"coefficient\":4,\"shift\":5,\"parity\":1,\"epsilon\":[1,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, k(j)=2j+1, 4/5, 1/13, 24, 98/135, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(1, k(j)=2j+1, 4/5, 1/13, 24, 98/135, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092054,"parameters":{"limit":[4,5],"coefficient":4,"shift":5,"parity":1,"epsilon":[1,13],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":4,\"epsilon\":[1,13],\"limit\":[4,5],\"parity\":1,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0055","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[5,6],\"coefficient\":5,\"shift\":6,\"parity\":0,\"epsilon\":[1,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 5/6, 1/14, 33, 65/72, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 5/6, 1/14, 33, 65/72, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092055,"parameters":{"limit":[5,6],"coefficient":5,"shift":6,"parity":0,"epsilon":[1,14],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":5,\"epsilon\":[1,14],\"limit\":[5,6],\"parity\":0,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0056","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[6,7],\"coefficient\":6,\"shift\":7,\"parity\":1,\"epsilon\":[1,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, k(j)=2j+1, 6/7, 1/15, 42, 255/322, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(1, k(j)=2j+1, 6/7, 1/15, 42, 255/322, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092056,"parameters":{"limit":[6,7],"coefficient":6,"shift":7,"parity":1,"epsilon":[1,15],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":6,\"epsilon\":[1,15],\"limit\":[6,7],\"parity\":1,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0057","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[7,8],\"coefficient\":7,\"shift\":8,\"parity\":0,\"epsilon\":[1,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 7/8, 1/16, 53, 427/456, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 7/8, 1/16, 53, 427/456, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092057,"parameters":{"limit":[7,8],"coefficient":7,"shift":8,"parity":0,"epsilon":[1,16],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":7,\"epsilon\":[1,16],\"limit\":[7,8],\"parity\":0,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0058","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[8,9],\"coefficient\":8,\"shift\":9,\"parity\":1,\"epsilon\":[1,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, k(j)=2j+1, 8/9, 1/17, 64, 172/207, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(1, k(j)=2j+1, 8/9, 1/17, 64, 172/207, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092058,"parameters":{"limit":[8,9],"coefficient":8,"shift":9,"parity":1,"epsilon":[1,17],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":8,\"epsilon\":[1,17],\"limit\":[8,9],\"parity\":1,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0059","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[9,10],\"coefficient\":9,\"shift\":10,\"parity\":0,\"epsilon\":[1,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 9/10, 1/18, 77, 783/820, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 9/10, 1/18, 77, 783/820, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092059,"parameters":{"limit":[9,10],"coefficient":9,"shift":10,"parity":0,"epsilon":[1,18],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":9,\"epsilon\":[1,18],\"limit\":[9,10],\"parity\":0,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0060","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[10,11],\"coefficient\":10,\"shift\":11,\"parity\":1,\"epsilon\":[1,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, k(j)=2j+1, 10/11, 1/19, 90, 905/1056, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(1, k(j)=2j+1, 10/11, 1/19, 90, 905/1056, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092060,"parameters":{"limit":[10,11],"coefficient":10,"shift":11,"parity":1,"epsilon":[1,19],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":10,\"epsilon\":[1,19],\"limit\":[10,11],\"parity\":1,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0061","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[11,12],\"coefficient\":11,\"shift\":12,\"parity\":0,\"epsilon\":[1,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 11/12, 1/20, 105, 143/148, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 11/12, 1/20, 105, 143/148, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092061,"parameters":{"limit":[11,12],"coefficient":11,"shift":12,"parity":0,"epsilon":[1,20],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":11,\"epsilon\":[1,20],\"limit\":[11,12],\"parity\":0,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0062","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[12,13],\"coefficient\":12,\"shift\":13,\"parity\":1,\"epsilon\":[1,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, k(j)=2j+1, 12/13, 1/21, 120, 1446/1651, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(1, k(j)=2j+1, 12/13, 1/21, 120, 1446/1651, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092062,"parameters":{"limit":[12,13],"coefficient":12,"shift":13,"parity":1,"epsilon":[1,21],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":12,\"epsilon\":[1,21],\"limit\":[12,13],\"parity\":1,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0063","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[13,14],\"coefficient\":13,\"shift\":14,\"parity\":0,\"epsilon\":[1,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 13/14, 1/22, 137, 1963/2016, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 13/14, 1/22, 137, 1963/2016, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092063,"parameters":{"limit":[13,14],"coefficient":13,"shift":14,"parity":0,"epsilon":[1,22],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":13,\"epsilon\":[1,22],\"limit\":[13,14],\"parity\":0,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0064","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[14,15],\"coefficient\":14,\"shift\":15,\"parity\":1,\"epsilon\":[1,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, k(j)=2j+1, 14/15, 1/23, 154, 721/810, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(1, k(j)=2j+1, 14/15, 1/23, 154, 721/810, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092064,"parameters":{"limit":[14,15],"coefficient":14,"shift":15,"parity":1,"epsilon":[1,23],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":14,\"epsilon\":[1,23],\"limit\":[14,15],\"parity\":1,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0065","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[15,16],\"coefficient\":15,\"shift\":16,\"parity\":0,\"epsilon\":[1,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 15/16, 1/24, 173, 2835/2896, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 15/16, 1/24, 173, 2835/2896, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092065,"parameters":{"limit":[15,16],"coefficient":15,"shift":16,"parity":0,"epsilon":[1,24],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":15,\"epsilon\":[1,24],\"limit\":[15,16],\"parity\":0,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0066","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[16,17],\"coefficient\":16,\"shift\":17,\"parity\":1,\"epsilon\":[1,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, k(j)=2j+1, 16/17, 1/25, 192, 3080/3417, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(1, k(j)=2j+1, 16/17, 1/25, 192, 3080/3417, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092066,"parameters":{"limit":[16,17],"coefficient":16,"shift":17,"parity":1,"epsilon":[1,25],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":16,\"epsilon\":[1,25],\"limit\":[16,17],\"parity\":1,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0067","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[17,18],\"coefficient\":17,\"shift\":18,\"parity\":0,\"epsilon\":[1,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 17/18, 1/26, 213, 1309/1332, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 17/18, 1/26, 213, 1309/1332, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092067,"parameters":{"limit":[17,18],"coefficient":17,"shift":18,"parity":0,"epsilon":[1,26],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":17,\"epsilon\":[1,26],\"limit\":[17,18],\"parity\":0,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0068","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[18,19],\"coefficient\":18,\"shift\":19,\"parity\":1,\"epsilon\":[1,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, k(j)=2j+1, 18/19, 1/27, 234, 4221/4636, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(1, k(j)=2j+1, 18/19, 1/27, 234, 4221/4636, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092068,"parameters":{"limit":[18,19],"coefficient":18,"shift":19,"parity":1,"epsilon":[1,27],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":18,\"epsilon\":[1,27],\"limit\":[18,19],\"parity\":1,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0069","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[19,20],\"coefficient\":19,\"shift\":20,\"parity\":0,\"epsilon\":[1,28]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 19/20, 1/28, 257, 5263/5340, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 19/20, 1/28, 257, 5263/5340, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092069,"parameters":{"limit":[19,20],"coefficient":19,"shift":20,"parity":0,"epsilon":[1,28],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":19,\"epsilon\":[1,28],\"limit\":[19,20],\"parity\":0,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0070","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[20,21],\"coefficient\":20,\"shift\":21,\"parity\":1,\"epsilon\":[1,29]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, k(j)=2j+1, 20/21, 1/29, 280, 1870/2037, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(1, k(j)=2j+1, 20/21, 1/29, 280, 1870/2037, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092070,"parameters":{"limit":[20,21],"coefficient":20,"shift":21,"parity":1,"epsilon":[1,29],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":20,\"epsilon\":[1,29],\"limit\":[20,21],\"parity\":1,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0071","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[21,22],\"coefficient\":21,\"shift\":22,\"parity\":0,\"epsilon\":[1,30]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 21/22, 1/30, 305, 6867/6952, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 21/22, 1/30, 305, 6867/6952, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092071,"parameters":{"limit":[21,22],"coefficient":21,"shift":22,"parity":0,"epsilon":[1,30],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":21,\"epsilon\":[1,30],\"limit\":[21,22],\"parity\":0,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0072","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[22,23],\"coefficient\":22,\"shift\":23,\"parity\":1,\"epsilon\":[1,31]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, k(j)=2j+1, 22/23, 1/31, 330, 7271/7866, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(1, k(j)=2j+1, 22/23, 1/31, 330, 7271/7866, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092072,"parameters":{"limit":[22,23],"coefficient":22,"shift":23,"parity":1,"epsilon":[1,31],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":22,\"epsilon\":[1,31],\"limit\":[22,23],\"parity\":1,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0073","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[23,24],\"coefficient\":23,\"shift\":24,\"parity\":0,\"epsilon\":[1,32]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 23/24, 1/32, 357, 2921/2952, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 23/24, 1/32, 357, 2921/2952, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092073,"parameters":{"limit":[23,24],"coefficient":23,"shift":24,"parity":0,"epsilon":[1,32],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":23,\"epsilon\":[1,32],\"limit\":[23,24],\"parity\":0,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0074","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[24,25],\"coefficient\":24,\"shift\":25,\"parity\":1,\"epsilon\":[1,33]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, k(j)=2j+1, 24/25, 1/33, 384, 9228/9925, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(1, k(j)=2j+1, 24/25, 1/33, 384, 9228/9925, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092074,"parameters":{"limit":[24,25],"coefficient":24,"shift":25,"parity":1,"epsilon":[1,33],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":24,\"epsilon\":[1,33],\"limit\":[24,25],\"parity\":1,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0075","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-oscillatory-tail；資料={\"limit\":[25,26],\"coefficient\":25,\"shift\":26,\"parity\":0,\"epsilon\":[1,34]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, k(j)=2j+0, 25/26, 1/34, 413, 10975/11076, 是, forall_j_ge_J_symbolic_subsequence_tail)","explanation_zh":"Wave6第三層核對=(0, k(j)=2j+0, 25/26, 1/34, 413, 10975/11076, 是, forall_j_ge_J_symbolic_subsequence_tail)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-oscillatory-tail","seed":202611092075,"parameters":{"limit":[25,26],"coefficient":25,"shift":26,"parity":0,"epsilon":[1,34],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-oscillatory-tail","semantic_template":"qf-mod40-bw-oscillatory-tail/{\"coefficient\":25,\"epsilon\":[1,34],\"limit\":[25,26],\"parity\":0,\"shift\":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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0076","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[0,1,0,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1, 0, 2], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([0, 1, 0, 2], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092076,"parameters":{"values":[0,1,0,2],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[0,1,0,2]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0077","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[1,2,1,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 2, 1, 3], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([1, 2, 1, 3], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092077,"parameters":{"values":[1,2,1,3],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[1,2,1,3]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0078","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[2,3,2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 3, 2, 4], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([2, 3, 2, 4], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092078,"parameters":{"values":[2,3,2,4],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[2,3,2,4]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0079","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[3,4,3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 4, 3, 5], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([3, 4, 3, 5], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092079,"parameters":{"values":[3,4,3,5],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[3,4,3,5]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0080","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[4,5,4,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 5, 4, 6], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([4, 5, 4, 6], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092080,"parameters":{"values":[4,5,4,6],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[4,5,4,6]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0081","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[5,6,5,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, 6, 5, 7], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([5, 6, 5, 7], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092081,"parameters":{"values":[5,6,5,7],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[5,6,5,7]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0082","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[6,7,6,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 7, 6, 8], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([6, 7, 6, 8], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092082,"parameters":{"values":[6,7,6,8],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[6,7,6,8]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0083","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[7,8,7,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, 8, 7, 9], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([7, 8, 7, 9], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092083,"parameters":{"values":[7,8,7,9],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[7,8,7,9]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0084","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[8,9,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 9, 8, 10], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([8, 9, 8, 10], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092084,"parameters":{"values":[8,9,8,10],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[8,9,8,10]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0085","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[9,10,9,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, 10, 9, 11], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([9, 10, 9, 11], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092085,"parameters":{"values":[9,10,9,11],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[9,10,9,11]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0086","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[10,11,10,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, 11, 10, 12], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([10, 11, 10, 12], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092086,"parameters":{"values":[10,11,10,12],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[10,11,10,12]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0087","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[11,12,11,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, 12, 11, 13], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([11, 12, 11, 13], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092087,"parameters":{"values":[11,12,11,13],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[11,12,11,13]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0088","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[12,13,12,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, 13, 12, 14], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([12, 13, 12, 14], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092088,"parameters":{"values":[12,13,12,14],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[12,13,12,14]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0089","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[13,14,13,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13, 14, 13, 15], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([13, 14, 13, 15], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092089,"parameters":{"values":[13,14,13,15],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[13,14,13,15]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0090","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[14,15,14,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14, 15, 14, 16], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([14, 15, 14, 16], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092090,"parameters":{"values":[14,15,14,16],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[14,15,14,16]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0091","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[15,16,15,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([15, 16, 15, 17], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([15, 16, 15, 17], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092091,"parameters":{"values":[15,16,15,17],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[15,16,15,17]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0092","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[16,17,16,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([16, 17, 16, 18], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([16, 17, 16, 18], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092092,"parameters":{"values":[16,17,16,18],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[16,17,16,18]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0093","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[17,18,17,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17, 18, 17, 19], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([17, 18, 17, 19], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092093,"parameters":{"values":[17,18,17,19],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[17,18,17,19]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0094","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[18,19,18,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([18, 19, 18, 20], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([18, 19, 18, 20], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092094,"parameters":{"values":[18,19,18,20],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[18,19,18,20]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0095","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[19,20,19,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([19, 20, 19, 21], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([19, 20, 19, 21], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092095,"parameters":{"values":[19,20,19,21],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[19,20,19,21]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0096","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[20,21,20,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([20, 21, 20, 22], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([20, 21, 20, 22], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092096,"parameters":{"values":[20,21,20,22],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[20,21,20,22]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0097","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[21,22,21,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([21, 22, 21, 23], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([21, 22, 21, 23], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092097,"parameters":{"values":[21,22,21,23],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[21,22,21,23]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0098","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[22,23,22,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([22, 23, 22, 24], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([22, 23, 22, 24], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092098,"parameters":{"values":[22,23,22,24],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[22,23,22,24]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0099","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[23,24,23,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([23, 24, 23, 25], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([23, 24, 23, 25], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092099,"parameters":{"values":[23,24,23,25],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[23,24,23,25]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、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-20261109-r02-bolzano-weierstrass-theorem-real-0100","batch_id":"batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations","language":"zh-Hant","source_mko_ids":["mko-bolzano-weierstrass-theorem-real"],"title_zh":"實數列Bolzano－Weierstrass定理","stem_zh":"whole-sequence bounded certificate與symbolic strictly increasing infinite index。操作=bw-finite-prefix-boundary；資料={\"values\":[24,25,24,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([24, 25, 24, 26], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)","explanation_zh":"Wave6第三層核對=([24, 25, 24, 26], 否, 否, 否, finite_prefix_cannot_certify_whole_sequence_boundedness_or_infinite_convergent_subsequence)；依完整明示資料獨立重算；convergent subsequence須有epsilon-tail certificate；finite prefix或finite cluster search不足；定理只保證某個subsequence，不保證原sequence收斂或limit unique。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["實數列Bolzano－Weierstrass定理"]},"variant":{"family_id":"qf-mod40-bw-finite-prefix-boundary","seed":202611092100,"parameters":{"values":[24,25,24,26],"domain":"Symbolic_bounded_real_sequence_subsequence","operation":"bw-finite-prefix-boundary","semantic_template":"qf-mod40-bw-finite-prefix-boundary/{\"values\":[24,25,24,26]}"}},"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-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}