{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0001","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[12,1],\"a\":-13,\"b\":-11,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, 1], -13, -11, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([12, 1], -13, -11, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162001,"parameters":{"coefficients":[12,1],"a":-13,"b":-11,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-13,1],\"b\":[-11,1],\"coefficients\":[[12,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0002","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[11,1],\"a\":-12,\"b\":-10,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, 1], -12, -10, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([11, 1], -12, -10, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162002,"parameters":{"coefficients":[11,1],"a":-12,"b":-10,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-12,1],\"b\":[-10,1],\"coefficients\":[[11,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0003","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[10,1],\"a\":-11,\"b\":-9,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, 1], -11, -9, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([10, 1], -11, -9, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162003,"parameters":{"coefficients":[10,1],"a":-11,"b":-9,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-11,1],\"b\":[-9,1],\"coefficients\":[[10,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0004","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[9,1],\"a\":-10,\"b\":-8,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, 1], -10, -8, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([9, 1], -10, -8, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162004,"parameters":{"coefficients":[9,1],"a":-10,"b":-8,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-10,1],\"b\":[-8,1],\"coefficients\":[[9,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0005","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[8,1],\"a\":-9,\"b\":-7,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 1], -9, -7, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([8, 1], -9, -7, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162005,"parameters":{"coefficients":[8,1],"a":-9,"b":-7,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-9,1],\"b\":[-7,1],\"coefficients\":[[8,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0006","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[7,1],\"a\":-8,\"b\":-6,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, 1], -8, -6, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([7, 1], -8, -6, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162006,"parameters":{"coefficients":[7,1],"a":-8,"b":-6,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-8,1],\"b\":[-6,1],\"coefficients\":[[7,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0007","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[6,1],\"a\":-7,\"b\":-5,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 1], -7, -5, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([6, 1], -7, -5, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162007,"parameters":{"coefficients":[6,1],"a":-7,"b":-5,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-7,1],\"b\":[-5,1],\"coefficients\":[[6,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0008","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[5,1],\"a\":-6,\"b\":-4,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, 1], -6, -4, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([5, 1], -6, -4, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162008,"parameters":{"coefficients":[5,1],"a":-6,"b":-4,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-6,1],\"b\":[-4,1],\"coefficients\":[[5,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0009","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[4,1],\"a\":-5,\"b\":-3,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 1], -5, -3, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([4, 1], -5, -3, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162009,"parameters":{"coefficients":[4,1],"a":-5,"b":-3,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-5,1],\"b\":[-3,1],\"coefficients\":[[4,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0010","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[3,1],\"a\":-4,\"b\":-2,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 1], -4, -2, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([3, 1], -4, -2, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162010,"parameters":{"coefficients":[3,1],"a":-4,"b":-2,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-4,1],\"b\":[-2,1],\"coefficients\":[[3,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0011","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[2,1],\"a\":-3,\"b\":-1,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 1], -3, -1, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([2, 1], -3, -1, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162011,"parameters":{"coefficients":[2,1],"a":-3,"b":-1,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-3,1],\"b\":[-1,1],\"coefficients\":[[2,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0012","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[1,1],\"a\":-2,\"b\":0,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 1], -2, 0, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([1, 1], -2, 0, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162012,"parameters":{"coefficients":[1,1],"a":-2,"b":0,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-2,1],\"b\":[0,1],\"coefficients\":[[1,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0013","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[0,1],\"a\":-1,\"b\":1,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([0, 1], -1, 1, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([0, 1], -1, 1, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162013,"parameters":{"coefficients":[0,1],"a":-1,"b":1,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[-1,1],\"b\":[1,1],\"coefficients\":[[0,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0014","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[-1,1],\"a\":0,\"b\":2,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-1, 1], 0, 2, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([-1, 1], 0, 2, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162014,"parameters":{"coefficients":[-1,1],"a":0,"b":2,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[0,1],\"b\":[2,1],\"coefficients\":[[-1,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0015","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[-2,1],\"a\":1,\"b\":3,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-2, 1], 1, 3, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([-2, 1], 1, 3, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162015,"parameters":{"coefficients":[-2,1],"a":1,"b":3,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[1,1],\"b\":[3,1],\"coefficients\":[[-2,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0016","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[-3,1],\"a\":2,\"b\":4,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-3, 1], 2, 4, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([-3, 1], 2, 4, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162016,"parameters":{"coefficients":[-3,1],"a":2,"b":4,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[2,1],\"b\":[4,1],\"coefficients\":[[-3,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0017","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[-4,1],\"a\":3,\"b\":5,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-4, 1], 3, 5, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([-4, 1], 3, 5, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162017,"parameters":{"coefficients":[-4,1],"a":3,"b":5,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[3,1],\"b\":[5,1],\"coefficients\":[[-4,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0018","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[-5,1],\"a\":4,\"b\":6,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-5, 1], 4, 6, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([-5, 1], 4, 6, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162018,"parameters":{"coefficients":[-5,1],"a":4,"b":6,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[4,1],\"b\":[6,1],\"coefficients\":[[-5,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0019","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[-6,1],\"a\":5,\"b\":7,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-6, 1], 5, 7, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([-6, 1], 5, 7, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162019,"parameters":{"coefficients":[-6,1],"a":5,"b":7,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[5,1],\"b\":[7,1],\"coefficients\":[[-6,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0020","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[-7,1],\"a\":6,\"b\":8,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-7, 1], 6, 8, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([-7, 1], 6, 8, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162020,"parameters":{"coefficients":[-7,1],"a":6,"b":8,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[6,1],\"b\":[8,1],\"coefficients\":[[-7,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0021","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[-8,1],\"a\":7,\"b\":9,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-8, 1], 7, 9, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([-8, 1], 7, 9, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162021,"parameters":{"coefficients":[-8,1],"a":7,"b":9,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[7,1],\"b\":[9,1],\"coefficients\":[[-8,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0022","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[-9,1],\"a\":8,\"b\":10,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-9, 1], 8, 10, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([-9, 1], 8, 10, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162022,"parameters":{"coefficients":[-9,1],"a":8,"b":10,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[8,1],\"b\":[10,1],\"coefficients\":[[-9,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0023","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[-10,1],\"a\":9,\"b\":11,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-10, 1], 9, 11, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([-10, 1], 9, 11, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162023,"parameters":{"coefficients":[-10,1],"a":9,"b":11,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[9,1],\"b\":[11,1],\"coefficients\":[[-10,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0024","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[-11,1],\"a\":10,\"b\":12,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-11, 1], 10, 12, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([-11, 1], 10, 12, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162024,"parameters":{"coefficients":[-11,1],"a":10,"b":12,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[10,1],\"b\":[12,1],\"coefficients\":[[-11,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0025","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-hypothesis；資料={\"coefficients\":[-12,1],\"a\":11,\"b\":13,\"target\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([-12, 1], 11, 13, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)","explanation_zh":"Wave6第十層核對=([-12, 1], 11, 13, -1, 1, 0, 是, 是, polynomial_continuity_plus_inclusive_endpoint_bracket)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-hypothesis","seed":202611162025,"parameters":{"coefficients":[-12,1],"a":11,"b":13,"target":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-hypothesis","semantic_template":"qf-mod47-ivt-hypothesis/{\"a\":[11,1],\"b\":[13,1],\"coefficients\":[[-12,1],[1,1]],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0026","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[12,1],\"a\":-13,\"b\":-11,\"target\":0,\"root\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(-12, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162026,"parameters":{"coefficients":[12,1],"a":-13,"b":-11,"target":0,"root":-12,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-13,1],\"b\":[-11,1],\"coefficients\":[[12,1],[1,1]],\"root\":[-12,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0027","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[11,1],\"a\":-12,\"b\":-10,\"target\":0,\"root\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(-11, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162027,"parameters":{"coefficients":[11,1],"a":-12,"b":-10,"target":0,"root":-11,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-12,1],\"b\":[-10,1],\"coefficients\":[[11,1],[1,1]],\"root\":[-11,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0028","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[10,1],\"a\":-11,\"b\":-9,\"target\":0,\"root\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(-10, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162028,"parameters":{"coefficients":[10,1],"a":-11,"b":-9,"target":0,"root":-10,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-11,1],\"b\":[-9,1],\"coefficients\":[[10,1],[1,1]],\"root\":[-10,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0029","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[9,1],\"a\":-10,\"b\":-8,\"target\":0,\"root\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(-9, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162029,"parameters":{"coefficients":[9,1],"a":-10,"b":-8,"target":0,"root":-9,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-10,1],\"b\":[-8,1],\"coefficients\":[[9,1],[1,1]],\"root\":[-9,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0030","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[8,1],\"a\":-9,\"b\":-7,\"target\":0,\"root\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(-8, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162030,"parameters":{"coefficients":[8,1],"a":-9,"b":-7,"target":0,"root":-8,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-9,1],\"b\":[-7,1],\"coefficients\":[[8,1],[1,1]],\"root\":[-8,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0031","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[7,1],\"a\":-8,\"b\":-6,\"target\":0,\"root\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(-7, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162031,"parameters":{"coefficients":[7,1],"a":-8,"b":-6,"target":0,"root":-7,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-8,1],\"b\":[-6,1],\"coefficients\":[[7,1],[1,1]],\"root\":[-7,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0032","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[6,1],\"a\":-7,\"b\":-5,\"target\":0,\"root\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(-6, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162032,"parameters":{"coefficients":[6,1],"a":-7,"b":-5,"target":0,"root":-6,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-7,1],\"b\":[-5,1],\"coefficients\":[[6,1],[1,1]],\"root\":[-6,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0033","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[5,1],\"a\":-6,\"b\":-4,\"target\":0,\"root\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(-5, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162033,"parameters":{"coefficients":[5,1],"a":-6,"b":-4,"target":0,"root":-5,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-6,1],\"b\":[-4,1],\"coefficients\":[[5,1],[1,1]],\"root\":[-5,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0034","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[4,1],\"a\":-5,\"b\":-3,\"target\":0,\"root\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(-4, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162034,"parameters":{"coefficients":[4,1],"a":-5,"b":-3,"target":0,"root":-4,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-5,1],\"b\":[-3,1],\"coefficients\":[[4,1],[1,1]],\"root\":[-4,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0035","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[3,1],\"a\":-4,\"b\":-2,\"target\":0,\"root\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(-3, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162035,"parameters":{"coefficients":[3,1],"a":-4,"b":-2,"target":0,"root":-3,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-4,1],\"b\":[-2,1],\"coefficients\":[[3,1],[1,1]],\"root\":[-3,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0036","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[2,1],\"a\":-3,\"b\":-1,\"target\":0,\"root\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(-2, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162036,"parameters":{"coefficients":[2,1],"a":-3,"b":-1,"target":0,"root":-2,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-3,1],\"b\":[-1,1],\"coefficients\":[[2,1],[1,1]],\"root\":[-2,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0037","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[1,1],\"a\":-2,\"b\":0,\"target\":0,\"root\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(-1, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162037,"parameters":{"coefficients":[1,1],"a":-2,"b":0,"target":0,"root":-1,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-2,1],\"b\":[0,1],\"coefficients\":[[1,1],[1,1]],\"root\":[-1,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0038","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[0,1],\"a\":-1,\"b\":1,\"target\":0,\"root\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(0, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162038,"parameters":{"coefficients":[0,1],"a":-1,"b":1,"target":0,"root":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[-1,1],\"b\":[1,1],\"coefficients\":[[0,1],[1,1]],\"root\":[0,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0039","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-1,1],\"a\":0,\"b\":2,\"target\":0,\"root\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(1, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162039,"parameters":{"coefficients":[-1,1],"a":0,"b":2,"target":0,"root":1,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[0,1],\"b\":[2,1],\"coefficients\":[[-1,1],[1,1]],\"root\":[1,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0040","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-2,1],\"a\":1,\"b\":3,\"target\":0,\"root\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(2, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162040,"parameters":{"coefficients":[-2,1],"a":1,"b":3,"target":0,"root":2,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[1,1],\"b\":[3,1],\"coefficients\":[[-2,1],[1,1]],\"root\":[2,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0041","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-3,1],\"a\":2,\"b\":4,\"target\":0,\"root\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(3, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162041,"parameters":{"coefficients":[-3,1],"a":2,"b":4,"target":0,"root":3,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[2,1],\"b\":[4,1],\"coefficients\":[[-3,1],[1,1]],\"root\":[3,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0042","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-4,1],\"a\":3,\"b\":5,\"target\":0,\"root\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(4, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162042,"parameters":{"coefficients":[-4,1],"a":3,"b":5,"target":0,"root":4,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[3,1],\"b\":[5,1],\"coefficients\":[[-4,1],[1,1]],\"root\":[4,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0043","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-5,1],\"a\":4,\"b\":6,\"target\":0,\"root\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(5, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162043,"parameters":{"coefficients":[-5,1],"a":4,"b":6,"target":0,"root":5,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[4,1],\"b\":[6,1],\"coefficients\":[[-5,1],[1,1]],\"root\":[5,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0044","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-6,1],\"a\":5,\"b\":7,\"target\":0,\"root\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(6, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162044,"parameters":{"coefficients":[-6,1],"a":5,"b":7,"target":0,"root":6,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[5,1],\"b\":[7,1],\"coefficients\":[[-6,1],[1,1]],\"root\":[6,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0045","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-7,1],\"a\":6,\"b\":8,\"target\":0,\"root\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(7, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162045,"parameters":{"coefficients":[-7,1],"a":6,"b":8,"target":0,"root":7,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[6,1],\"b\":[8,1],\"coefficients\":[[-7,1],[1,1]],\"root\":[7,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0046","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-8,1],\"a\":7,\"b\":9,\"target\":0,\"root\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(8, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162046,"parameters":{"coefficients":[-8,1],"a":7,"b":9,"target":0,"root":8,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[7,1],\"b\":[9,1],\"coefficients\":[[-8,1],[1,1]],\"root\":[8,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0047","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-9,1],\"a\":8,\"b\":10,\"target\":0,\"root\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(9, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162047,"parameters":{"coefficients":[-9,1],"a":8,"b":10,"target":0,"root":9,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[8,1],\"b\":[10,1],\"coefficients\":[[-9,1],[1,1]],\"root\":[9,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0048","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-10,1],\"a\":9,\"b\":11,\"target\":0,\"root\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(10, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162048,"parameters":{"coefficients":[-10,1],"a":9,"b":11,"target":0,"root":10,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[9,1],\"b\":[11,1],\"coefficients\":[[-10,1],[1,1]],\"root\":[10,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0049","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-11,1],\"a\":10,\"b\":12,\"target\":0,\"root\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(11, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162049,"parameters":{"coefficients":[-11,1],"a":10,"b":12,"target":0,"root":11,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[10,1],\"b\":[12,1],\"coefficients\":[[-11,1],[1,1]],\"root\":[11,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0050","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-exact-root；資料={\"coefficients\":[-12,1],\"a\":11,\"b\":13,\"target\":0,\"root\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)","explanation_zh":"Wave6第十層核對=(12, 0, 是, 是, 否, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-exact-root","seed":202611162050,"parameters":{"coefficients":[-12,1],"a":11,"b":13,"target":0,"root":12,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-exact-root","semantic_template":"qf-mod47-ivt-exact-root/{\"a\":[11,1],\"b\":[13,1],\"coefficients\":[[-12,1],[1,1]],\"root\":[12,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0051","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[13,1],\"a\":-13,\"b\":-11,\"target\":0,\"root\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-13, -11, 0, -13, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-13, -11, 0, -13, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162051,"parameters":{"coefficients":[13,1],"a":-13,"b":-11,"target":0,"root":-13,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-13,1],\"b\":[-11,1],\"coefficients\":[[13,1],[1,1]],\"root\":[-13,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0052","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[12,1],\"a\":-12,\"b\":-10,\"target\":0,\"root\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, -10, 0, -12, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-12, -10, 0, -12, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162052,"parameters":{"coefficients":[12,1],"a":-12,"b":-10,"target":0,"root":-12,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-12,1],\"b\":[-10,1],\"coefficients\":[[12,1],[1,1]],\"root\":[-12,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0053","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[11,1],\"a\":-11,\"b\":-9,\"target\":0,\"root\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, -9, 0, -11, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-11, -9, 0, -11, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162053,"parameters":{"coefficients":[11,1],"a":-11,"b":-9,"target":0,"root":-11,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-11,1],\"b\":[-9,1],\"coefficients\":[[11,1],[1,1]],\"root\":[-11,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0054","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[10,1],\"a\":-10,\"b\":-8,\"target\":0,\"root\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, -8, 0, -10, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-10, -8, 0, -10, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162054,"parameters":{"coefficients":[10,1],"a":-10,"b":-8,"target":0,"root":-10,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-10,1],\"b\":[-8,1],\"coefficients\":[[10,1],[1,1]],\"root\":[-10,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0055","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[9,1],\"a\":-9,\"b\":-7,\"target\":0,\"root\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, -7, 0, -9, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-9, -7, 0, -9, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162055,"parameters":{"coefficients":[9,1],"a":-9,"b":-7,"target":0,"root":-9,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-9,1],\"b\":[-7,1],\"coefficients\":[[9,1],[1,1]],\"root\":[-9,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0056","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[8,1],\"a\":-8,\"b\":-6,\"target\":0,\"root\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, -6, 0, -8, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-8, -6, 0, -8, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162056,"parameters":{"coefficients":[8,1],"a":-8,"b":-6,"target":0,"root":-8,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-8,1],\"b\":[-6,1],\"coefficients\":[[8,1],[1,1]],\"root\":[-8,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0057","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[7,1],\"a\":-7,\"b\":-5,\"target\":0,\"root\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, -5, 0, -7, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-7, -5, 0, -7, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162057,"parameters":{"coefficients":[7,1],"a":-7,"b":-5,"target":0,"root":-7,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-7,1],\"b\":[-5,1],\"coefficients\":[[7,1],[1,1]],\"root\":[-7,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0058","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[6,1],\"a\":-6,\"b\":-4,\"target\":0,\"root\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, -4, 0, -6, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-6, -4, 0, -6, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162058,"parameters":{"coefficients":[6,1],"a":-6,"b":-4,"target":0,"root":-6,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-6,1],\"b\":[-4,1],\"coefficients\":[[6,1],[1,1]],\"root\":[-6,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0059","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[5,1],\"a\":-5,\"b\":-3,\"target\":0,\"root\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, -3, 0, -5, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-5, -3, 0, -5, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162059,"parameters":{"coefficients":[5,1],"a":-5,"b":-3,"target":0,"root":-5,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-5,1],\"b\":[-3,1],\"coefficients\":[[5,1],[1,1]],\"root\":[-5,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0060","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[4,1],\"a\":-4,\"b\":-2,\"target\":0,\"root\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, -2, 0, -4, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-4, -2, 0, -4, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162060,"parameters":{"coefficients":[4,1],"a":-4,"b":-2,"target":0,"root":-4,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-4,1],\"b\":[-2,1],\"coefficients\":[[4,1],[1,1]],\"root\":[-4,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0061","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[3,1],\"a\":-3,\"b\":-1,\"target\":0,\"root\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, -1, 0, -3, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-3, -1, 0, -3, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162061,"parameters":{"coefficients":[3,1],"a":-3,"b":-1,"target":0,"root":-3,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-3,1],\"b\":[-1,1],\"coefficients\":[[3,1],[1,1]],\"root\":[-3,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0062","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[2,1],\"a\":-2,\"b\":0,\"target\":0,\"root\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, 0, 0, -2, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-2, 0, 0, -2, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162062,"parameters":{"coefficients":[2,1],"a":-2,"b":0,"target":0,"root":-2,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-2,1],\"b\":[0,1],\"coefficients\":[[2,1],[1,1]],\"root\":[-2,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0063","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[1,1],\"a\":-1,\"b\":1,\"target\":0,\"root\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, 1, 0, -1, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(-1, 1, 0, -1, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162063,"parameters":{"coefficients":[1,1],"a":-1,"b":1,"target":0,"root":-1,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[-1,1],\"b\":[1,1],\"coefficients\":[[1,1],[1,1]],\"root\":[-1,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0064","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[0,1],\"a\":0,\"b\":2,\"target\":0,\"root\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 0, 0, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(0, 2, 0, 0, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162064,"parameters":{"coefficients":[0,1],"a":0,"b":2,"target":0,"root":0,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[0,1],\"b\":[2,1],\"coefficients\":[[0,1],[1,1]],\"root\":[0,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0065","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[-1,1],\"a\":1,\"b\":3,\"target\":0,\"root\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 0, 1, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(1, 3, 0, 1, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162065,"parameters":{"coefficients":[-1,1],"a":1,"b":3,"target":0,"root":1,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[1,1],\"b\":[3,1],\"coefficients\":[[-1,1],[1,1]],\"root\":[1,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0066","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[-2,1],\"a\":2,\"b\":4,\"target\":0,\"root\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, 0, 2, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(2, 4, 0, 2, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162066,"parameters":{"coefficients":[-2,1],"a":2,"b":4,"target":0,"root":2,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[2,1],\"b\":[4,1],\"coefficients\":[[-2,1],[1,1]],\"root\":[2,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0067","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[-3,1],\"a\":3,\"b\":5,\"target\":0,\"root\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 0, 3, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(3, 5, 0, 3, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162067,"parameters":{"coefficients":[-3,1],"a":3,"b":5,"target":0,"root":3,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[3,1],\"b\":[5,1],\"coefficients\":[[-3,1],[1,1]],\"root\":[3,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0068","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[-4,1],\"a\":4,\"b\":6,\"target\":0,\"root\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 6, 0, 4, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(4, 6, 0, 4, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162068,"parameters":{"coefficients":[-4,1],"a":4,"b":6,"target":0,"root":4,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[4,1],\"b\":[6,1],\"coefficients\":[[-4,1],[1,1]],\"root\":[4,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0069","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[-5,1],\"a\":5,\"b\":7,\"target\":0,\"root\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 7, 0, 5, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(5, 7, 0, 5, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162069,"parameters":{"coefficients":[-5,1],"a":5,"b":7,"target":0,"root":5,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[5,1],\"b\":[7,1],\"coefficients\":[[-5,1],[1,1]],\"root\":[5,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0070","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[-6,1],\"a\":6,\"b\":8,\"target\":0,\"root\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 8, 0, 6, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(6, 8, 0, 6, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162070,"parameters":{"coefficients":[-6,1],"a":6,"b":8,"target":0,"root":6,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[6,1],\"b\":[8,1],\"coefficients\":[[-6,1],[1,1]],\"root\":[6,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0071","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[-7,1],\"a\":7,\"b\":9,\"target\":0,\"root\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 9, 0, 7, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(7, 9, 0, 7, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162071,"parameters":{"coefficients":[-7,1],"a":7,"b":9,"target":0,"root":7,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[7,1],\"b\":[9,1],\"coefficients\":[[-7,1],[1,1]],\"root\":[7,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0072","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[-8,1],\"a\":8,\"b\":10,\"target\":0,\"root\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 10, 0, 8, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(8, 10, 0, 8, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162072,"parameters":{"coefficients":[-8,1],"a":8,"b":10,"target":0,"root":8,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[8,1],\"b\":[10,1],\"coefficients\":[[-8,1],[1,1]],\"root\":[8,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0073","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[-9,1],\"a\":9,\"b\":11,\"target\":0,\"root\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 11, 0, 9, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(9, 11, 0, 9, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162073,"parameters":{"coefficients":[-9,1],"a":9,"b":11,"target":0,"root":9,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[9,1],\"b\":[11,1],\"coefficients\":[[-9,1],[1,1]],\"root\":[9,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0074","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[-10,1],\"a\":10,\"b\":12,\"target\":0,\"root\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 12, 0, 10, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(10, 12, 0, 10, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162074,"parameters":{"coefficients":[-10,1],"a":10,"b":12,"target":0,"root":10,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[10,1],\"b\":[12,1],\"coefficients\":[[-10,1],[1,1]],\"root\":[10,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0075","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-endpoint-bracket；資料={\"coefficients\":[-11,1],\"a\":11,\"b\":13,\"target\":0,\"root\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 13, 0, 11, 是, 是, 是, endpoints_are_included)","explanation_zh":"Wave6第十層核對=(11, 13, 0, 11, 是, 是, 是, endpoints_are_included)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-endpoint-bracket","seed":202611162075,"parameters":{"coefficients":[-11,1],"a":11,"b":13,"target":0,"root":11,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-endpoint-bracket","semantic_template":"qf-mod47-ivt-endpoint-bracket/{\"a\":[11,1],\"b\":[13,1],\"coefficients\":[[-11,1],[1,1]],\"root\":[11,1],\"target\":[0,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0076","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"discontinuous_jump\",\"scale\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_jump, f(x)=-1_for_x<0_and_1_for_x>=0, -1, 1, 0, 是, 否, 否)","explanation_zh":"Wave6第十層核對=(discontinuous_jump, f(x)=-1_for_x<0_and_1_for_x>=0, -1, 1, 0, 是, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162076,"parameters":{"mode":"discontinuous_jump","scale":1,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"discontinuous_jump\",\"scale\":[1,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0077","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonbracket\",\"scale\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonbracket, f(x)=2*x, 0, 1, 4, 否, 否)","explanation_zh":"Wave6第十層核對=(nonbracket, f(x)=2*x, 0, 1, 4, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162077,"parameters":{"mode":"nonbracket","scale":2,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonbracket\",\"scale\":[2,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0078","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonunique\",\"scale\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonunique, f(x)=x^2, -3, 3, 9, 是, two_roots_possible, 否)","explanation_zh":"Wave6第十層核對=(nonunique, f(x)=x^2, -3, 3, 9, 是, two_roots_possible, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162078,"parameters":{"mode":"nonunique","scale":3,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonunique\",\"scale\":[3,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0079","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"discontinuous_jump\",\"scale\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_jump, f(x)=-4_for_x<0_and_4_for_x>=0, -1, 1, 0, 是, 否, 否)","explanation_zh":"Wave6第十層核對=(discontinuous_jump, f(x)=-4_for_x<0_and_4_for_x>=0, -1, 1, 0, 是, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162079,"parameters":{"mode":"discontinuous_jump","scale":4,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"discontinuous_jump\",\"scale\":[4,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0080","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonbracket\",\"scale\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonbracket, f(x)=5*x, 0, 1, 10, 否, 否)","explanation_zh":"Wave6第十層核對=(nonbracket, f(x)=5*x, 0, 1, 10, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162080,"parameters":{"mode":"nonbracket","scale":5,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonbracket\",\"scale\":[5,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0081","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonunique\",\"scale\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonunique, f(x)=x^2, -6, 6, 36, 是, two_roots_possible, 否)","explanation_zh":"Wave6第十層核對=(nonunique, f(x)=x^2, -6, 6, 36, 是, two_roots_possible, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162081,"parameters":{"mode":"nonunique","scale":6,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonunique\",\"scale\":[6,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0082","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"discontinuous_jump\",\"scale\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_jump, f(x)=-7_for_x<0_and_7_for_x>=0, -1, 1, 0, 是, 否, 否)","explanation_zh":"Wave6第十層核對=(discontinuous_jump, f(x)=-7_for_x<0_and_7_for_x>=0, -1, 1, 0, 是, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162082,"parameters":{"mode":"discontinuous_jump","scale":7,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"discontinuous_jump\",\"scale\":[7,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0083","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonbracket\",\"scale\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonbracket, f(x)=8*x, 0, 1, 16, 否, 否)","explanation_zh":"Wave6第十層核對=(nonbracket, f(x)=8*x, 0, 1, 16, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162083,"parameters":{"mode":"nonbracket","scale":8,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonbracket\",\"scale\":[8,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0084","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonunique\",\"scale\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonunique, f(x)=x^2, -9, 9, 81, 是, two_roots_possible, 否)","explanation_zh":"Wave6第十層核對=(nonunique, f(x)=x^2, -9, 9, 81, 是, two_roots_possible, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162084,"parameters":{"mode":"nonunique","scale":9,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonunique\",\"scale\":[9,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0085","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"discontinuous_jump\",\"scale\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_jump, f(x)=-10_for_x<0_and_10_for_x>=0, -1, 1, 0, 是, 否, 否)","explanation_zh":"Wave6第十層核對=(discontinuous_jump, f(x)=-10_for_x<0_and_10_for_x>=0, -1, 1, 0, 是, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162085,"parameters":{"mode":"discontinuous_jump","scale":10,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"discontinuous_jump\",\"scale\":[10,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0086","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonbracket\",\"scale\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonbracket, f(x)=11*x, 0, 1, 22, 否, 否)","explanation_zh":"Wave6第十層核對=(nonbracket, f(x)=11*x, 0, 1, 22, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162086,"parameters":{"mode":"nonbracket","scale":11,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonbracket\",\"scale\":[11,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0087","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonunique\",\"scale\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonunique, f(x)=x^2, -12, 12, 144, 是, two_roots_possible, 否)","explanation_zh":"Wave6第十層核對=(nonunique, f(x)=x^2, -12, 12, 144, 是, two_roots_possible, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162087,"parameters":{"mode":"nonunique","scale":12,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonunique\",\"scale\":[12,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0088","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"discontinuous_jump\",\"scale\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_jump, f(x)=-13_for_x<0_and_13_for_x>=0, -1, 1, 0, 是, 否, 否)","explanation_zh":"Wave6第十層核對=(discontinuous_jump, f(x)=-13_for_x<0_and_13_for_x>=0, -1, 1, 0, 是, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162088,"parameters":{"mode":"discontinuous_jump","scale":13,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"discontinuous_jump\",\"scale\":[13,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0089","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonbracket\",\"scale\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonbracket, f(x)=14*x, 0, 1, 28, 否, 否)","explanation_zh":"Wave6第十層核對=(nonbracket, f(x)=14*x, 0, 1, 28, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162089,"parameters":{"mode":"nonbracket","scale":14,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonbracket\",\"scale\":[14,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0090","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonunique\",\"scale\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonunique, f(x)=x^2, -15, 15, 225, 是, two_roots_possible, 否)","explanation_zh":"Wave6第十層核對=(nonunique, f(x)=x^2, -15, 15, 225, 是, two_roots_possible, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162090,"parameters":{"mode":"nonunique","scale":15,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonunique\",\"scale\":[15,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0091","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"discontinuous_jump\",\"scale\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_jump, f(x)=-16_for_x<0_and_16_for_x>=0, -1, 1, 0, 是, 否, 否)","explanation_zh":"Wave6第十層核對=(discontinuous_jump, f(x)=-16_for_x<0_and_16_for_x>=0, -1, 1, 0, 是, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162091,"parameters":{"mode":"discontinuous_jump","scale":16,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"discontinuous_jump\",\"scale\":[16,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0092","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonbracket\",\"scale\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonbracket, f(x)=17*x, 0, 1, 34, 否, 否)","explanation_zh":"Wave6第十層核對=(nonbracket, f(x)=17*x, 0, 1, 34, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162092,"parameters":{"mode":"nonbracket","scale":17,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonbracket\",\"scale\":[17,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0093","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonunique\",\"scale\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonunique, f(x)=x^2, -18, 18, 324, 是, two_roots_possible, 否)","explanation_zh":"Wave6第十層核對=(nonunique, f(x)=x^2, -18, 18, 324, 是, two_roots_possible, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162093,"parameters":{"mode":"nonunique","scale":18,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonunique\",\"scale\":[18,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0094","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"discontinuous_jump\",\"scale\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_jump, f(x)=-19_for_x<0_and_19_for_x>=0, -1, 1, 0, 是, 否, 否)","explanation_zh":"Wave6第十層核對=(discontinuous_jump, f(x)=-19_for_x<0_and_19_for_x>=0, -1, 1, 0, 是, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162094,"parameters":{"mode":"discontinuous_jump","scale":19,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"discontinuous_jump\",\"scale\":[19,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0095","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonbracket\",\"scale\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonbracket, f(x)=20*x, 0, 1, 40, 否, 否)","explanation_zh":"Wave6第十層核對=(nonbracket, f(x)=20*x, 0, 1, 40, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162095,"parameters":{"mode":"nonbracket","scale":20,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonbracket\",\"scale\":[20,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0096","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonunique\",\"scale\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonunique, f(x)=x^2, -21, 21, 441, 是, two_roots_possible, 否)","explanation_zh":"Wave6第十層核對=(nonunique, f(x)=x^2, -21, 21, 441, 是, two_roots_possible, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162096,"parameters":{"mode":"nonunique","scale":21,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonunique\",\"scale\":[21,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0097","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"discontinuous_jump\",\"scale\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_jump, f(x)=-22_for_x<0_and_22_for_x>=0, -1, 1, 0, 是, 否, 否)","explanation_zh":"Wave6第十層核對=(discontinuous_jump, f(x)=-22_for_x<0_and_22_for_x>=0, -1, 1, 0, 是, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162097,"parameters":{"mode":"discontinuous_jump","scale":22,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"discontinuous_jump\",\"scale\":[22,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0098","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonbracket\",\"scale\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonbracket, f(x)=23*x, 0, 1, 46, 否, 否)","explanation_zh":"Wave6第十層核對=(nonbracket, f(x)=23*x, 0, 1, 46, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162098,"parameters":{"mode":"nonbracket","scale":23,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonbracket\",\"scale\":[23,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0099","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"nonunique\",\"scale\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nonunique, f(x)=x^2, -24, 24, 576, 是, two_roots_possible, 否)","explanation_zh":"Wave6第十層核對=(nonunique, f(x)=x^2, -24, 24, 576, 是, two_roots_possible, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162099,"parameters":{"mode":"nonunique","scale":24,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"nonunique\",\"scale\":[24,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261116-r02-intermediate-value-theorem-0100","batch_id":"batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-intermediate-value-theorem"],"title_zh":"介值定理","stem_zh":"a<b、continuous interval function與inclusive target bracket責任。操作=ivt-discontinuity-boundary；資料={\"mode\":\"discontinuous_jump\",\"scale\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(discontinuous_jump, f(x)=-25_for_x<0_and_25_for_x>=0, -1, 1, 0, 是, 否, 否)","explanation_zh":"Wave6第十層核對=(discontinuous_jump, f(x)=-25_for_x<0_and_25_for_x>=0, -1, 1, 0, 是, 否, 否)；依完整明示資料獨立重算；exact root witness只證存在且endpoint允許；不保證unique或closed form；discontinuous sign change與finite samples不能套IVT。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["介值定理"]},"variant":{"family_id":"qf-mod47-ivt-discontinuity-boundary","seed":202611162100,"parameters":{"mode":"discontinuous_jump","scale":25,"domain":"Exact_polynomial_intermediate_value","operation":"ivt-discontinuity-boundary","semantic_template":"qf-mod47-ivt-discontinuity-boundary/{\"mode\":\"discontinuous_jump\",\"scale\":[25,1]}"}},"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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}