{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0001","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,2],[1,3],[1,4],[1,5],[1,6]],\"epsilon\":[1,3],\"N\":0}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/2, 1/3, 1/4, 1/5, 1/6], 1/3, 0, [[0, 4, 1/3], [4, 0, 1/3]], 否, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/2, 1/3, 1/4, 1/5, 1/6], 1/3, 0, [[0, 4, 1/3], [4, 0, 1/3]], 否, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263001,"parameters":{"sequence":[[1,2],[1,3],[1,4],[1,5],[1,6]],"epsilon":[1,3],"N":0,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":0,\"epsilon\":[1,3],\"sequence\":[[1,2],[1,3],[1,4],[1,5],[1,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0002","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,3],[1,4],[1,5],[1,6],[1,7]],\"epsilon\":[1,4],\"N\":1}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/3, 1/4, 1/5, 1/6, 1/7], 1/4, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/3, 1/4, 1/5, 1/6, 1/7], 1/4, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263002,"parameters":{"sequence":[[1,3],[1,4],[1,5],[1,6],[1,7]],"epsilon":[1,4],"N":1,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":1,\"epsilon\":[1,4],\"sequence\":[[1,3],[1,4],[1,5],[1,6],[1,7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0003","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,4],[1,5],[1,6],[1,7],[1,8]],\"epsilon\":[1,5],\"N\":2}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 1/5, 1/6, 1/7, 1/8], 1/5, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/4, 1/5, 1/6, 1/7, 1/8], 1/5, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263003,"parameters":{"sequence":[[1,4],[1,5],[1,6],[1,7],[1,8]],"epsilon":[1,5],"N":2,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":2,\"epsilon\":[1,5],\"sequence\":[[1,4],[1,5],[1,6],[1,7],[1,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0004","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,5],[1,6],[1,7],[1,8],[1,9]],\"epsilon\":[1,6],\"N\":0}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/5, 1/6, 1/7, 1/8, 1/9], 1/6, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/5, 1/6, 1/7, 1/8, 1/9], 1/6, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263004,"parameters":{"sequence":[[1,5],[1,6],[1,7],[1,8],[1,9]],"epsilon":[1,6],"N":0,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":0,\"epsilon\":[1,6],\"sequence\":[[1,5],[1,6],[1,7],[1,8],[1,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0005","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,6],[1,7],[1,8],[1,9],[1,10]],\"epsilon\":[1,7],\"N\":1}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/6, 1/7, 1/8, 1/9, 1/10], 1/7, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/6, 1/7, 1/8, 1/9, 1/10], 1/7, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263005,"parameters":{"sequence":[[1,6],[1,7],[1,8],[1,9],[1,10]],"epsilon":[1,7],"N":1,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":1,\"epsilon\":[1,7],\"sequence\":[[1,6],[1,7],[1,8],[1,9],[1,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0006","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,7],[1,8],[1,9],[1,10],[1,11]],\"epsilon\":[1,3],\"N\":2}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/7, 1/8, 1/9, 1/10, 1/11], 1/3, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/7, 1/8, 1/9, 1/10, 1/11], 1/3, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263006,"parameters":{"sequence":[[1,7],[1,8],[1,9],[1,10],[1,11]],"epsilon":[1,3],"N":2,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":2,\"epsilon\":[1,3],\"sequence\":[[1,7],[1,8],[1,9],[1,10],[1,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0007","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,8],[1,9],[1,10],[1,11],[1,12]],\"epsilon\":[1,4],\"N\":0}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/8, 1/9, 1/10, 1/11, 1/12], 1/4, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/8, 1/9, 1/10, 1/11, 1/12], 1/4, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263007,"parameters":{"sequence":[[1,8],[1,9],[1,10],[1,11],[1,12]],"epsilon":[1,4],"N":0,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":0,\"epsilon\":[1,4],\"sequence\":[[1,8],[1,9],[1,10],[1,11],[1,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0008","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,9],[1,10],[1,11],[1,12],[1,13]],\"epsilon\":[1,5],\"N\":1}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/9, 1/10, 1/11, 1/12, 1/13], 1/5, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/9, 1/10, 1/11, 1/12, 1/13], 1/5, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263008,"parameters":{"sequence":[[1,9],[1,10],[1,11],[1,12],[1,13]],"epsilon":[1,5],"N":1,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":1,\"epsilon\":[1,5],\"sequence\":[[1,9],[1,10],[1,11],[1,12],[1,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0009","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,10],[1,11],[1,12],[1,13],[1,14]],\"epsilon\":[1,6],\"N\":2}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/10, 1/11, 1/12, 1/13, 1/14], 1/6, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/10, 1/11, 1/12, 1/13, 1/14], 1/6, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263009,"parameters":{"sequence":[[1,10],[1,11],[1,12],[1,13],[1,14]],"epsilon":[1,6],"N":2,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":2,\"epsilon\":[1,6],\"sequence\":[[1,10],[1,11],[1,12],[1,13],[1,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0010","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,11],[1,12],[1,13],[1,14],[1,15]],\"epsilon\":[1,7],\"N\":0}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/11, 1/12, 1/13, 1/14, 1/15], 1/7, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/11, 1/12, 1/13, 1/14, 1/15], 1/7, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263010,"parameters":{"sequence":[[1,11],[1,12],[1,13],[1,14],[1,15]],"epsilon":[1,7],"N":0,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":0,\"epsilon\":[1,7],\"sequence\":[[1,11],[1,12],[1,13],[1,14],[1,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0011","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,12],[1,13],[1,14],[1,15],[1,16]],\"epsilon\":[1,3],\"N\":1}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/12, 1/13, 1/14, 1/15, 1/16], 1/3, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/12, 1/13, 1/14, 1/15, 1/16], 1/3, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263011,"parameters":{"sequence":[[1,12],[1,13],[1,14],[1,15],[1,16]],"epsilon":[1,3],"N":1,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":1,\"epsilon\":[1,3],\"sequence\":[[1,12],[1,13],[1,14],[1,15],[1,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0012","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,13],[1,14],[1,15],[1,16],[1,17]],\"epsilon\":[1,4],\"N\":2}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/13, 1/14, 1/15, 1/16, 1/17], 1/4, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/13, 1/14, 1/15, 1/16, 1/17], 1/4, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263012,"parameters":{"sequence":[[1,13],[1,14],[1,15],[1,16],[1,17]],"epsilon":[1,4],"N":2,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":2,\"epsilon\":[1,4],\"sequence\":[[1,13],[1,14],[1,15],[1,16],[1,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0013","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,14],[1,15],[1,16],[1,17],[1,18]],\"epsilon\":[1,5],\"N\":0}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/14, 1/15, 1/16, 1/17, 1/18], 1/5, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/14, 1/15, 1/16, 1/17, 1/18], 1/5, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263013,"parameters":{"sequence":[[1,14],[1,15],[1,16],[1,17],[1,18]],"epsilon":[1,5],"N":0,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":0,\"epsilon\":[1,5],\"sequence\":[[1,14],[1,15],[1,16],[1,17],[1,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0014","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,15],[1,16],[1,17],[1,18],[1,19]],\"epsilon\":[1,6],\"N\":1}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/15, 1/16, 1/17, 1/18, 1/19], 1/6, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/15, 1/16, 1/17, 1/18, 1/19], 1/6, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263014,"parameters":{"sequence":[[1,15],[1,16],[1,17],[1,18],[1,19]],"epsilon":[1,6],"N":1,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":1,\"epsilon\":[1,6],\"sequence\":[[1,15],[1,16],[1,17],[1,18],[1,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0015","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,16],[1,17],[1,18],[1,19],[1,20]],\"epsilon\":[1,7],\"N\":2}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/16, 1/17, 1/18, 1/19, 1/20], 1/7, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/16, 1/17, 1/18, 1/19, 1/20], 1/7, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263015,"parameters":{"sequence":[[1,16],[1,17],[1,18],[1,19],[1,20]],"epsilon":[1,7],"N":2,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":2,\"epsilon\":[1,7],\"sequence\":[[1,16],[1,17],[1,18],[1,19],[1,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0016","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,17],[1,18],[1,19],[1,20],[1,21]],\"epsilon\":[1,3],\"N\":0}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/17, 1/18, 1/19, 1/20, 1/21], 1/3, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/17, 1/18, 1/19, 1/20, 1/21], 1/3, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263016,"parameters":{"sequence":[[1,17],[1,18],[1,19],[1,20],[1,21]],"epsilon":[1,3],"N":0,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":0,\"epsilon\":[1,3],\"sequence\":[[1,17],[1,18],[1,19],[1,20],[1,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0017","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,18],[1,19],[1,20],[1,21],[1,22]],\"epsilon\":[1,4],\"N\":1}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/18, 1/19, 1/20, 1/21, 1/22], 1/4, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/18, 1/19, 1/20, 1/21, 1/22], 1/4, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263017,"parameters":{"sequence":[[1,18],[1,19],[1,20],[1,21],[1,22]],"epsilon":[1,4],"N":1,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":1,\"epsilon\":[1,4],\"sequence\":[[1,18],[1,19],[1,20],[1,21],[1,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0018","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,19],[1,20],[1,21],[1,22],[1,23]],\"epsilon\":[1,5],\"N\":2}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/19, 1/20, 1/21, 1/22, 1/23], 1/5, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/19, 1/20, 1/21, 1/22, 1/23], 1/5, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263018,"parameters":{"sequence":[[1,19],[1,20],[1,21],[1,22],[1,23]],"epsilon":[1,5],"N":2,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":2,\"epsilon\":[1,5],\"sequence\":[[1,19],[1,20],[1,21],[1,22],[1,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0019","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,20],[1,21],[1,22],[1,23],[1,24]],\"epsilon\":[1,6],\"N\":0}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/20, 1/21, 1/22, 1/23, 1/24], 1/6, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/20, 1/21, 1/22, 1/23, 1/24], 1/6, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263019,"parameters":{"sequence":[[1,20],[1,21],[1,22],[1,23],[1,24]],"epsilon":[1,6],"N":0,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":0,\"epsilon\":[1,6],\"sequence\":[[1,20],[1,21],[1,22],[1,23],[1,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0020","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,21],[1,22],[1,23],[1,24],[1,25]],\"epsilon\":[1,7],\"N\":1}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/21, 1/22, 1/23, 1/24, 1/25], 1/7, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/21, 1/22, 1/23, 1/24, 1/25], 1/7, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263020,"parameters":{"sequence":[[1,21],[1,22],[1,23],[1,24],[1,25]],"epsilon":[1,7],"N":1,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":1,\"epsilon\":[1,7],\"sequence\":[[1,21],[1,22],[1,23],[1,24],[1,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0021","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,22],[1,23],[1,24],[1,25],[1,26]],\"epsilon\":[1,3],\"N\":2}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/22, 1/23, 1/24, 1/25, 1/26], 1/3, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/22, 1/23, 1/24, 1/25, 1/26], 1/3, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263021,"parameters":{"sequence":[[1,22],[1,23],[1,24],[1,25],[1,26]],"epsilon":[1,3],"N":2,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":2,\"epsilon\":[1,3],\"sequence\":[[1,22],[1,23],[1,24],[1,25],[1,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0022","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,23],[1,24],[1,25],[1,26],[1,27]],\"epsilon\":[1,4],\"N\":0}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/23, 1/24, 1/25, 1/26, 1/27], 1/4, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/23, 1/24, 1/25, 1/26, 1/27], 1/4, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263022,"parameters":{"sequence":[[1,23],[1,24],[1,25],[1,26],[1,27]],"epsilon":[1,4],"N":0,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":0,\"epsilon\":[1,4],\"sequence\":[[1,23],[1,24],[1,25],[1,26],[1,27]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0023","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,24],[1,25],[1,26],[1,27],[1,28]],\"epsilon\":[1,5],\"N\":1}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/24, 1/25, 1/26, 1/27, 1/28], 1/5, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/24, 1/25, 1/26, 1/27, 1/28], 1/5, 1, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263023,"parameters":{"sequence":[[1,24],[1,25],[1,26],[1,27],[1,28]],"epsilon":[1,5],"N":1,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":1,\"epsilon\":[1,5],\"sequence\":[[1,24],[1,25],[1,26],[1,27],[1,28]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0024","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,25],[1,26],[1,27],[1,28],[1,29]],\"epsilon\":[1,6],\"N\":2}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/25, 1/26, 1/27, 1/28, 1/29], 1/6, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/25, 1/26, 1/27, 1/28, 1/29], 1/6, 2, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263024,"parameters":{"sequence":[[1,25],[1,26],[1,27],[1,28],[1,29]],"epsilon":[1,6],"N":2,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":2,\"epsilon\":[1,6],\"sequence\":[[1,25],[1,26],[1,27],[1,28],[1,29]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0025","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-finite-tail-audit；資料={\"sequence\":[[1,26],[1,27],[1,28],[1,29],[1,30]],\"epsilon\":[1,7],\"N\":0}。請依 [sequence,epsilon,N,bad_pairs,finite_tail_pass,general_cauchy,boundary] 欄序回答。","answer":{"kind":"expression","value":"([1/26, 1/27, 1/28, 1/29, 1/30], 1/7, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)","explanation_zh":"結構深度核對=([1/26, 1/27, 1/28, 1/29, 1/30], 1/7, 0, [], 是, 否, 有限尾段不足以證一般Cauchy)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263025,"parameters":{"sequence":[[1,26],[1,27],[1,28],[1,29],[1,30]],"epsilon":[1,7],"N":0,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-finite-tail-audit","semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":0,\"epsilon\":[1,7],\"sequence\":[[1,26],[1,27],[1,28],[1,29],[1,30]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0026","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[-12,1],[1,2],[-3,1]],\"tail\":[-5,1]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([-12, 1/2, -3], -5, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([-12, 1/2, -3], -5, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263026,"parameters":{"prefix":[[-12,1],[1,2],[-3,1]],"tail":[-5,1],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-12,1],[1,2],[-3,1]],\"tail\":[-5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0027","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[-11,1],[2,2],[-2,2]],\"tail\":[-4,2]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([-11, 1, -1], -2, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([-11, 1, -1], -2, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263027,"parameters":{"prefix":[[-11,1],[2,2],[-2,2]],"tail":[-4,2],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-11,1],[2,2],[-2,2]],\"tail\":[-4,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0028","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[-10,1],[3,2],[-1,3]],\"tail\":[-3,3]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([-10, 3/2, -1/3], -1, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([-10, 3/2, -1/3], -1, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263028,"parameters":{"prefix":[[-10,1],[3,2],[-1,3]],"tail":[-3,3],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-10,1],[3,2],[-1,3]],\"tail\":[-3,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0029","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[-9,1],[4,2],[0,4]],\"tail\":[-2,4]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([-9, 2, 0], -1/2, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([-9, 2, 0], -1/2, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263029,"parameters":{"prefix":[[-9,1],[4,2],[0,4]],"tail":[-2,4],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-9,1],[4,2],[0,4]],\"tail\":[-2,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0030","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[-8,1],[5,2],[1,5]],\"tail\":[-1,5]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([-8, 5/2, 1/5], -1/5, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([-8, 5/2, 1/5], -1/5, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263030,"parameters":{"prefix":[[-8,1],[5,2],[1,5]],"tail":[-1,5],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-8,1],[5,2],[1,5]],\"tail\":[-1,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0031","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[-7,1],[6,2],[2,1]],\"tail\":[0,6]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([-7, 3, 2], 0, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([-7, 3, 2], 0, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263031,"parameters":{"prefix":[[-7,1],[6,2],[2,1]],"tail":[0,6],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-7,1],[6,2],[2,1]],\"tail\":[0,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0032","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[-6,1],[7,2],[3,2]],\"tail\":[1,7]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([-6, 7/2, 3/2], 1/7, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([-6, 7/2, 3/2], 1/7, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263032,"parameters":{"prefix":[[-6,1],[7,2],[3,2]],"tail":[1,7],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-6,1],[7,2],[3,2]],\"tail\":[1,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0033","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[-5,1],[8,2],[-3,3]],\"tail\":[2,1]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([-5, 4, -1], 2, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([-5, 4, -1], 2, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263033,"parameters":{"prefix":[[-5,1],[8,2],[-3,3]],"tail":[2,1],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-5,1],[8,2],[-3,3]],\"tail\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0034","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[-4,1],[9,2],[-2,4]],\"tail\":[3,2]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([-4, 9/2, -1/2], 3/2, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([-4, 9/2, -1/2], 3/2, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263034,"parameters":{"prefix":[[-4,1],[9,2],[-2,4]],"tail":[3,2],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-4,1],[9,2],[-2,4]],\"tail\":[3,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0035","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[-3,1],[10,2],[-1,5]],\"tail\":[4,3]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([-3, 5, -1/5], 4/3, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([-3, 5, -1/5], 4/3, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263035,"parameters":{"prefix":[[-3,1],[10,2],[-1,5]],"tail":[4,3],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-3,1],[10,2],[-1,5]],\"tail\":[4,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0036","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[-2,1],[11,2],[0,1]],\"tail\":[5,4]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([-2, 11/2, 0], 5/4, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([-2, 11/2, 0], 5/4, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263036,"parameters":{"prefix":[[-2,1],[11,2],[0,1]],"tail":[5,4],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-2,1],[11,2],[0,1]],\"tail\":[5,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0037","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[-1,1],[12,2],[1,2]],\"tail\":[-5,5]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([-1, 6, 1/2], -1, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([-1, 6, 1/2], -1, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263037,"parameters":{"prefix":[[-1,1],[12,2],[1,2]],"tail":[-5,5],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-1,1],[12,2],[1,2]],\"tail\":[-5,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0038","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[0,1],[13,2],[2,3]],\"tail\":[-4,6]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([0, 13/2, 2/3], -2/3, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([0, 13/2, 2/3], -2/3, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263038,"parameters":{"prefix":[[0,1],[13,2],[2,3]],"tail":[-4,6],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[0,1],[13,2],[2,3]],\"tail\":[-4,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0039","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[1,1],[14,2],[3,4]],\"tail\":[-3,7]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([1, 7, 3/4], -3/7, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([1, 7, 3/4], -3/7, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263039,"parameters":{"prefix":[[1,1],[14,2],[3,4]],"tail":[-3,7],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[1,1],[14,2],[3,4]],\"tail\":[-3,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0040","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[2,1],[15,2],[-3,5]],\"tail\":[-2,1]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([2, 15/2, -3/5], -2, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([2, 15/2, -3/5], -2, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263040,"parameters":{"prefix":[[2,1],[15,2],[-3,5]],"tail":[-2,1],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[2,1],[15,2],[-3,5]],\"tail\":[-2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0041","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[3,1],[16,2],[-2,1]],\"tail\":[-1,2]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([3, 8, -2], -1/2, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([3, 8, -2], -1/2, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263041,"parameters":{"prefix":[[3,1],[16,2],[-2,1]],"tail":[-1,2],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[3,1],[16,2],[-2,1]],\"tail\":[-1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0042","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[4,1],[17,2],[-1,2]],\"tail\":[0,3]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([4, 17/2, -1/2], 0, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([4, 17/2, -1/2], 0, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263042,"parameters":{"prefix":[[4,1],[17,2],[-1,2]],"tail":[0,3],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[4,1],[17,2],[-1,2]],\"tail\":[0,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0043","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[5,1],[18,2],[0,3]],\"tail\":[1,4]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([5, 9, 0], 1/4, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([5, 9, 0], 1/4, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263043,"parameters":{"prefix":[[5,1],[18,2],[0,3]],"tail":[1,4],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[5,1],[18,2],[0,3]],\"tail\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0044","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[6,1],[19,2],[1,4]],\"tail\":[2,5]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([6, 19/2, 1/4], 2/5, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([6, 19/2, 1/4], 2/5, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263044,"parameters":{"prefix":[[6,1],[19,2],[1,4]],"tail":[2,5],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[6,1],[19,2],[1,4]],\"tail\":[2,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0045","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[7,1],[20,2],[2,5]],\"tail\":[3,6]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([7, 10, 2/5], 1/2, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([7, 10, 2/5], 1/2, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263045,"parameters":{"prefix":[[7,1],[20,2],[2,5]],"tail":[3,6],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[7,1],[20,2],[2,5]],\"tail\":[3,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0046","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[8,1],[21,2],[3,1]],\"tail\":[4,7]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([8, 21/2, 3], 4/7, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([8, 21/2, 3], 4/7, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263046,"parameters":{"prefix":[[8,1],[21,2],[3,1]],"tail":[4,7],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[8,1],[21,2],[3,1]],\"tail\":[4,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0047","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[9,1],[22,2],[-3,2]],\"tail\":[5,1]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([9, 11, -3/2], 5, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([9, 11, -3/2], 5, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263047,"parameters":{"prefix":[[9,1],[22,2],[-3,2]],"tail":[5,1],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[9,1],[22,2],[-3,2]],\"tail\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0048","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[10,1],[23,2],[-2,3]],\"tail\":[-5,2]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([10, 23/2, -2/3], -5/2, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([10, 23/2, -2/3], -5/2, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263048,"parameters":{"prefix":[[10,1],[23,2],[-2,3]],"tail":[-5,2],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[10,1],[23,2],[-2,3]],\"tail\":[-5,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0049","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[11,1],[24,2],[-1,4]],\"tail\":[-4,3]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([11, 12, -1/4], -4/3, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([11, 12, -1/4], -4/3, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263049,"parameters":{"prefix":[[11,1],[24,2],[-1,4]],"tail":[-4,3],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[11,1],[24,2],[-1,4]],\"tail\":[-4,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0050","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-ultimately-constant-certificate；資料={\"prefix\":[[12,1],[25,2],[0,5]],\"tail\":[-3,4]}。請依 [prefix,tail,N,witness,cauchy] 欄序回答。","answer":{"kind":"expression","value":"([12, 25/2, 0], -3/4, 3, 任意epsilon>0取N=prefix長度, 是)","explanation_zh":"結構深度核對=([12, 25/2, 0], -3/4, 3, 任意epsilon>0取N=prefix長度, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263050,"parameters":{"prefix":[[12,1],[25,2],[0,5]],"tail":[-3,4],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-ultimately-constant-certificate","semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[12,1],[25,2],[0,5]],\"tail\":[-3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0051","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":1,\"epsilon\":[1,3]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(1, 1/3, 3, 1/4, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(1, 1/3, 3, 1/4, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263051,"parameters":{"shift":1,"epsilon":[1,3],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,3],\"shift\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0052","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":2,\"epsilon\":[1,4]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(2, 1/4, 3, 1/5, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(2, 1/4, 3, 1/5, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263052,"parameters":{"shift":2,"epsilon":[1,4],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,4],\"shift\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0053","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":3,\"epsilon\":[1,5]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(3, 1/5, 3, 1/6, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(3, 1/5, 3, 1/6, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263053,"parameters":{"shift":3,"epsilon":[1,5],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,5],\"shift\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0054","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":4,\"epsilon\":[1,6]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(4, 1/6, 3, 1/7, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(4, 1/6, 3, 1/7, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263054,"parameters":{"shift":4,"epsilon":[1,6],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,6],\"shift\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0055","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":5,\"epsilon\":[1,7]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(5, 1/7, 3, 1/8, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(5, 1/7, 3, 1/8, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263055,"parameters":{"shift":5,"epsilon":[1,7],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,7],\"shift\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0056","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":6,\"epsilon\":[1,8]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(6, 1/8, 3, 1/9, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(6, 1/8, 3, 1/9, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263056,"parameters":{"shift":6,"epsilon":[1,8],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,8],\"shift\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0057","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":7,\"epsilon\":[1,9]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(7, 1/9, 3, 1/10, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(7, 1/9, 3, 1/10, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263057,"parameters":{"shift":7,"epsilon":[1,9],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,9],\"shift\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0058","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":8,\"epsilon\":[1,10]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(8, 1/10, 3, 1/11, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(8, 1/10, 3, 1/11, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263058,"parameters":{"shift":8,"epsilon":[1,10],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,10],\"shift\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0059","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":9,\"epsilon\":[1,11]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(9, 1/11, 3, 1/12, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(9, 1/11, 3, 1/12, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263059,"parameters":{"shift":9,"epsilon":[1,11],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,11],\"shift\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0060","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":10,\"epsilon\":[1,12]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(10, 1/12, 3, 1/13, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(10, 1/12, 3, 1/13, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263060,"parameters":{"shift":10,"epsilon":[1,12],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,12],\"shift\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0061","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":11,\"epsilon\":[1,13]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(11, 1/13, 3, 1/14, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(11, 1/13, 3, 1/14, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263061,"parameters":{"shift":11,"epsilon":[1,13],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,13],\"shift\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0062","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":12,\"epsilon\":[1,14]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(12, 1/14, 3, 1/15, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(12, 1/14, 3, 1/15, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263062,"parameters":{"shift":12,"epsilon":[1,14],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,14],\"shift\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0063","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":13,\"epsilon\":[1,15]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(13, 1/15, 3, 1/16, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(13, 1/15, 3, 1/16, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263063,"parameters":{"shift":13,"epsilon":[1,15],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,15],\"shift\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0064","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":14,\"epsilon\":[1,16]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(14, 1/16, 3, 1/17, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(14, 1/16, 3, 1/17, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263064,"parameters":{"shift":14,"epsilon":[1,16],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,16],\"shift\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0065","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":15,\"epsilon\":[1,17]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(15, 1/17, 3, 1/18, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(15, 1/17, 3, 1/18, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263065,"parameters":{"shift":15,"epsilon":[1,17],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,17],\"shift\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0066","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":16,\"epsilon\":[1,18]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(16, 1/18, 3, 1/19, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(16, 1/18, 3, 1/19, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263066,"parameters":{"shift":16,"epsilon":[1,18],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,18],\"shift\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0067","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":17,\"epsilon\":[1,19]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(17, 1/19, 3, 1/20, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(17, 1/19, 3, 1/20, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263067,"parameters":{"shift":17,"epsilon":[1,19],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,19],\"shift\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0068","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":18,\"epsilon\":[1,20]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(18, 1/20, 3, 1/21, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(18, 1/20, 3, 1/21, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263068,"parameters":{"shift":18,"epsilon":[1,20],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,20],\"shift\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0069","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":19,\"epsilon\":[1,21]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(19, 1/21, 3, 1/22, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(19, 1/21, 3, 1/22, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263069,"parameters":{"shift":19,"epsilon":[1,21],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,21],\"shift\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0070","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":20,\"epsilon\":[1,22]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(20, 1/22, 3, 1/23, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(20, 1/22, 3, 1/23, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263070,"parameters":{"shift":20,"epsilon":[1,22],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,22],\"shift\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0071","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":21,\"epsilon\":[1,23]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(21, 1/23, 3, 1/24, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(21, 1/23, 3, 1/24, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263071,"parameters":{"shift":21,"epsilon":[1,23],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,23],\"shift\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0072","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":22,\"epsilon\":[1,24]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(22, 1/24, 3, 1/25, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(22, 1/24, 3, 1/25, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263072,"parameters":{"shift":22,"epsilon":[1,24],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,24],\"shift\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0073","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":23,\"epsilon\":[1,25]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(23, 1/25, 3, 1/26, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(23, 1/25, 3, 1/26, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263073,"parameters":{"shift":23,"epsilon":[1,25],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,25],\"shift\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0074","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":24,\"epsilon\":[1,26]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(24, 1/26, 3, 1/27, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(24, 1/26, 3, 1/27, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263074,"parameters":{"shift":24,"epsilon":[1,26],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,26],\"shift\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0075","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-reciprocal-epsilon-witness；資料={\"shift\":25,\"epsilon\":[1,27]}。請依 [shift,epsilon,N,bound,witness,cauchy_for_epsilon] 欄序回答。","answer":{"kind":"expression","value":"(25, 1/27, 3, 1/28, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)","explanation_zh":"結構深度核對=(25, 1/27, 3, 1/28, m,n>=N implies distance<=1/(N+shift)<epsilon, 是)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263075,"parameters":{"shift":25,"epsilon":[1,27],"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-reciprocal-epsilon-witness","semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,27],\"shift\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0076","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":1,\"N\":1}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 2, 1, 2, 是, 否)","explanation_zh":"結構深度核對=(1, 1, 2, 1, 2, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263076,"parameters":{"amplitude":1,"N":1,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":1,\"amplitude\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0077","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":2,\"N\":3}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(2, 3, 4, 2, 4, 是, 否)","explanation_zh":"結構深度核對=(2, 3, 4, 2, 4, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263077,"parameters":{"amplitude":2,"N":3,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":3,\"amplitude\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0078","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":3,\"N\":5}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 6, 3, 6, 是, 否)","explanation_zh":"結構深度核對=(3, 5, 6, 3, 6, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263078,"parameters":{"amplitude":3,"N":5,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":5,\"amplitude\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0079","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":4,\"N\":7}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(4, 7, 8, 4, 8, 是, 否)","explanation_zh":"結構深度核對=(4, 7, 8, 4, 8, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263079,"parameters":{"amplitude":4,"N":7,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":7,\"amplitude\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0080","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":5,\"N\":9}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(5, 9, 10, 5, 10, 是, 否)","explanation_zh":"結構深度核對=(5, 9, 10, 5, 10, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263080,"parameters":{"amplitude":5,"N":9,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":9,\"amplitude\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0081","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":6,\"N\":11}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(6, 11, 12, 6, 12, 是, 否)","explanation_zh":"結構深度核對=(6, 11, 12, 6, 12, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263081,"parameters":{"amplitude":6,"N":11,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":11,\"amplitude\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0082","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":7,\"N\":13}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(7, 13, 14, 7, 14, 是, 否)","explanation_zh":"結構深度核對=(7, 13, 14, 7, 14, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263082,"parameters":{"amplitude":7,"N":13,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":13,\"amplitude\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0083","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":8,\"N\":15}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(8, 15, 16, 8, 16, 是, 否)","explanation_zh":"結構深度核對=(8, 15, 16, 8, 16, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263083,"parameters":{"amplitude":8,"N":15,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":15,\"amplitude\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0084","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":9,\"N\":17}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(9, 17, 18, 9, 18, 是, 否)","explanation_zh":"結構深度核對=(9, 17, 18, 9, 18, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263084,"parameters":{"amplitude":9,"N":17,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":17,\"amplitude\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0085","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":10,\"N\":19}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(10, 19, 20, 10, 20, 是, 否)","explanation_zh":"結構深度核對=(10, 19, 20, 10, 20, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263085,"parameters":{"amplitude":10,"N":19,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":19,\"amplitude\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0086","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":11,\"N\":21}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(11, 21, 22, 11, 22, 是, 否)","explanation_zh":"結構深度核對=(11, 21, 22, 11, 22, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263086,"parameters":{"amplitude":11,"N":21,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":21,\"amplitude\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0087","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":12,\"N\":23}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(12, 23, 24, 12, 24, 是, 否)","explanation_zh":"結構深度核對=(12, 23, 24, 12, 24, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263087,"parameters":{"amplitude":12,"N":23,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":23,\"amplitude\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0088","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":13,\"N\":25}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(13, 25, 26, 13, 26, 是, 否)","explanation_zh":"結構深度核對=(13, 25, 26, 13, 26, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263088,"parameters":{"amplitude":13,"N":25,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":25,\"amplitude\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0089","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":14,\"N\":27}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(14, 27, 28, 14, 28, 是, 否)","explanation_zh":"結構深度核對=(14, 27, 28, 14, 28, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263089,"parameters":{"amplitude":14,"N":27,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":27,\"amplitude\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0090","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":15,\"N\":29}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(15, 29, 30, 15, 30, 是, 否)","explanation_zh":"結構深度核對=(15, 29, 30, 15, 30, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263090,"parameters":{"amplitude":15,"N":29,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":29,\"amplitude\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0091","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":16,\"N\":31}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(16, 31, 32, 16, 32, 是, 否)","explanation_zh":"結構深度核對=(16, 31, 32, 16, 32, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263091,"parameters":{"amplitude":16,"N":31,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":31,\"amplitude\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0092","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":17,\"N\":33}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(17, 33, 34, 17, 34, 是, 否)","explanation_zh":"結構深度核對=(17, 33, 34, 17, 34, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263092,"parameters":{"amplitude":17,"N":33,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":33,\"amplitude\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0093","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":18,\"N\":35}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(18, 35, 36, 18, 36, 是, 否)","explanation_zh":"結構深度核對=(18, 35, 36, 18, 36, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263093,"parameters":{"amplitude":18,"N":35,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":35,\"amplitude\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0094","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":19,\"N\":37}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(19, 37, 38, 19, 38, 是, 否)","explanation_zh":"結構深度核對=(19, 37, 38, 19, 38, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263094,"parameters":{"amplitude":19,"N":37,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":37,\"amplitude\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0095","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":20,\"N\":39}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(20, 39, 40, 20, 40, 是, 否)","explanation_zh":"結構深度核對=(20, 39, 40, 20, 40, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263095,"parameters":{"amplitude":20,"N":39,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":39,\"amplitude\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0096","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":21,\"N\":41}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(21, 41, 42, 21, 42, 是, 否)","explanation_zh":"結構深度核對=(21, 41, 42, 21, 42, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263096,"parameters":{"amplitude":21,"N":41,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":41,\"amplitude\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0097","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":22,\"N\":43}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(22, 43, 44, 22, 44, 是, 否)","explanation_zh":"結構深度核對=(22, 43, 44, 22, 44, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263097,"parameters":{"amplitude":22,"N":43,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":43,\"amplitude\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0098","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":23,\"N\":45}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(23, 45, 46, 23, 46, 是, 否)","explanation_zh":"結構深度核對=(23, 45, 46, 23, 46, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263098,"parameters":{"amplitude":23,"N":45,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":45,\"amplitude\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0099","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":24,\"N\":47}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(24, 47, 48, 24, 48, 是, 否)","explanation_zh":"結構深度核對=(24, 47, 48, 24, 48, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263099,"parameters":{"amplitude":24,"N":47,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":47,\"amplitude\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261026-r03-cauchy-sequence-0100","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","language":"zh-Hant","source_mko_ids":["mko-cauchy-sequence"],"title_zh":"Cauchy序列","stem_zh":"保留forall epsilon exists N forall m,n>=N。操作=cauchy-alternating-counterexample；資料={\"amplitude\":25,\"N\":49}。請依 [amplitude,N,next_index,epsilon,distance,counterexample,cauchy] 欄序回答。","answer":{"kind":"expression","value":"(25, 49, 50, 25, 50, 是, 否)","explanation_zh":"結構深度核對=(25, 49, 50, 25, 50, 是, 否)；依完整明示資料重算；完整有限／精確切片不自動升格為任意無限定理；有限尾段本身不證Cauchy；只接受最終常值或明示界證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Cauchy序列"]},"variant":{"family_id":"qf-mod26-cauchy-alternating-counterexample","seed":202610263100,"parameters":{"amplitude":25,"N":49,"domain":"Certified_real_Cauchy_sequence_slices","operation":"cauchy-alternating-counterexample","semantic_template":"qf-mod26-cauchy-alternating-counterexample/{\"N\":49,\"amplitude\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、一般無限定理、獨立性、一般圖同構、不限維分析、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}