{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0001","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":1,\"exponent\":2,\"lower\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x^2, [1,infinity), 1, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=1/x^2, [1,infinity), 1, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124001,"parameters":{"scale":1,"exponent":2,"lower":1,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":2,\"lower\":1,\"scale\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0002","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":2,\"exponent\":3,\"lower\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=2/x^3, [2,infinity), 1/4, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=2/x^3, [2,infinity), 1/4, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124002,"parameters":{"scale":2,"exponent":3,"lower":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":3,\"lower\":2,\"scale\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0003","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":3,\"exponent\":4,\"lower\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=3/x^4, [3,infinity), 1/27, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=3/x^4, [3,infinity), 1/27, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124003,"parameters":{"scale":3,"exponent":4,"lower":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":4,\"lower\":3,\"scale\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0004","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":4,\"exponent\":5,\"lower\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=4/x^5, [4,infinity), 1/256, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=4/x^5, [4,infinity), 1/256, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124004,"parameters":{"scale":4,"exponent":5,"lower":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":5,\"lower\":4,\"scale\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0005","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":5,\"exponent\":2,\"lower\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=5/x^2, [5,infinity), 1, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=5/x^2, [5,infinity), 1, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124005,"parameters":{"scale":5,"exponent":2,"lower":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":2,\"lower\":5,\"scale\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0006","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":6,\"exponent\":3,\"lower\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=6/x^3, [1,infinity), 3, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=6/x^3, [1,infinity), 3, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124006,"parameters":{"scale":6,"exponent":3,"lower":1,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":3,\"lower\":1,\"scale\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0007","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":7,\"exponent\":4,\"lower\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=7/x^4, [2,infinity), 7/24, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=7/x^4, [2,infinity), 7/24, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124007,"parameters":{"scale":7,"exponent":4,"lower":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":4,\"lower\":2,\"scale\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0008","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":8,\"exponent\":5,\"lower\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=8/x^5, [3,infinity), 2/81, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=8/x^5, [3,infinity), 2/81, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124008,"parameters":{"scale":8,"exponent":5,"lower":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":5,\"lower\":3,\"scale\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0009","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":9,\"exponent\":2,\"lower\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=9/x^2, [4,infinity), 9/4, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=9/x^2, [4,infinity), 9/4, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124009,"parameters":{"scale":9,"exponent":2,"lower":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":2,\"lower\":4,\"scale\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0010","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":10,\"exponent\":3,\"lower\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=10/x^3, [5,infinity), 1/5, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=10/x^3, [5,infinity), 1/5, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124010,"parameters":{"scale":10,"exponent":3,"lower":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":3,\"lower\":5,\"scale\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0011","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":11,\"exponent\":4,\"lower\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=11/x^4, [1,infinity), 11/3, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=11/x^4, [1,infinity), 11/3, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124011,"parameters":{"scale":11,"exponent":4,"lower":1,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":4,\"lower\":1,\"scale\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0012","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":12,\"exponent\":5,\"lower\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=12/x^5, [2,infinity), 3/16, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=12/x^5, [2,infinity), 3/16, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124012,"parameters":{"scale":12,"exponent":5,"lower":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":5,\"lower\":2,\"scale\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0013","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":13,\"exponent\":2,\"lower\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=13/x^2, [3,infinity), 13/3, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=13/x^2, [3,infinity), 13/3, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124013,"parameters":{"scale":13,"exponent":2,"lower":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":2,\"lower\":3,\"scale\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0014","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":14,\"exponent\":3,\"lower\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=14/x^3, [4,infinity), 7/16, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=14/x^3, [4,infinity), 7/16, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124014,"parameters":{"scale":14,"exponent":3,"lower":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":3,\"lower\":4,\"scale\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0015","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":15,\"exponent\":4,\"lower\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=15/x^4, [5,infinity), 1/25, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=15/x^4, [5,infinity), 1/25, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124015,"parameters":{"scale":15,"exponent":4,"lower":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":4,\"lower\":5,\"scale\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0016","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":16,\"exponent\":5,\"lower\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=16/x^5, [1,infinity), 4, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=16/x^5, [1,infinity), 4, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124016,"parameters":{"scale":16,"exponent":5,"lower":1,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":5,\"lower\":1,\"scale\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0017","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":17,\"exponent\":2,\"lower\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=17/x^2, [2,infinity), 17/2, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=17/x^2, [2,infinity), 17/2, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124017,"parameters":{"scale":17,"exponent":2,"lower":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":2,\"lower\":2,\"scale\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0018","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":18,\"exponent\":3,\"lower\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=18/x^3, [3,infinity), 1, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=18/x^3, [3,infinity), 1, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124018,"parameters":{"scale":18,"exponent":3,"lower":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":3,\"lower\":3,\"scale\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0019","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":19,\"exponent\":4,\"lower\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=19/x^4, [4,infinity), 19/192, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=19/x^4, [4,infinity), 19/192, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124019,"parameters":{"scale":19,"exponent":4,"lower":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":4,\"lower\":4,\"scale\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0020","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":20,\"exponent\":5,\"lower\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=20/x^5, [5,infinity), 1/125, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=20/x^5, [5,infinity), 1/125, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124020,"parameters":{"scale":20,"exponent":5,"lower":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":5,\"lower\":5,\"scale\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0021","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":21,\"exponent\":2,\"lower\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=21/x^2, [1,infinity), 21, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=21/x^2, [1,infinity), 21, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124021,"parameters":{"scale":21,"exponent":2,"lower":1,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":2,\"lower\":1,\"scale\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0022","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":22,\"exponent\":3,\"lower\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=22/x^3, [2,infinity), 11/4, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=22/x^3, [2,infinity), 11/4, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124022,"parameters":{"scale":22,"exponent":3,"lower":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":3,\"lower\":2,\"scale\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0023","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":23,\"exponent\":4,\"lower\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=23/x^4, [3,infinity), 23/81, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=23/x^4, [3,infinity), 23/81, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124023,"parameters":{"scale":23,"exponent":4,"lower":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":4,\"lower\":3,\"scale\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0024","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":24,\"exponent\":5,\"lower\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=24/x^5, [4,infinity), 3/128, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=24/x^5, [4,infinity), 3/128, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124024,"parameters":{"scale":24,"exponent":5,"lower":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":5,\"lower\":4,\"scale\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0025","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-one-sided-tail；資料={\"scale\":25,\"exponent\":2,\"lower\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=25/x^2, [5,infinity), 5, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)","explanation_zh":"Wave6第六層核對=(f(x)=25/x^2, [5,infinity), 5, 是, limit_of_integrals_from_lower_to_b_as_b_tends_to_infinity)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-one-sided-tail","seed":202611124025,"parameters":{"scale":25,"exponent":2,"lower":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-one-sided-tail","semantic_template":"qf-mod43-improper-one-sided-tail/{\"exponent\":2,\"lower\":5,\"scale\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0026","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":1,\"right_scale\":2,\"left_power_numerator\":0,\"left_power_denominator\":1,\"right_power_numerator\":1,\"right_power_denominator\":2,\"singularity\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-13,-11], -12, left:1*(c-x)^(-0/1), right:2*(x-c)^(-1/2), 1, 4, 是, 是, 5, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-13,-11], -12, left:1*(c-x)^(-0/1), right:2*(x-c)^(-1/2), 1, 4, 是, 是, 5, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124026,"parameters":{"left_scale":1,"right_scale":2,"left_power_numerator":0,"left_power_denominator":1,"right_power_numerator":1,"right_power_denominator":2,"singularity":-12,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":1,\"left_power_numerator\":0,\"left_scale\":1,\"right_power_denominator\":2,\"right_power_numerator\":1,\"right_scale\":2,\"singularity\":-12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0027","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":2,\"right_scale\":3,\"left_power_numerator\":1,\"left_power_denominator\":2,\"right_power_numerator\":2,\"right_power_denominator\":3,\"singularity\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-12,-10], -11, left:2*(c-x)^(-1/2), right:3*(x-c)^(-2/3), 4, 9, 是, 是, 13, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-12,-10], -11, left:2*(c-x)^(-1/2), right:3*(x-c)^(-2/3), 4, 9, 是, 是, 13, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124027,"parameters":{"left_scale":2,"right_scale":3,"left_power_numerator":1,"left_power_denominator":2,"right_power_numerator":2,"right_power_denominator":3,"singularity":-11,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":2,\"left_power_numerator\":1,\"left_scale\":2,\"right_power_denominator\":3,\"right_power_numerator\":2,\"right_scale\":3,\"singularity\":-11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0028","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":3,\"right_scale\":4,\"left_power_numerator\":2,\"left_power_denominator\":3,\"right_power_numerator\":3,\"right_power_denominator\":4,\"singularity\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-11,-9], -10, left:3*(c-x)^(-2/3), right:4*(x-c)^(-3/4), 9, 16, 是, 是, 25, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-11,-9], -10, left:3*(c-x)^(-2/3), right:4*(x-c)^(-3/4), 9, 16, 是, 是, 25, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124028,"parameters":{"left_scale":3,"right_scale":4,"left_power_numerator":2,"left_power_denominator":3,"right_power_numerator":3,"right_power_denominator":4,"singularity":-10,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":3,\"left_power_numerator\":2,\"left_scale\":3,\"right_power_denominator\":4,\"right_power_numerator\":3,\"right_scale\":4,\"singularity\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0029","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":4,\"right_scale\":5,\"left_power_numerator\":3,\"left_power_denominator\":4,\"right_power_numerator\":0,\"right_power_denominator\":1,\"singularity\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-10,-8], -9, left:4*(c-x)^(-3/4), right:5*(x-c)^(-0/1), 16, 5, 是, 是, 21, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-10,-8], -9, left:4*(c-x)^(-3/4), right:5*(x-c)^(-0/1), 16, 5, 是, 是, 21, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124029,"parameters":{"left_scale":4,"right_scale":5,"left_power_numerator":3,"left_power_denominator":4,"right_power_numerator":0,"right_power_denominator":1,"singularity":-9,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":4,\"left_power_numerator\":3,\"left_scale\":4,\"right_power_denominator\":1,\"right_power_numerator\":0,\"right_scale\":5,\"singularity\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0030","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":5,\"right_scale\":6,\"left_power_numerator\":0,\"left_power_denominator\":1,\"right_power_numerator\":1,\"right_power_denominator\":2,\"singularity\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-9,-7], -8, left:5*(c-x)^(-0/1), right:6*(x-c)^(-1/2), 5, 12, 是, 是, 17, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-9,-7], -8, left:5*(c-x)^(-0/1), right:6*(x-c)^(-1/2), 5, 12, 是, 是, 17, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124030,"parameters":{"left_scale":5,"right_scale":6,"left_power_numerator":0,"left_power_denominator":1,"right_power_numerator":1,"right_power_denominator":2,"singularity":-8,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":1,\"left_power_numerator\":0,\"left_scale\":5,\"right_power_denominator\":2,\"right_power_numerator\":1,\"right_scale\":6,\"singularity\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0031","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":6,\"right_scale\":7,\"left_power_numerator\":1,\"left_power_denominator\":2,\"right_power_numerator\":2,\"right_power_denominator\":3,\"singularity\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-8,-6], -7, left:6*(c-x)^(-1/2), right:7*(x-c)^(-2/3), 12, 21, 是, 是, 33, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-8,-6], -7, left:6*(c-x)^(-1/2), right:7*(x-c)^(-2/3), 12, 21, 是, 是, 33, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124031,"parameters":{"left_scale":6,"right_scale":7,"left_power_numerator":1,"left_power_denominator":2,"right_power_numerator":2,"right_power_denominator":3,"singularity":-7,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":2,\"left_power_numerator\":1,\"left_scale\":6,\"right_power_denominator\":3,\"right_power_numerator\":2,\"right_scale\":7,\"singularity\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0032","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":7,\"right_scale\":8,\"left_power_numerator\":2,\"left_power_denominator\":3,\"right_power_numerator\":3,\"right_power_denominator\":4,\"singularity\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-7,-5], -6, left:7*(c-x)^(-2/3), right:8*(x-c)^(-3/4), 21, 32, 是, 是, 53, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-7,-5], -6, left:7*(c-x)^(-2/3), right:8*(x-c)^(-3/4), 21, 32, 是, 是, 53, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124032,"parameters":{"left_scale":7,"right_scale":8,"left_power_numerator":2,"left_power_denominator":3,"right_power_numerator":3,"right_power_denominator":4,"singularity":-6,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":3,\"left_power_numerator\":2,\"left_scale\":7,\"right_power_denominator\":4,\"right_power_numerator\":3,\"right_scale\":8,\"singularity\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0033","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":8,\"right_scale\":9,\"left_power_numerator\":3,\"left_power_denominator\":4,\"right_power_numerator\":0,\"right_power_denominator\":1,\"singularity\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-6,-4], -5, left:8*(c-x)^(-3/4), right:9*(x-c)^(-0/1), 32, 9, 是, 是, 41, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-6,-4], -5, left:8*(c-x)^(-3/4), right:9*(x-c)^(-0/1), 32, 9, 是, 是, 41, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124033,"parameters":{"left_scale":8,"right_scale":9,"left_power_numerator":3,"left_power_denominator":4,"right_power_numerator":0,"right_power_denominator":1,"singularity":-5,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":4,\"left_power_numerator\":3,\"left_scale\":8,\"right_power_denominator\":1,\"right_power_numerator\":0,\"right_scale\":9,\"singularity\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0034","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":9,\"right_scale\":10,\"left_power_numerator\":0,\"left_power_denominator\":1,\"right_power_numerator\":1,\"right_power_denominator\":2,\"singularity\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-5,-3], -4, left:9*(c-x)^(-0/1), right:10*(x-c)^(-1/2), 9, 20, 是, 是, 29, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-5,-3], -4, left:9*(c-x)^(-0/1), right:10*(x-c)^(-1/2), 9, 20, 是, 是, 29, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124034,"parameters":{"left_scale":9,"right_scale":10,"left_power_numerator":0,"left_power_denominator":1,"right_power_numerator":1,"right_power_denominator":2,"singularity":-4,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":1,\"left_power_numerator\":0,\"left_scale\":9,\"right_power_denominator\":2,\"right_power_numerator\":1,\"right_scale\":10,\"singularity\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0035","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":10,\"right_scale\":11,\"left_power_numerator\":1,\"left_power_denominator\":2,\"right_power_numerator\":2,\"right_power_denominator\":3,\"singularity\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-4,-2], -3, left:10*(c-x)^(-1/2), right:11*(x-c)^(-2/3), 20, 33, 是, 是, 53, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-4,-2], -3, left:10*(c-x)^(-1/2), right:11*(x-c)^(-2/3), 20, 33, 是, 是, 53, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124035,"parameters":{"left_scale":10,"right_scale":11,"left_power_numerator":1,"left_power_denominator":2,"right_power_numerator":2,"right_power_denominator":3,"singularity":-3,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":2,\"left_power_numerator\":1,\"left_scale\":10,\"right_power_denominator\":3,\"right_power_numerator\":2,\"right_scale\":11,\"singularity\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0036","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":11,\"right_scale\":12,\"left_power_numerator\":2,\"left_power_denominator\":3,\"right_power_numerator\":3,\"right_power_denominator\":4,\"singularity\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-3,-1], -2, left:11*(c-x)^(-2/3), right:12*(x-c)^(-3/4), 33, 48, 是, 是, 81, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-3,-1], -2, left:11*(c-x)^(-2/3), right:12*(x-c)^(-3/4), 33, 48, 是, 是, 81, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124036,"parameters":{"left_scale":11,"right_scale":12,"left_power_numerator":2,"left_power_denominator":3,"right_power_numerator":3,"right_power_denominator":4,"singularity":-2,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":3,\"left_power_numerator\":2,\"left_scale\":11,\"right_power_denominator\":4,\"right_power_numerator\":3,\"right_scale\":12,\"singularity\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0037","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":12,\"right_scale\":13,\"left_power_numerator\":3,\"left_power_denominator\":4,\"right_power_numerator\":0,\"right_power_denominator\":1,\"singularity\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2,0], -1, left:12*(c-x)^(-3/4), right:13*(x-c)^(-0/1), 48, 13, 是, 是, 61, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-2,0], -1, left:12*(c-x)^(-3/4), right:13*(x-c)^(-0/1), 48, 13, 是, 是, 61, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124037,"parameters":{"left_scale":12,"right_scale":13,"left_power_numerator":3,"left_power_denominator":4,"right_power_numerator":0,"right_power_denominator":1,"singularity":-1,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":4,\"left_power_numerator\":3,\"left_scale\":12,\"right_power_denominator\":1,\"right_power_numerator\":0,\"right_scale\":13,\"singularity\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0038","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":13,\"right_scale\":14,\"left_power_numerator\":0,\"left_power_denominator\":1,\"right_power_numerator\":1,\"right_power_denominator\":2,\"singularity\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-1,1], 0, left:13*(c-x)^(-0/1), right:14*(x-c)^(-1/2), 13, 28, 是, 是, 41, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([-1,1], 0, left:13*(c-x)^(-0/1), right:14*(x-c)^(-1/2), 13, 28, 是, 是, 41, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124038,"parameters":{"left_scale":13,"right_scale":14,"left_power_numerator":0,"left_power_denominator":1,"right_power_numerator":1,"right_power_denominator":2,"singularity":0,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":1,\"left_power_numerator\":0,\"left_scale\":13,\"right_power_denominator\":2,\"right_power_numerator\":1,\"right_scale\":14,\"singularity\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0039","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":14,\"right_scale\":15,\"left_power_numerator\":1,\"left_power_denominator\":2,\"right_power_numerator\":2,\"right_power_denominator\":3,\"singularity\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0,2], 1, left:14*(c-x)^(-1/2), right:15*(x-c)^(-2/3), 28, 45, 是, 是, 73, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([0,2], 1, left:14*(c-x)^(-1/2), right:15*(x-c)^(-2/3), 28, 45, 是, 是, 73, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124039,"parameters":{"left_scale":14,"right_scale":15,"left_power_numerator":1,"left_power_denominator":2,"right_power_numerator":2,"right_power_denominator":3,"singularity":1,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":2,\"left_power_numerator\":1,\"left_scale\":14,\"right_power_denominator\":3,\"right_power_numerator\":2,\"right_scale\":15,\"singularity\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0040","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":15,\"right_scale\":16,\"left_power_numerator\":2,\"left_power_denominator\":3,\"right_power_numerator\":3,\"right_power_denominator\":4,\"singularity\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1,3], 2, left:15*(c-x)^(-2/3), right:16*(x-c)^(-3/4), 45, 64, 是, 是, 109, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([1,3], 2, left:15*(c-x)^(-2/3), right:16*(x-c)^(-3/4), 45, 64, 是, 是, 109, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124040,"parameters":{"left_scale":15,"right_scale":16,"left_power_numerator":2,"left_power_denominator":3,"right_power_numerator":3,"right_power_denominator":4,"singularity":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":3,\"left_power_numerator\":2,\"left_scale\":15,\"right_power_denominator\":4,\"right_power_numerator\":3,\"right_scale\":16,\"singularity\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0041","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":16,\"right_scale\":17,\"left_power_numerator\":3,\"left_power_denominator\":4,\"right_power_numerator\":0,\"right_power_denominator\":1,\"singularity\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2,4], 3, left:16*(c-x)^(-3/4), right:17*(x-c)^(-0/1), 64, 17, 是, 是, 81, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([2,4], 3, left:16*(c-x)^(-3/4), right:17*(x-c)^(-0/1), 64, 17, 是, 是, 81, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124041,"parameters":{"left_scale":16,"right_scale":17,"left_power_numerator":3,"left_power_denominator":4,"right_power_numerator":0,"right_power_denominator":1,"singularity":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":4,\"left_power_numerator\":3,\"left_scale\":16,\"right_power_denominator\":1,\"right_power_numerator\":0,\"right_scale\":17,\"singularity\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0042","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":17,\"right_scale\":18,\"left_power_numerator\":0,\"left_power_denominator\":1,\"right_power_numerator\":1,\"right_power_denominator\":2,\"singularity\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3,5], 4, left:17*(c-x)^(-0/1), right:18*(x-c)^(-1/2), 17, 36, 是, 是, 53, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([3,5], 4, left:17*(c-x)^(-0/1), right:18*(x-c)^(-1/2), 17, 36, 是, 是, 53, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124042,"parameters":{"left_scale":17,"right_scale":18,"left_power_numerator":0,"left_power_denominator":1,"right_power_numerator":1,"right_power_denominator":2,"singularity":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":1,\"left_power_numerator\":0,\"left_scale\":17,\"right_power_denominator\":2,\"right_power_numerator\":1,\"right_scale\":18,\"singularity\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0043","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":18,\"right_scale\":19,\"left_power_numerator\":1,\"left_power_denominator\":2,\"right_power_numerator\":2,\"right_power_denominator\":3,\"singularity\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4,6], 5, left:18*(c-x)^(-1/2), right:19*(x-c)^(-2/3), 36, 57, 是, 是, 93, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([4,6], 5, left:18*(c-x)^(-1/2), right:19*(x-c)^(-2/3), 36, 57, 是, 是, 93, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124043,"parameters":{"left_scale":18,"right_scale":19,"left_power_numerator":1,"left_power_denominator":2,"right_power_numerator":2,"right_power_denominator":3,"singularity":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":2,\"left_power_numerator\":1,\"left_scale\":18,\"right_power_denominator\":3,\"right_power_numerator\":2,\"right_scale\":19,\"singularity\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0044","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":19,\"right_scale\":20,\"left_power_numerator\":2,\"left_power_denominator\":3,\"right_power_numerator\":3,\"right_power_denominator\":4,\"singularity\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5,7], 6, left:19*(c-x)^(-2/3), right:20*(x-c)^(-3/4), 57, 80, 是, 是, 137, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([5,7], 6, left:19*(c-x)^(-2/3), right:20*(x-c)^(-3/4), 57, 80, 是, 是, 137, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124044,"parameters":{"left_scale":19,"right_scale":20,"left_power_numerator":2,"left_power_denominator":3,"right_power_numerator":3,"right_power_denominator":4,"singularity":6,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":3,\"left_power_numerator\":2,\"left_scale\":19,\"right_power_denominator\":4,\"right_power_numerator\":3,\"right_scale\":20,\"singularity\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0045","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":20,\"right_scale\":21,\"left_power_numerator\":3,\"left_power_denominator\":4,\"right_power_numerator\":0,\"right_power_denominator\":1,\"singularity\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6,8], 7, left:20*(c-x)^(-3/4), right:21*(x-c)^(-0/1), 80, 21, 是, 是, 101, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([6,8], 7, left:20*(c-x)^(-3/4), right:21*(x-c)^(-0/1), 80, 21, 是, 是, 101, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124045,"parameters":{"left_scale":20,"right_scale":21,"left_power_numerator":3,"left_power_denominator":4,"right_power_numerator":0,"right_power_denominator":1,"singularity":7,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":4,\"left_power_numerator\":3,\"left_scale\":20,\"right_power_denominator\":1,\"right_power_numerator\":0,\"right_scale\":21,\"singularity\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0046","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":21,\"right_scale\":22,\"left_power_numerator\":0,\"left_power_denominator\":1,\"right_power_numerator\":1,\"right_power_denominator\":2,\"singularity\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7,9], 8, left:21*(c-x)^(-0/1), right:22*(x-c)^(-1/2), 21, 44, 是, 是, 65, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([7,9], 8, left:21*(c-x)^(-0/1), right:22*(x-c)^(-1/2), 21, 44, 是, 是, 65, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124046,"parameters":{"left_scale":21,"right_scale":22,"left_power_numerator":0,"left_power_denominator":1,"right_power_numerator":1,"right_power_denominator":2,"singularity":8,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":1,\"left_power_numerator\":0,\"left_scale\":21,\"right_power_denominator\":2,\"right_power_numerator\":1,\"right_scale\":22,\"singularity\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0047","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":22,\"right_scale\":23,\"left_power_numerator\":1,\"left_power_denominator\":2,\"right_power_numerator\":2,\"right_power_denominator\":3,\"singularity\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8,10], 9, left:22*(c-x)^(-1/2), right:23*(x-c)^(-2/3), 44, 69, 是, 是, 113, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([8,10], 9, left:22*(c-x)^(-1/2), right:23*(x-c)^(-2/3), 44, 69, 是, 是, 113, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124047,"parameters":{"left_scale":22,"right_scale":23,"left_power_numerator":1,"left_power_denominator":2,"right_power_numerator":2,"right_power_denominator":3,"singularity":9,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":2,\"left_power_numerator\":1,\"left_scale\":22,\"right_power_denominator\":3,\"right_power_numerator\":2,\"right_scale\":23,\"singularity\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0048","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":23,\"right_scale\":24,\"left_power_numerator\":2,\"left_power_denominator\":3,\"right_power_numerator\":3,\"right_power_denominator\":4,\"singularity\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9,11], 10, left:23*(c-x)^(-2/3), right:24*(x-c)^(-3/4), 69, 96, 是, 是, 165, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([9,11], 10, left:23*(c-x)^(-2/3), right:24*(x-c)^(-3/4), 69, 96, 是, 是, 165, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124048,"parameters":{"left_scale":23,"right_scale":24,"left_power_numerator":2,"left_power_denominator":3,"right_power_numerator":3,"right_power_denominator":4,"singularity":10,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":3,\"left_power_numerator\":2,\"left_scale\":23,\"right_power_denominator\":4,\"right_power_numerator\":3,\"right_scale\":24,\"singularity\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0049","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":24,\"right_scale\":25,\"left_power_numerator\":3,\"left_power_denominator\":4,\"right_power_numerator\":0,\"right_power_denominator\":1,\"singularity\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10,12], 11, left:24*(c-x)^(-3/4), right:25*(x-c)^(-0/1), 96, 25, 是, 是, 121, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([10,12], 11, left:24*(c-x)^(-3/4), right:25*(x-c)^(-0/1), 96, 25, 是, 是, 121, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124049,"parameters":{"left_scale":24,"right_scale":25,"left_power_numerator":3,"left_power_denominator":4,"right_power_numerator":0,"right_power_denominator":1,"singularity":11,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":4,\"left_power_numerator\":3,\"left_scale\":24,\"right_power_denominator\":1,\"right_power_numerator\":0,\"right_scale\":25,\"singularity\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0050","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-interior-singular-split；資料={\"left_scale\":25,\"right_scale\":26,\"left_power_numerator\":0,\"left_power_denominator\":1,\"right_power_numerator\":1,\"right_power_denominator\":2,\"singularity\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11,13], 12, left:25*(c-x)^(-0/1), right:26*(x-c)^(-1/2), 25, 52, 是, 是, 77, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)","explanation_zh":"Wave6第六層核對=([11,13], 12, left:25*(c-x)^(-0/1), right:26*(x-c)^(-1/2), 25, 52, 是, 是, 77, interior_singularity_is_split_and_both_one_sided_limits_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-interior-singular-split","seed":202611124050,"parameters":{"left_scale":25,"right_scale":26,"left_power_numerator":0,"left_power_denominator":1,"right_power_numerator":1,"right_power_denominator":2,"singularity":12,"domain":"Exact_symbolic_improper_integral","operation":"improper-interior-singular-split","semantic_template":"qf-mod43-improper-interior-singular-split/{\"left_power_denominator\":1,\"left_power_numerator\":0,\"left_scale\":25,\"right_power_denominator\":2,\"right_power_numerator\":1,\"right_scale\":26,\"singularity\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0051","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":1,\"right_scale\":2,\"exponent\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:1/abs(x)^2;-1<x<1:0;x>=1:2/x^2, (-infinity,infinity), 1, 2, 是, 是, 3, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:1/abs(x)^2;-1<x<1:0;x>=1:2/x^2, (-infinity,infinity), 1, 2, 是, 是, 3, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124051,"parameters":{"left_scale":1,"right_scale":2,"exponent":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":2,\"left_scale\":1,\"right_scale\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0052","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":2,\"right_scale\":3,\"exponent\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:2/abs(x)^3;-1<x<1:0;x>=1:3/x^3, (-infinity,infinity), 1, 3/2, 是, 是, 5/2, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:2/abs(x)^3;-1<x<1:0;x>=1:3/x^3, (-infinity,infinity), 1, 3/2, 是, 是, 5/2, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124052,"parameters":{"left_scale":2,"right_scale":3,"exponent":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":3,\"left_scale\":2,\"right_scale\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0053","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":3,\"right_scale\":4,\"exponent\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:3/abs(x)^4;-1<x<1:0;x>=1:4/x^4, (-infinity,infinity), 1, 4/3, 是, 是, 7/3, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:3/abs(x)^4;-1<x<1:0;x>=1:4/x^4, (-infinity,infinity), 1, 4/3, 是, 是, 7/3, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124053,"parameters":{"left_scale":3,"right_scale":4,"exponent":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":4,\"left_scale\":3,\"right_scale\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0054","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":4,\"right_scale\":5,\"exponent\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:4/abs(x)^5;-1<x<1:0;x>=1:5/x^5, (-infinity,infinity), 1, 5/4, 是, 是, 9/4, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:4/abs(x)^5;-1<x<1:0;x>=1:5/x^5, (-infinity,infinity), 1, 5/4, 是, 是, 9/4, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124054,"parameters":{"left_scale":4,"right_scale":5,"exponent":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":5,\"left_scale\":4,\"right_scale\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0055","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":5,\"right_scale\":6,\"exponent\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:5/abs(x)^2;-1<x<1:0;x>=1:6/x^2, (-infinity,infinity), 5, 6, 是, 是, 11, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:5/abs(x)^2;-1<x<1:0;x>=1:6/x^2, (-infinity,infinity), 5, 6, 是, 是, 11, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124055,"parameters":{"left_scale":5,"right_scale":6,"exponent":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":2,\"left_scale\":5,\"right_scale\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0056","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":6,\"right_scale\":7,\"exponent\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:6/abs(x)^3;-1<x<1:0;x>=1:7/x^3, (-infinity,infinity), 3, 7/2, 是, 是, 13/2, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:6/abs(x)^3;-1<x<1:0;x>=1:7/x^3, (-infinity,infinity), 3, 7/2, 是, 是, 13/2, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124056,"parameters":{"left_scale":6,"right_scale":7,"exponent":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":3,\"left_scale\":6,\"right_scale\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0057","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":7,\"right_scale\":8,\"exponent\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:7/abs(x)^4;-1<x<1:0;x>=1:8/x^4, (-infinity,infinity), 7/3, 8/3, 是, 是, 5, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:7/abs(x)^4;-1<x<1:0;x>=1:8/x^4, (-infinity,infinity), 7/3, 8/3, 是, 是, 5, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124057,"parameters":{"left_scale":7,"right_scale":8,"exponent":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":4,\"left_scale\":7,\"right_scale\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0058","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":8,\"right_scale\":9,\"exponent\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:8/abs(x)^5;-1<x<1:0;x>=1:9/x^5, (-infinity,infinity), 2, 9/4, 是, 是, 17/4, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:8/abs(x)^5;-1<x<1:0;x>=1:9/x^5, (-infinity,infinity), 2, 9/4, 是, 是, 17/4, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124058,"parameters":{"left_scale":8,"right_scale":9,"exponent":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":5,\"left_scale\":8,\"right_scale\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0059","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":9,\"right_scale\":10,\"exponent\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:9/abs(x)^2;-1<x<1:0;x>=1:10/x^2, (-infinity,infinity), 9, 10, 是, 是, 19, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:9/abs(x)^2;-1<x<1:0;x>=1:10/x^2, (-infinity,infinity), 9, 10, 是, 是, 19, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124059,"parameters":{"left_scale":9,"right_scale":10,"exponent":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":2,\"left_scale\":9,\"right_scale\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0060","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":10,\"right_scale\":11,\"exponent\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:10/abs(x)^3;-1<x<1:0;x>=1:11/x^3, (-infinity,infinity), 5, 11/2, 是, 是, 21/2, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:10/abs(x)^3;-1<x<1:0;x>=1:11/x^3, (-infinity,infinity), 5, 11/2, 是, 是, 21/2, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124060,"parameters":{"left_scale":10,"right_scale":11,"exponent":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":3,\"left_scale\":10,\"right_scale\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0061","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":11,\"right_scale\":12,\"exponent\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:11/abs(x)^4;-1<x<1:0;x>=1:12/x^4, (-infinity,infinity), 11/3, 4, 是, 是, 23/3, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:11/abs(x)^4;-1<x<1:0;x>=1:12/x^4, (-infinity,infinity), 11/3, 4, 是, 是, 23/3, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124061,"parameters":{"left_scale":11,"right_scale":12,"exponent":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":4,\"left_scale\":11,\"right_scale\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0062","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":12,\"right_scale\":13,\"exponent\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:12/abs(x)^5;-1<x<1:0;x>=1:13/x^5, (-infinity,infinity), 3, 13/4, 是, 是, 25/4, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:12/abs(x)^5;-1<x<1:0;x>=1:13/x^5, (-infinity,infinity), 3, 13/4, 是, 是, 25/4, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124062,"parameters":{"left_scale":12,"right_scale":13,"exponent":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":5,\"left_scale\":12,\"right_scale\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0063","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":13,\"right_scale\":14,\"exponent\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:13/abs(x)^2;-1<x<1:0;x>=1:14/x^2, (-infinity,infinity), 13, 14, 是, 是, 27, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:13/abs(x)^2;-1<x<1:0;x>=1:14/x^2, (-infinity,infinity), 13, 14, 是, 是, 27, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124063,"parameters":{"left_scale":13,"right_scale":14,"exponent":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":2,\"left_scale\":13,\"right_scale\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0064","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":14,\"right_scale\":15,\"exponent\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:14/abs(x)^3;-1<x<1:0;x>=1:15/x^3, (-infinity,infinity), 7, 15/2, 是, 是, 29/2, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:14/abs(x)^3;-1<x<1:0;x>=1:15/x^3, (-infinity,infinity), 7, 15/2, 是, 是, 29/2, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124064,"parameters":{"left_scale":14,"right_scale":15,"exponent":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":3,\"left_scale\":14,\"right_scale\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0065","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":15,\"right_scale\":16,\"exponent\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:15/abs(x)^4;-1<x<1:0;x>=1:16/x^4, (-infinity,infinity), 5, 16/3, 是, 是, 31/3, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:15/abs(x)^4;-1<x<1:0;x>=1:16/x^4, (-infinity,infinity), 5, 16/3, 是, 是, 31/3, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124065,"parameters":{"left_scale":15,"right_scale":16,"exponent":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":4,\"left_scale\":15,\"right_scale\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0066","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":16,\"right_scale\":17,\"exponent\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:16/abs(x)^5;-1<x<1:0;x>=1:17/x^5, (-infinity,infinity), 4, 17/4, 是, 是, 33/4, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:16/abs(x)^5;-1<x<1:0;x>=1:17/x^5, (-infinity,infinity), 4, 17/4, 是, 是, 33/4, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124066,"parameters":{"left_scale":16,"right_scale":17,"exponent":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":5,\"left_scale\":16,\"right_scale\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0067","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":17,\"right_scale\":18,\"exponent\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:17/abs(x)^2;-1<x<1:0;x>=1:18/x^2, (-infinity,infinity), 17, 18, 是, 是, 35, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:17/abs(x)^2;-1<x<1:0;x>=1:18/x^2, (-infinity,infinity), 17, 18, 是, 是, 35, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124067,"parameters":{"left_scale":17,"right_scale":18,"exponent":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":2,\"left_scale\":17,\"right_scale\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0068","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":18,\"right_scale\":19,\"exponent\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:18/abs(x)^3;-1<x<1:0;x>=1:19/x^3, (-infinity,infinity), 9, 19/2, 是, 是, 37/2, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:18/abs(x)^3;-1<x<1:0;x>=1:19/x^3, (-infinity,infinity), 9, 19/2, 是, 是, 37/2, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124068,"parameters":{"left_scale":18,"right_scale":19,"exponent":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":3,\"left_scale\":18,\"right_scale\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0069","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":19,\"right_scale\":20,\"exponent\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:19/abs(x)^4;-1<x<1:0;x>=1:20/x^4, (-infinity,infinity), 19/3, 20/3, 是, 是, 13, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:19/abs(x)^4;-1<x<1:0;x>=1:20/x^4, (-infinity,infinity), 19/3, 20/3, 是, 是, 13, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124069,"parameters":{"left_scale":19,"right_scale":20,"exponent":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":4,\"left_scale\":19,\"right_scale\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0070","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":20,\"right_scale\":21,\"exponent\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:20/abs(x)^5;-1<x<1:0;x>=1:21/x^5, (-infinity,infinity), 5, 21/4, 是, 是, 41/4, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:20/abs(x)^5;-1<x<1:0;x>=1:21/x^5, (-infinity,infinity), 5, 21/4, 是, 是, 41/4, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124070,"parameters":{"left_scale":20,"right_scale":21,"exponent":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":5,\"left_scale\":20,\"right_scale\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0071","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":21,\"right_scale\":22,\"exponent\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:21/abs(x)^2;-1<x<1:0;x>=1:22/x^2, (-infinity,infinity), 21, 22, 是, 是, 43, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:21/abs(x)^2;-1<x<1:0;x>=1:22/x^2, (-infinity,infinity), 21, 22, 是, 是, 43, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124071,"parameters":{"left_scale":21,"right_scale":22,"exponent":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":2,\"left_scale\":21,\"right_scale\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0072","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":22,\"right_scale\":23,\"exponent\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:22/abs(x)^3;-1<x<1:0;x>=1:23/x^3, (-infinity,infinity), 11, 23/2, 是, 是, 45/2, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:22/abs(x)^3;-1<x<1:0;x>=1:23/x^3, (-infinity,infinity), 11, 23/2, 是, 是, 45/2, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124072,"parameters":{"left_scale":22,"right_scale":23,"exponent":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":3,\"left_scale\":22,\"right_scale\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0073","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":23,\"right_scale\":24,\"exponent\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:23/abs(x)^4;-1<x<1:0;x>=1:24/x^4, (-infinity,infinity), 23/3, 8, 是, 是, 47/3, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:23/abs(x)^4;-1<x<1:0;x>=1:24/x^4, (-infinity,infinity), 23/3, 8, 是, 是, 47/3, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124073,"parameters":{"left_scale":23,"right_scale":24,"exponent":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":4,\"left_scale\":23,\"right_scale\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0074","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":24,\"right_scale\":25,\"exponent\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:24/abs(x)^5;-1<x<1:0;x>=1:25/x^5, (-infinity,infinity), 6, 25/4, 是, 是, 49/4, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:24/abs(x)^5;-1<x<1:0;x>=1:25/x^5, (-infinity,infinity), 6, 25/4, 是, 是, 49/4, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124074,"parameters":{"left_scale":24,"right_scale":25,"exponent":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":5,\"left_scale\":24,\"right_scale\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0075","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-two-sided-independent；資料={\"left_scale\":25,\"right_scale\":26,\"exponent\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(x<=-1:25/abs(x)^2;-1<x<1:0;x>=1:26/x^2, (-infinity,infinity), 25, 26, 是, 是, 51, both_one_sided_tail_integrals_converge_independently)","explanation_zh":"Wave6第六層核對=(x<=-1:25/abs(x)^2;-1<x<1:0;x>=1:26/x^2, (-infinity,infinity), 25, 26, 是, 是, 51, both_one_sided_tail_integrals_converge_independently)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-two-sided-independent","seed":202611124075,"parameters":{"left_scale":25,"right_scale":26,"exponent":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-two-sided-independent","semantic_template":"qf-mod43-improper-two-sided-independent/{\"exponent\":2,\"left_scale\":25,\"right_scale\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0076","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":2,\"epsilon_denominator\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 2, 1/2, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 2, 1/2, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124076,"parameters":{"cutoff":2,"epsilon_denominator":2,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":2,\"epsilon_denominator\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0077","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":3,\"epsilon_denominator\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 3, 1/3, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 3, 1/3, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124077,"parameters":{"cutoff":3,"epsilon_denominator":3,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":3,\"epsilon_denominator\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0078","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":4,\"epsilon_denominator\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 4, 1/4, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 4, 1/4, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124078,"parameters":{"cutoff":4,"epsilon_denominator":4,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":4,\"epsilon_denominator\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0079","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":5,\"epsilon_denominator\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 5, 1/5, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 5, 1/5, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124079,"parameters":{"cutoff":5,"epsilon_denominator":5,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":5,\"epsilon_denominator\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0080","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":6,\"epsilon_denominator\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 6, 1/6, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 6, 1/6, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124080,"parameters":{"cutoff":6,"epsilon_denominator":6,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":6,\"epsilon_denominator\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0081","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":7,\"epsilon_denominator\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 7, 1/7, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 7, 1/7, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124081,"parameters":{"cutoff":7,"epsilon_denominator":7,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":7,\"epsilon_denominator\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0082","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":8,\"epsilon_denominator\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 8, 1/8, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 8, 1/8, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124082,"parameters":{"cutoff":8,"epsilon_denominator":8,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":8,\"epsilon_denominator\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0083","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":9,\"epsilon_denominator\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 9, 1/9, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 9, 1/9, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124083,"parameters":{"cutoff":9,"epsilon_denominator":9,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":9,\"epsilon_denominator\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0084","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":10,\"epsilon_denominator\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 10, 1/10, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 10, 1/10, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124084,"parameters":{"cutoff":10,"epsilon_denominator":10,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":10,\"epsilon_denominator\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0085","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":11,\"epsilon_denominator\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 11, 1/11, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 11, 1/11, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124085,"parameters":{"cutoff":11,"epsilon_denominator":11,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":11,\"epsilon_denominator\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0086","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":12,\"epsilon_denominator\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 12, 1/12, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 12, 1/12, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124086,"parameters":{"cutoff":12,"epsilon_denominator":12,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":12,\"epsilon_denominator\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0087","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":13,\"epsilon_denominator\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 13, 1/13, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 13, 1/13, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124087,"parameters":{"cutoff":13,"epsilon_denominator":13,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":13,\"epsilon_denominator\":13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0088","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":14,\"epsilon_denominator\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 14, 1/14, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 14, 1/14, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124088,"parameters":{"cutoff":14,"epsilon_denominator":14,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":14,\"epsilon_denominator\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0089","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":15,\"epsilon_denominator\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 15, 1/15, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 15, 1/15, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124089,"parameters":{"cutoff":15,"epsilon_denominator":15,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":15,\"epsilon_denominator\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0090","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":16,\"epsilon_denominator\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 16, 1/16, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 16, 1/16, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124090,"parameters":{"cutoff":16,"epsilon_denominator":16,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":16,\"epsilon_denominator\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0091","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":17,\"epsilon_denominator\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 17, 1/17, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 17, 1/17, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124091,"parameters":{"cutoff":17,"epsilon_denominator":17,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":17,\"epsilon_denominator\":17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0092","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":18,\"epsilon_denominator\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 18, 1/18, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 18, 1/18, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124092,"parameters":{"cutoff":18,"epsilon_denominator":18,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":18,\"epsilon_denominator\":18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0093","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":19,\"epsilon_denominator\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 19, 1/19, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 19, 1/19, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124093,"parameters":{"cutoff":19,"epsilon_denominator":19,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":19,\"epsilon_denominator\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0094","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":20,\"epsilon_denominator\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 20, 1/20, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 20, 1/20, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124094,"parameters":{"cutoff":20,"epsilon_denominator":20,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":20,\"epsilon_denominator\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0095","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":21,\"epsilon_denominator\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 21, 1/21, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 21, 1/21, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124095,"parameters":{"cutoff":21,"epsilon_denominator":21,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":21,\"epsilon_denominator\":21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0096","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":22,\"epsilon_denominator\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 22, 1/22, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 22, 1/22, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124096,"parameters":{"cutoff":22,"epsilon_denominator":22,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":22,\"epsilon_denominator\":22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0097","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":23,\"epsilon_denominator\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 23, 1/23, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 23, 1/23, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124097,"parameters":{"cutoff":23,"epsilon_denominator":23,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":23,\"epsilon_denominator\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0098","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":24,\"epsilon_denominator\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 24, 1/24, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 24, 1/24, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124098,"parameters":{"cutoff":24,"epsilon_denominator":24,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":24,\"epsilon_denominator\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0099","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":25,\"epsilon_denominator\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 25, 1/25, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 25, 1/25, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124099,"parameters":{"cutoff":25,"epsilon_denominator":25,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":25,\"epsilon_denominator\":25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261112-r04-improper-integral-0100","batch_id":"batch-20261014-advance-20261112-r04-improper-integral-foundations","language":"zh-Hant","source_mko_ids":["mko-improper-integral"],"title_zh":"瑕積分","stem_zh":"題幹明示integrand、區間、cutoff與每個無界端或奇點的one-sided limit責任。操作=improper-principal-value-boundary；資料={\"cutoff\":26,\"epsilon_denominator\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(f(x)=1/x, (-infinity,infinity) with singularity at 0, 26, 1/26, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)","explanation_zh":"Wave6第六層核對=(f(x)=1/x, (-infinity,infinity) with singularity at 0, 26, 1/26, 0, 是, 否, 否, symmetric_principal_value_is_zero_but_the_required_one_sided_limits_diverge_and_finite_cutoffs_do_not_prove_convergence)；依完整明示資料獨立重算；one-sided family固定scale/x^p於[lower,infinity)；two-sided family固定zero-middle piecewise tails；interior family在[c-1,c+1]逐側明示powers；two-sided improper integral與interior singularity都要求各分段獨立收斂；symmetric principal value或有限cutoffs不足。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["瑕積分"]},"variant":{"family_id":"qf-mod43-improper-principal-value-boundary","seed":202611124100,"parameters":{"cutoff":26,"epsilon_denominator":26,"domain":"Exact_symbolic_improper_integral","operation":"improper-principal-value-boundary","semantic_template":"qf-mod43-improper-principal-value-boundary/{\"cutoff\":26,\"epsilon_denominator\":26}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}